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California Institute of Technology

The California Institute of Technology (branded as Caltech) is a private research university in Pasadena, California. The university is responsible for many modern scientific advances and is among a small group of institutes of technology in the United States that are devoted to the instruction of pure and applied sciences.

The institution was founded as a preparatory and vocational school by Amos G. Throop in 1891 and began attracting influential scientists such as George Ellery Hale, Arthur Amos Noyes, and Robert Andrews Millikan in the early 20th century. The vocational and preparatory schools were disbanded and spun off in 1910, and the college assumed its present name in 1920. In 1934, Caltech was elected to the Association of American Universities, and the antecedents of NASA's Jet Propulsion Laboratory, which Caltech continues to manage and operate, were established between 1936 and 1943 under Theodore von Kármán.

Caltech has six academic divisions with strong emphasis on science and engineering, managing $332 million in sponsored research in 2011. Its 124 acre primary campus is located approximately 11 mi northeast of downtown Los Angeles, in Pasadena. First-year students are required to live on campus, and 95% of undergraduates remain in the on-campus housing system at Caltech. Student life is governed by an honor code that allows faculty to assign take-home examinations. The Caltech Beavers compete in 13 intercollegiate sports in the NCAA Division III's Southern California Intercollegiate Athletic Conference (SCIAC).

Scientists and engineers at or from the university have played an essential role in many modern scientific breakthroughs and innovations, including advances in space research, sustainability science, quantum physics, earthquake monitoring, protein engineering, and soft robotics. , there are 79 Nobel laureates who have been affiliated with Caltech, making it the institution with the highest number of Nobelists per capita in America. This includes 46 alumni and faculty members (47 prizes, with chemist Linus Pauling being the only individual in history to win two unshared prizes). In addition, four Fields Medalists and six Turing Award winners have been affiliated with Caltech.

Throop College

Caltech started as a vocational school founded in present-day Old Pasadena on Fair Oaks Avenue and Chestnut Street on September 23, 1891, by local businessman and politician Amos G. Throop. The school was known successively as Throop University, Throop Polytechnic Institute (and Manual Training School) and Throop College of Technology before acquiring its current name in 1920. The vocational school was disbanded and the preparatory program was split off to form the independent Polytechnic School in 1907.

At a time when scientific research in the United States was still in its infancy, George Ellery Hale, a solar astronomer from the University of Chicago, founded the Mount Wilson Observatory in 1904. He joined Throop's board of trustees in 1907, and soon began developing the university, and the whole of Pasadena, into a major scientific and cultural destination. He engineered the appointment of James A. B. Scherer, a literary scholar untutored in science but very capable in administration and fund-raising, to Throop's presidency in 1908. Scherer persuaded retired businessman and trustee Charles W. Gates to donate $25,000 in seed money to build Gates Laboratory, the first science building on campus.

World Wars

In 1910, Throop moved to its current site. Arthur Fleming donated the land for the permanent campus site. Theodore Roosevelt delivered an address at Throop Institute on March 21, 1911, and he declared:

"I want to see institutions like Throop turn out perhaps ninety-nine of every hundred students as men who are to do given pieces of industrial work better than any one else can do them; I want to see those men do the kind of work that is now being done on the Panama Canal and on the great irrigation projects in the interior of this country—and the one-hundredth man I want to see with the kind of cultural scientific training that will make him and his fellows the matrix out of which you can occasionally develop a man like your great astronomer, George Ellery Hale."

Also in 1911, a bill was introduced in the California Legislature calling for the establishment of a publicly funded "California Institute of Technology," with an initial budget of a million dollars, ten times the budget of Throop at the time. The board of trustees offered to turn Throop over to the state, but the presidents of Stanford University and the University of California, Berkeley successfully lobbied to defeat the bill, which allowed Throop to develop as the only scientific research-oriented educational institute in southern California, public or private, until the onset of World War II necessitated the broader development of research-based science education. The promise of Throop attracted physical chemist Arthur Amos Noyes from MIT to develop the institution and assist in establishing it as a center for science and technology.

With the onset of World War I, Hale organized the National Research Council to coordinate and support scientific work on military problems. While he supported the idea of federal appropriations for science, he took exception to a federal bill that would have funded engineering research at land-grant colleges, and instead sought to raise a $1 million national research fund entirely from private sources. To that end, as Hale wrote in The New York Times:

Throop College of Technology, in Pasadena California has recently afforded a striking illustration of one way in which the Research Council can secure co-operation and advance scientific investigation. This institution, with its able investigators and excellent research laboratories, could be of great service in any broad scheme of cooperation. President Scherer, hearing of the formation of the council, immediately offered to take part in its work, and with this object, he secured within three days an additional research endowment of one hundred thousand dollars.

Through the National Research Council, Hale simultaneously lobbied for science to play a larger role in national affairs, and for Throop to play a national role in science. The new funds were designated for physics research, and ultimately led to the establishment of the Norman Bridge Laboratory, which attracted experimental physicist Robert Andrews Millikan from the University of Chicago in 1917. During the course of the war, Hale, Noyes and Millikan worked together in Washington on the NRC. Subsequently, they continued their partnership in developing Caltech.

Under the leadership of Hale, Noyes, and Millikan (aided by the booming economy of Southern California), Caltech grew to national prominence in the 1920s and concentrated on the development of Roosevelt's "Hundredth Man". On November 29, 1921, the trustees declared it to be the express policy of the institute to pursue scientific research of the greatest importance and at the same time "to continue to conduct thorough courses in engineering and pure science, basing the work of these courses on exceptionally strong instruction in the fundamental sciences of mathematics, physics, and chemistry; broadening and enriching the curriculum by a liberal amount of instruction in such subjects as English, history, and economics; and vitalizing all the work of the Institute by the infusion in generous measure of the spirit of research". In 1923, Millikan was awarded the Nobel Prize in Physics. In 1925, the school established a department of geology and hired William Bennett Munro, then chairman of the division of History, Government, and Economics at Harvard University, to create a division of humanities and social sciences at Caltech. In 1928, a division of biology was established under the leadership of Thomas Hunt Morgan, the most distinguished biologist in the United States at the time, and discoverer of the role of genes and the chromosome in heredity. In 1930, Kerckhoff Marine Laboratory was established in Corona del Mar under the care of Professor George MacGinitie. In 1926, a graduate school of aeronautics was created, which eventually attracted Theodore von Kármán. Kármán later helped create the Jet Propulsion Laboratory, and played an integral part in establishing Caltech as one of the world's centers for rocket science. In 1928, construction of the Palomar Observatory began.

Millikan served as "Chairman of the Executive Council" (effectively Caltech's president) from 1921 to 1945, and his influence was such that the institute was occasionally referred to as "Millikan's School". Millikan initiated a visiting-scholars program soon after joining Caltech. Notable scientists who accepted his invitation include Paul Dirac, Erwin Schrödinger, Werner Heisenberg, Hendrik Lorentz and Niels Bohr. Albert Einstein arrived on the Caltech campus for the first time in 1931 to polish up his Theory of General Relativity, and he returned to Caltech subsequently as a visiting professor in 1932 and 1933.

During World War II, Caltech was one of 131 colleges and universities nationally that took part in the V-12 Navy College Training Program which offered students a path to a Navy commission. The United States Navy also maintained a naval training school for aeronautical engineering, resident inspectors of ordinance and naval material, and a liaison officer to the National Defense Research Committee on campus.

Project Vista

From April to December 1951, Caltech was the host of a federal classified study, Project Vista. The selection of Caltech as host for the project was based on the university's expertise in rocketry and nuclear physics. In response to the war in Korea and the pressure from the Soviet Union, the project was Caltech's way of assisting the federal government in its effort to increase national security. The project was created to study new ways of improving the relationship between tactical air support and ground troops. The Army, Air Force, and Navy sponsored the project; however, it was under contract with the Army. The study was named after the hotel, Vista del Arroyo Hotel, which housed the study. The study operated under a committee with the supervision of President Lee A. DuBridge. William A. Fowler, a professor at Caltech, was selected as research director. More than a fourth of Caltech's faculty and a group of outside scientists staffed the project. Moreover, the number increases if one takes into account visiting scientists, military liaisons, secretarial, and security staff. In compensation for its participation, the university received about $750,000.

Post-war growth
From the 1950s to 1980s, Caltech was the home of Murray Gell-Mann and Richard Feynman, whose work was central to the establishment of the Standard Model of particle physics. Feynman was also widely known outside the physics community as an exceptional teacher and a colorful, unconventional character.

During Lee A. DuBridge's tenure as Caltech's president (1946–1969), Caltech's faculty doubled and the campus tripled in size. DuBridge, unlike his predecessors, welcomed federal funding of science. New research fields flourished, including chemical biology, planetary science, nuclear astrophysics, and geochemistry. A 200-inch telescope was dedicated on nearby Palomar Mountain in 1948 and remained the world's most powerful optical telescope for over forty years.

Caltech opened its doors to female undergraduates during the presidency of Harold Brown in 1970, and they made up 14% of the entering class. The portion of female undergraduates has been increasing since then.

Protests by Caltech students are rare. The earliest was a 1968 protest outside the NBC Burbank studios, in response to rumors that NBC was to cancel Star Trek. In 1973, the students from Dabney House protested a presidential visit with a sign on the library bearing the simple phrase "Impeach Nixon". The following week, Ross McCollum, president of the National Oil Company, wrote an open letter to Dabney House stating that in light of their actions he had decided not to donate one million dollars to Caltech. The Dabney family, being Republicans, disowned Dabney House after hearing of the protest.

21st century
Since 2000, the Einstein Papers Project has been located at Caltech. The project was established in 1986 to assemble, preserve, translate, and publish papers selected from the literary estate of Albert Einstein and from other collections.

In fall 2008, the freshman class was 42% female, a record for Caltech's undergraduate enrollment. In the same year, the Institute concluded a six-year-long fund-raising campaign. The campaign raised more than $1.4 billion from about 16,000 donors. Nearly half of the funds went into the support of Caltech programs and projects.

In 2010, Caltech, in partnership with Lawrence Berkeley National Laboratory and headed by Professor Nathan Lewis, established a DOE Energy Innovation Hub aimed at developing revolutionary methods to generate fuels directly from sunlight. This hub, the Joint Center for Artificial Photosynthesis, will receive up to $122 million in federal funding over five years.

Since 2012, Caltech began to offer classes through massive open online courses (MOOCs) under Coursera, from 2013, edX, and bootcamps.

Jean-Lou Chameau, the eighth president, announced on February 19, 2013, that he would be stepping down to accept the presidency at King Abdullah University of Science and Technology. Thomas F. Rosenbaum was announced to be the ninth president of Caltech on October 24, 2013, and his term began on July 1, 2014.

In 2019, Caltech received a gift of $750 million for sustainability research from the Resnick family of The Wonderful Company. The gift is the largest ever for environmental sustainability research and the second-largest private donation to a US academic institution (after Bloomberg's gift of $1.8 billion to Johns Hopkins University in 2018).

On account of President Robert A. Millikan's affiliation with the Human Betterment Foundation, in January 2021, the Caltech Board of Trustees authorized the removal of Millikan's name (and the names of five other historical figures affiliated with the Foundation), from campus buildings.

Campus
Caltech's 124 acre primary campus is located in Pasadena, California, approximately 11 mi northeast of downtown Los Angeles. It is within walking distance of Old Town Pasadena and the Pasadena Playhouse District and therefore the two locations are frequent getaways for Caltech students.

In 1917 Hale hired architect Bertram Goodhue to produce a master plan for the 22 acre campus. Goodhue conceived the overall layout of the campus and designed the physics building, Dabney Hall, and several other structures, in which he sought to be consistent with the local climate, the character of the school, and Hale's educational philosophy. Goodhue's designs for Caltech were also influenced by the traditional Spanish mission architecture of Southern California.

During the 1960s, Caltech underwent considerable expansion, in part due to the philanthropy of alumnus Arnold O. Beckman. In 1953, Beckman was asked to join the Caltech Board of Trustees. In 1964, he became its chairman. Over the next few years, as Caltech's president emeritus David Baltimore describes it, Arnold Beckman and his wife Mabel "shaped the destiny of Caltech".

In 1971 a magnitude-6.6 earthquake in San Fernando caused some damage to the Caltech campus. Engineers who evaluated the damage found that two historic buildings dating from the early days of the Institute—Throop Hall and the Goodhue-designed Culbertson Auditorium—had cracked.

New additions to the campus include the Cahill Center for Astronomy and Astrophysics and the Walter and Leonore Annenberg Center for Information Science and Technology, which opened in 2009, and the Warren and Katherine Schlinger Laboratory for Chemistry and Chemical Engineering followed in March 2010. The institute also concluded an upgrading of the South Houses in 2006. In late 2010, Caltech completed a 1.3 MW solar array projected to produce approximately 1.6 GWh in 2011.

Organization and administration

Caltech is incorporated as a non-profit corporation and is governed by a privately appointed 46-member board of trustees who serve five-year terms of office and retire at the age of 72. The trustees elect a president to serve as the chief executive officer of the institute and administer the affairs on the institute on behalf of the board, a provost who serves as the chief academic officer of the institute below the president, and ten other vice presidential and other senior positions. Thomas F. Rosenbaum became the ninth president of Caltech in 2014. Caltech's endowment is governed by a permanent trustee committee and administered by an investment office.

The institute is organized into six primary academic divisions: Biology and Biological Engineering (founded 1927), Chemistry and Chemical Engineering (founded 1926), Engineering and Applied Science (founded 1926), Geological and Planetary Sciences (founded 1927), Humanities and Social Sciences (founded 1926), Physics, Mathematics, and Astronomy (founded 1926). Given Caltech's historical prestige and the small size of its faculty in many major fields, the institution is exceptionally careful in selecting candidates. This rigorous process can result in some positions remaining unfilled for several years until the right candidate is found. Caltech dedicates significant resources to attract top-tier faculty and provides them with substantial financial support to foster their research and academic endeavors. The voting faculty of Caltech include all professors, instructors, research associates and fellows, and the University Librarian. Faculty are responsible for establishing admission requirements, academic standards, and curricula. The Faculty Board is the faculty's representative body and consists of 18 elected faculty representatives as well as other senior administration officials. Full-time professors are expected to teach classes, conduct research, advise students, and perform administrative work such as serving on committees.

Founded in 1930s, the Jet Propulsion Laboratory (JPL) is a federally funded research and development center (FFRDC) owned by NASA and operated as a division of Caltech through a contract between NASA and Caltech. In 2008, JPL spent over $1.6 billion on research and development and employed over 5,000 project-related and support employees. The JPL Director also serves as a Caltech Vice President and is responsible to the President of the Institute for the management of the laboratory.

Academics
Caltech is a small four-year, highly residential research university with slightly more students in graduate programs than undergraduate. The institute has been accredited by the Western Association of Schools and Colleges since 1949. Caltech is on the quarter system: the fall term starts in late September and ends before Christmas, the second term starts after New Year's Day and ends in mid-March, and the third term starts in late March or early April and ends in early June.

Rankings
Caltech is consistently ranked within the top ten universities in the world, and within the top four in the United States, by major global ranking systems. In 2021, Caltech ranked 6th globally based on aggregate world university rankings of THE, QS, and ARWU. For 2022, U.S. News & World Report ranked Caltech as tied for 9th in the United States among national universities overall, 11th for most innovative, and 15th for best value. U.S. News & World Report also ranked the graduate programs in chemistry and earth sciences first among national universities.

Caltech was ranked 1st internationally between 2011 and 2016 by the Times Higher Education World University Rankings. Caltech was ranked as the best university in the world in two categories: Engineering & Technology and Physical Sciences. It was also found to have the highest faculty citation rate in the world.

Undergraduate admissions
Admission to Caltech is extremely rigorous. Prior to going test blind, Caltech students had the highest test scores in the nation. In admissions for the Class of 2026 (entering 2022), Caltech was ranked the hardest college in America to gain acceptance to by admit rate, at an all-time low of 2.7%. For the freshmen who enrolled in 2019 (Class of 2023) the middle 50% range of SAT were 740–780 for evidence-based reading and writing and 790–800 for math, and 1530–1570 total. The middle 50% range ACT Composite score was 35–36. The SAT Math Level 2 middle 50% range was 800–800. The middle 50% range for the SAT Physics Subject Test was 760–800; SAT Chemistry Subject Test was 760–800; SAT Biology Subject Tests was 760–800. In June 2020, Caltech announced a test-blind policy where they would not require nor consider test scores for the next two years. The moratorium was extended twice, starting July 2021, but was subsequently cancelled starting with the Class of 2029. The institute is need-blind for domestic applicants.

For the Class of 2027 (enrolled Fall 2023), Caltech received 13,136 applications and accepted 412 applicants for a 3.14% admit rate; 270 enrolled. The subsequent year, for the Class of 2028, Caltech reduced the number of seats by almost one hundred, accepting 315 applicants out of approximately 13,000 total applications. For the Class of 2025, 32% were of underrepresented ancestry (which includes students who self-identify as American Indian/Alaska Native, Hispanic/Latino, Black/African American, and/or Native Hawaiian/Pacific Islander), and 6% were foreign students.

Tuition and financial aid
Undergraduate tuition for the 2021–2022 school year was $56,394 and total annual costs were estimated to be $79,947 excluding the Caltech Student Health Insurance Plan. In 2012–2013, Caltech awarded $17.1 million in need-based aid, $438k in non-need-based aid, and $2.51 million in self-help support to enrolled undergraduate students. The average financial aid package of all students eligible for aid was $38,756 and students graduated with an average debt of $15,090.

Undergraduate program
The full-time, four-year undergraduate program emphasizes instruction in the arts and sciences and has high graduate coexistence. Caltech offers 28 majors (called "options") and 12 minors across all six academic divisions. Caltech also offers interdisciplinary programs in Applied Physics, Biochemistry, Bioengineering, Computation and Neural Systems, Control and Dynamical Systems, Environmental Science and Engineering, Geobiology and Astrobiology, Geochemistry, and Planetary Astronomy. The most popular options are Chemical Engineering, Computer Science, Electrical Engineering, Mechanical Engineering and Physics. The most popular majors of the class of 2023 were Computer Science, Mechanical Engineering, Physics, and Electrical Engineering. Prior to the entering class of 2013, Caltech required students to take a core curriculum of five terms of mathematics, five terms of physics, two terms of chemistry, one term of biology, two terms of lab courses, one term of scientific communication, three terms of physical education, and 12 terms of humanities and social science. Since 2013, only three terms each of mathematics and physics have been required by the institute, with the remaining two terms each required by certain options.

A typical class is worth 9 academic units and given the extensive core curriculum requirements in addition to individual options' degree requirements, students need to take an average of 40.5 units per term (more than four classes) to graduate in four years. 36 units is the minimum full-time load, 48 units is considered a heavy load, and registrations above 51 units require an overload petition. Approximately 20 percent of students double-major. This is achievable since the humanities and social sciences majors have been designed to be done in conjunction with a science major. Although choosing two options in the same division is discouraged, it is still possible.

First-year students are enrolled in first-term classes based upon results of placement exams in math, physics, chemistry, and writing and take all classes in their first two terms on a Pass/Fail basis. There is little competition; collaboration on homework is encouraged and the honor system encourages take-home tests and flexible homework schedules. Caltech offers co-operative programs with other schools, such as the Pasadena Art Center College of Design and Occidental College.

According to a PayScale study, Caltech graduates earn a median early career salary of $83,400 and $143,100 mid-career, placing them in the top 5 among graduates of US colleges and universities. The average net return on investment over a period of 20 years is $887,000, the tenth-highest among US colleges.

Caltech offers Army and Air Force ROTC in cooperation with the University of Southern California.

Graduate program

The graduate instructional programs emphasize doctoral studies and are dominated by science, technology, engineering, and mathematics fields. The institute offers graduate degree programs for the Master of Science, Engineer's Degree, Doctor of Philosophy, BS/MS and MD/PhD, with the majority of students in the PhD program. The most popular options are Chemistry, Physics, Biology, Electrical Engineering, and Chemical Engineering.

Admission to Caltech's graduate study is highly competitive, with the faculty evaluating factors such as academic preparation, experience, research interests, and recommendations from teachers or mentors. A key aspect of the admission process is the matching of faculty and an applicant's research interests. GRE tests (general and advanced subject) are no longer required, and in most programs, scores will not be considered for admission, though some departments offer the option to submit self-reported scores.

Initially, most new graduate students are assigned a temporary advisor, allowing time to select a permanent advisor. To aid in this, some graduate options include rotations in research labs, facilitating a better match with faculty research groups that align with the students' scientific interests. Up to three rotations in the first year are allowed in some options.

Caltech provides on-campus housing options for incoming graduate students. All new graduate students are guaranteed housing in their first year, with a variety of living experiences available to suit different needs. Approximately half of Caltech's graduate student population resides in campus housing. The application period for new student housing opens on April 15, 2023, and closes on April 30, 2023. Post-first year, students participate in a housing lottery, with the results announced two months prior to the contract end date, aiding in planning for those who need to seek off-campus housing.

The research facilities at Caltech are available to graduate students, but there are opportunities for students to work in facilities of other universities, research centers (such as NASA's Jet Propulsion Laboratory), and private industries. The graduate student to faculty ratio is 4:1. A joint program also exists between Caltech and the Keck School of Medicine of the University of Southern California, the UCLA David Geffen School of Medicine, and the Kaiser Permanente Bernard J. Tyson School of Medicine, which grants MD/PhD degrees. Students in this program do their preclinical and clinical work at USC, UCLA, or KPSOM, and their PhD work with any member of the Caltech faculty, including the Biology, Chemistry, and Engineering and Applied Sciences Divisions. The MD degree would be from the student's respective medical school and the PhD would be awarded from Caltech.

Approximately 99 percent of doctoral students have full financial support. Financial support for graduate students comes in the form of fellowships, research assistantships, teaching assistantships or a combination of fellowship and assistantship support.

Graduate students are bound by the same honor code as the undergraduates, allowing for take-home examinations. The Graduate Honor Council oversees any violations of the code.

Research

Caltech is classified among "R1: Doctoral Universities – Very High Research Activity". Caltech was elected to the Association of American Universities in 1934 and remains a research university with "very high" research activity, primarily in STEM fields. Caltech manages research expenditures of $270 million annually, 66th among all universities in the U.S. and 17th among private institutions without medical schools for 2008. The largest federal agencies contributing to research are NASA, National Science Foundation, Department of Health and Human Services, Department of Defense, and Department of Energy. Caltech received $144 million in federal funding for the physical sciences, $40.8 million for the life sciences, $33.5 million for engineering, $14.4 million for environmental sciences, $7.16 million for computer sciences, and $1.97 million for mathematical sciences in 2008.

The institute was awarded an all-time high funding of $357 million in 2009. Active funding from the National Science Foundation Directorate of Mathematical and Physical Science (MPS) for Caltech stands at $343 million, the highest for any educational institution in the nation, and higher than the total funds allocated to any state except California and New York.

In 2005, Caltech had 739000 sqft dedicated to research: 330000 sqft to physical sciences, 163000 sqft to engineering, and 160000 sqft to biological sciences.

In addition to managing JPL, Caltech also operates the Palomar Observatory in San Diego County, the Owens Valley Radio Observatory in Bishop, California, the Submillimeter Observatory and W. M. Keck Observatory at the Mauna Kea Observatory, the Laser Interferometer Gravitational-Wave Observatory at Livingston, Louisiana and Richland, Washington, and Kerckhoff Marine Laboratory in Corona del Mar, California. The Institute launched the Kavli Nanoscience Institute at Caltech in 2006, the Keck Institute for Space Studies in 2008, and is also the current home for the Einstein Papers Project. The Spitzer Science Center (SSC), part of the Infrared Processing and Analysis Center located on the Caltech campus, is the data analysis and community support center for NASA's Spitzer Space Telescope.

Caltech partnered with UCLA to establish a Joint Center for Translational Medicine (UCLA-Caltech JCTM), which conducts experimental research into clinical applications, including the diagnosis and treatment of diseases such as cancer.

Caltech operates several TCCON stations as part of an international collaborative effort of measuring greenhouse gases globally. One station is on campus.

Undergraduates at Caltech are also encouraged to participate in research. About 80% of the class of 2010 did research through the annual Summer Undergraduate Research Fellowships (SURF) program at least once during their stay, and many continued during the school year. Students write and submit SURF proposals for research projects in collaboration with professors, and about 70 percent of applicants are awarded SURFs. The program is open to both Caltech and non-Caltech undergraduate students. It serves as preparation for graduate school and helps to explain why Caltech has the highest percentage of alumni who go on to receive a PhD of all the major universities.

The licensing and transferring of technology to the commercial sector is managed by the Office of Technology Transfer (OTT). OTT protects and manages the intellectual property developed by faculty members, students, other researchers, and JPL technologists. Caltech receives more invention disclosures per faculty member than any other university in the nation. , 1891 patents were granted to Caltech researchers since 1969.

Students
Caltech enrolled 987 undergraduate students and 1,410 graduate students for the 2021–2022 school year. Women made up 45% of the undergraduate and 33% of the graduate student body. The racial demographics of the school substantially differ from those of the nation as a whole.

The four-year graduation rate is 79% and the six-year rate is 92%, which is low compared to most leading U.S. universities, but substantially higher than it was in the 1960s and 1970s. Students majoring in STEM fields traditionally have graduation rates below 70%.

House system
During the early 20th century, a Caltech committee visited several universities and decided to transform the undergraduate housing system from fraternities to a house system. Four South Houses (or Hovses, as styled in the stone engravings) were built: Blacker House, Dabney House, Fleming House and Ricketts House. In the 1960s, three North Houses were built: Lloyd House, Page House, and Ruddock House, and during the 1990s, Avery House. The four South Houses closed for renovation in 2005 and reopened in 2006. The latest addition to residential life at Caltech is Bechtel Residence, which opened in 2018. It is not affiliated with the house system. All first- and second-year students live on campus in the house system or in the Bechtel Residence.

On account of Albert B. Ruddock's affiliation with the Human Betterment Foundation, in January 2021, the Caltech Board of Trustees authorized the removal of Ruddock's name from campus buildings. Ruddock House was renamed as the Grant D. Venerable House.

Athletics

Caltech has athletic teams in baseball, men's and women's basketball, cross country, men's and women's soccer, swimming and diving, men's and women's tennis, track and field, women's volleyball, and men's and women's water polo. Caltech's mascot is the Beaver, a homage to nature's engineer. Its teams are members of the NCAA Division III and compete in the Southern California Intercollegiate Athletic Conference (SCIAC), which Caltech co-founded in 1915.

On January 6, 2007, the Beavers' men's basketball team snapped a 207-game losing streak to Division III schools, beating Bard College 81–52. It was their first Division III victory since 1996. Until their win over Occidental College on February 22, 2011 the team had not won a game in SCIAC play since 1985. Ryan Elmquist's free throw with 3.3 seconds in regulation gave the Beavers the victory. The documentary film Quantum Hoops concerns the events of the Beavers' 2005–06 season.

On January 13, 2007, the Caltech women's basketball team snapped a 50-game losing streak, defeating the Pomona-Pitzer Sagehens 55–53. The women's program, which entered the SCIAC in 2002, garnered their first conference win. On the bench as honorary coach for the evening was Robert Grubbs, 2005 Nobel laureate in Chemistry. The team went on to beat Whittier College on February 10, for its second SCIAC win, and placed its first member on the All Conference team.

In 2007, 2008, and 2009, the women's table tennis team (a club team) competed in nationals. The women's Ultimate club team, known as "Snatch", has also been very successful in recent years, ranking 44 of over 200 college teams in the Ultimate Player's Association.

On February 2, 2013, the Caltech baseball team ended a 228-game losing streak, the team's first win in nearly 10 years.

The track and field team's home venue is at the South Athletic Field in Tournament Park, the site of the first Rose Bowl Game.

The school also sponsored an intercollegiate football team from 1973 through 1977, and played part of its home schedule at the Rose Bowl.

Performing and visual arts
The Caltech/Occidental College Orchestra is a full seventy-piece orchestra composed of students, faculty, and staff at Caltech and nearby Occidental College. The orchestra gives three pairs of concerts annually, at both Caltech and Occidental College. There are also two Caltech Jazz Bands and a Concert Band, as well as an active chamber music program. For vocal music, Caltech has a mixed-voice Glee Club and the smaller Chamber Singers. The theater program at Caltech is known as TACIT, or Theater Arts at the California Institute of Technology. There are two to three plays organized by TACIT per year, and they were involved in the production of the PHD Movie, released in 2011.

Annual events
Every Halloween, Dabney House conducts the infamous "Millikan pumpkin-drop experiment" from the top of Millikan Library, the highest point on campus. According to tradition, a claim was once made that the shattering of a pumpkin frozen in liquid nitrogen and dropped from a sufficient height would produce a triboluminescent spark. This yearly event involves a crowd of observers, who try to spot the elusive spark. The title of the event is an oblique reference to the famous Millikan oil-drop experiment which measured e, the elemental unit of electrical charge.

On Ditch Day, the seniors ditch school, leaving behind elaborately designed tasks and traps at the doors of their rooms to prevent underclassmen from entering. Over the years this has evolved to the point where many seniors spend months designing mechanical, electrical, and software obstacles to confound the underclassmen. Each group of seniors designs a "stack" to be solved by a handful of underclassmen. The faculty have been drawn into the event as well, and cancel all classes on Ditch Day so the underclassmen can participate in what has become a highlight of the academic year.

Another long-standing tradition is the playing of Wagner's "Ride of the Valkyries" at 7:00 each morning during finals week with the largest, loudest speakers available. The playing of that piece is not allowed at any other time (except if one happens to be listening to the entire 14 hours and 5 minutes of The Ring Cycle), and any offender is dragged into the showers to be drenched in cold water fully dressed.

Pranks

Caltech students have been known for their many pranks (also known as "RFs").

The two most famous in recent history are the changing of the Hollywood Sign to read "Caltech", by judiciously covering up certain parts of the letters, and the changing of the scoreboard to read Caltech 38, MIT 9 during the 1984 Rose Bowl Game. But the most famous of all occurred during the 1961 Rose Bowl Game, where Caltech students altered the flip-cards that were raised by the stadium attendees to display "Caltech", and several other "unintended" messages. This event is now referred to as the Great Rose Bowl Hoax.

In recent years, pranking has been officially encouraged by Tom Mannion, Caltech's Assistant VP for Student Affairs and Campus Life. "The grand old days of pranking have gone away at Caltech, and that's what we are trying to bring back," reported the Boston Globe.

In December 2011, Caltech students went to New York and pulled a prank in Manhattan's Greenwich Village. The prank involved making The Cube sculpture look like the Aperture Science Weighted Companion Cube from the video game Portal.

Caltech pranks have been documented in three Legends of Caltech books, the most recent of which was edited by alumni Autumn Looijen '99 and Mason Porter '98 and published in May 2007.

Rivalry with MIT
In 2005, a group of Caltech students pulled a string of pranks during MIT's Campus Preview Weekend for admitted students. These include covering up the word Massachusetts in the "Massachusetts Institute of Technology" engraving on the main building façade with a banner so that it read "That Other Institute of Technology". A group of MIT hackers responded by altering the banner so that the inscription read "The Only Institute of Technology." Caltech students also passed out T-shirts to MIT's incoming freshman class that had MIT written on the front and "...because not everyone can go to Caltech" along with an image of a palm tree on the back.

MIT retaliated in April 2006, when students posing as the Howe & Ser (Howitzer) Moving Company stole the 130-year-old, 1.7-ton Fleming House cannon and moved it over 3,000 miles to their campus in Cambridge, Massachusetts for their 2006 Campus Preview Weekend, repeating a similar prank performed by nearby Harvey Mudd College in 1986. Thirty members of Fleming House traveled to MIT and reclaimed their cannon on April 10, 2006.

On April 13, 2007 (Friday the 13th), a group of students from The California Tech, Caltech's campus newspaper, arrived and distributed fake copies of The Tech, MIT's campus newspaper, while prospective students were visiting for their Campus Preview Weekend. Articles included "MIT Invents the Interweb", "Architects Deem Campus 'Unfortunate, and "Infinite Corridor Not Actually Infinite".

In December 2009, some Caltech students declared that MIT had been sold and had become the Caltech East campus. A "sold" banner was hung on front of the MIT dome building and a "Welcome to Caltech East: School of the Humanities" banner over the Massachusetts Avenue Entrance. Newspapers and T-shirts were distributed, and door labels and fliers in the infinite corridor were put up in accordance with the "curriculum change."

In September 2010, MIT students attempted to put a TARDIS, the time machine from the BBC's Doctor Who, onto a roof. Caught in mid-act, the prank was aborted. In January 2011, Caltech students in conjunction with MIT students helped put the TARDIS on top of Baxter. Caltech students then moved the TARDIS to UC Berkeley and Stanford.

In April 2014, during MIT's Campus Preview Weekend, a group of Caltech students handed out mugs emblazoned with the MIT logo on the front and the words "The Institute of Technology" on the back. When heated, the mugs turn orange, display a palm tree, and read "Caltech The Hotter Institute of Technology." Identical mugs continue to be sold at the Caltech campus store.

Honor code
Life in the Caltech community is governed by the honor code, which simply states: "No member of the Caltech community shall take unfair advantage of any other member of the Caltech community." This is enforced by a Board of Control, which consists of undergraduate students, and by a similar body at the graduate level, called the Graduate Honor Council.

The honor code aims at promoting an atmosphere of respect and trust that allows Caltech students to enjoy privileges that make for a more relaxed atmosphere. For example, the honor code allows professors to make the majority of exams as take-home, allowing students to take them on their own schedule and in their preferred environment.

Through the late 1990s, the only exception to the honor code, implemented earlier in the decade in response to changes in federal regulations, concerned the sexual harassment policy. Today, there are myriad exceptions to the honor code in the form of new Institute policies such as the fire policy and alcohol policy. Although both policies are presented in the Honor System Handbook given to new members of the Caltech community, some undergraduates regard them as a slight against the honor code and the implicit trust and respect it represents within the community. In recent years, the Student Affairs Office has also taken up pursuing investigations independently of the Board of Control and Conduct Review Committee, an implicit violation of both the honor code and written disciplinary policy that has contributed to further erosion of trust between some parts of the undergraduate community and the administration.

Notable people
As of October 2022, Caltech has 46 Nobel laureates to its name awarded to 30 alumni (26 graduates and 4 postdocs), including 5 Caltech professors who are also alumni (Carl D. Anderson, Linus Pauling, William A. Fowler, Edward B. Lewis, and Kip Thorne), and 16 non-alumni professors (14 at the time of the award, not including David Baltimore and Renato Dulbecco). The total number of Nobel Prizes is 47 because Pauling received prizes in both Chemistry and Peace. Eight faculty and alumni have received a Crafoord Prize from the Royal Swedish Academy of Sciences, while 58 have been awarded the U.S. National Medal of Science, and 11 have received the National Medal of Technology. One alumnus, Stanislav Smirnov, won the Fields Medal in 2010. Other distinguished researchers have been affiliated with Caltech as postdoctoral scholars (for example, Barbara McClintock, James D. Watson, Sheldon Glashow and John Gurdon) or visiting professors (for example, Albert Einstein, Stephen Hawking and Edward Witten).

Alumni
There are 22,930 total living alumni in the U.S. and around the world. As of October 2022, 30 alumni and 16 non-alumni faculty have won the Nobel Prize. The Turing Award, the "Nobel Prize of Computer Science", has been awarded to six alumni, and one has won the Fields Medal.

Many alumni have participated in scientific research. Some have concentrated their studies on the very small universe of atoms and molecules. Nobel laureate Carl D. Anderson (BS 1927, PhD 1930) proved the existence of positrons and muons, Nobel laureate Edwin McMillan (BS 1928, MS 1929) synthesized the first transuranium element, Nobel laureate Leo James Rainwater (BS 1939) investigated the non-spherical shapes of atomic nuclei, and Nobel laureate Douglas D. Osheroff (BS 1967) studied the superfluid nature of helium-3. Donald Knuth (PhD 1963), the "father" of the analysis of algorithms, wrote The Art of Computer Programming and created the TeX computer typesetting system, which is commonly used in the scientific community. Bruce Reznick (BS 1973) is a mathematician noted for his contributions to number theory and the combinatorial-algebraic-analytic investigations of polynomials. Narendra Karmarkar (MS 1979) is known for the interior point method, a polynomial algorithm for linear programming known as Karmarkar's algorithm.

Other alumni have turned their gaze to the universe. C. Gordon Fullerton (BS 1957, MS 1958) piloted the third Space Shuttle mission. Astronaut (and later, United States Senator) Harrison Schmitt (BS 1957) was the only geologist to have walked on the surface of the Moon. Astronomer Eugene Merle Shoemaker (BS 1947, MS 1948) co-discovered Comet Shoemaker-Levy 9 (a comet which crashed into the planet Jupiter) and was the first person buried on the Moon (by having his ashes crashed into the Moon). Astronomer George O. Abell (BS 1951, MS 1952, PhD 1957) while a grad student at Caltech participated in the National Geographic Society-Palomar Sky Survey. This ultimately resulted in the publication of the Abell Catalogue of Clusters of Galaxies, the definitive work in the field.

Undergraduate alumni founded, or co-founded, companies such as LCD manufacturer Varitronix, Hotmail, Compaq, MathWorks (which created Matlab), and database provider Imply, while graduate students founded, or co-founded, companies such as Intel, TRW, and the non-profit educational organization, the Exploratorium.

Arnold Beckman (PhD 1928) invented the pH meter and commercialized it with the founding of Beckman Instruments. His success with that company enabled him to provide seed funding for William Shockley (BS 1932), who had co-invented semiconductor transistors and wanted to commercialize them. Shockley became the founding Director of the Shockley Semiconductor Laboratory division of Beckman Instruments. Shockley had previously worked at Bell Labs, whose first president was another alumnus, Frank Jewett (BS 1898). Because his aging mother lived in Palo Alto, California, Shockley established his laboratory near her in Mountain View, California. Shockley was a co-recipient of the Nobel Prize in physics in 1956, but his aggressive management style and odd personality at the Shockley Lab became unbearable. In late 1957, eight of his researchers resigned and with support from Sherman Fairchild formed Fairchild Semiconductor. Among the "traitorous eight" was Gordon E. Moore (PhD 1954), who later left Fairchild to co-found Intel. Other offspring companies of Fairchild Semiconductor include National Semiconductor and Advanced Micro Devices, which in turn spawned more technology companies in the area. Shockley's decision to use silicon instead of germanium as the semiconductor material, coupled with the abundance of silicon semiconductor related companies in the area, gave rise to the term "Silicon Valley" to describe that geographic region surrounding Palo Alto.

Caltech alumni also held public offices, with Mustafa A.G. Abushagur (PhD 1984) the Deputy Prime Minister of Libya and Prime Minister-Elect of Libya, James Fletcher (PhD 1948) the 4th and 7th Administrator of NASA, Steven Koonin (PhD 1972) the Undersecretary of Energy for Science, and Regina Dugan (PhD 1993) the 19th director of DARPA. The 20th director for DARPA, Arati Prabhakar, is also a Caltech alumna (PhD 1984) as well as Charles Elachi (Phd 1971), former director of the Jet Propulsion Lab. Arvind Virmani is a former Chief Economic Adviser to the Government of India. In 2013, President Obama announced the nomination of France Cordova (PhD 1979) as the director of the National Science Foundation and Ellen Williams (PhD 1982) as the director for ARPA-E.

Faculty and staff

Richard Feynman was among the most well-known physicists associated with Caltech, having published the Feynman Lectures on Physics, an undergraduate physics text, and popular science texts such as Six Easy Pieces for the general audience. The promotion of physics made him a public figure of science, although his Nobel-winning work in quantum electrodynamics was already very established in the scientific community. Murray Gell-Mann, a Nobel-winning physicist, introduced a classification of hadrons and went on to postulate the existence of quarks, which is currently accepted as part of the Standard Model. Long-time Caltech President Robert Andrews Millikan was the first to calculate the charge of the electron with his well-known oil-drop experiment, while Richard Chace Tolman is remembered for his contributions to cosmology and statistical mechanics. 2004 Nobel Prize in Physics winner H. David Politzer is a current professor at Caltech, as is astrophysicist and author Kip Thorne and eminent mathematician Barry Simon. Linus Pauling pioneered quantum chemistry and molecular biology, and went on to discover the nature of the chemical bond in 1939. Seismologist Charles Richter, also an alumnus, developed the magnitude scale that bears his name, the Richter magnitude scale for measuring the power of earthquakes. One of the founders of the geochemistry department, Clair Patterson was the first to accurately determine the age of the Earth via lead:uranium ratio in meteorites. In engineering, Theodore von Kármán made many key advances in aerodynamics, notably his work on supersonic and hypersonic airflow characterization. A repeating pattern of swirling vortices is named after him, the von Kármán vortex street. Participants in von Kármán's GALCIT project included Frank Malina, who helped develop the WAC Corporal, which was the first U.S. rocket to reach the edge of space, Jack Parsons, a pioneer in the development of liquid and solid rocket fuels who designed the first castable composite-based rocket motor, and Qian Xuesen, who was dubbed the "Father of Chinese Rocketry". More recently, Michael Brown, a professor of planetary astronomy, discovered many trans-Neptunian objects, most notably the dwarf planet Eris, which prompted the International Astronomical Union to redefine the term "planet".

David Baltimore, the Robert A. Millikan Professor of Biology, and Alice Huang, Senior Faculty Associate in Biology, served as the presidents of AAAS from 2007 to 2008 and 2010 to 2011, respectively.

33% of the faculty are members of the National Academy of Sciences or Engineering and/or fellows of the American Academy of Arts and Sciences. This is the highest percentage of any faculty in the country with the exception of the graduate institution Rockefeller University.

The average salary for assistant professors at Caltech is $111,300, associate professors $121,300, and full professors $172,800. Caltech faculty are active in applied physics, astronomy and astrophysics, biology, biochemistry, biological engineering, chemical engineering, computer science, geology, mechanical engineering, and physics.

Presidents

 * James Augustin Brown Scherer (1908–1920) (president of Throop College of Technology before the name change)
 * Robert A. Millikan (1921–1945), experimental physicist, Nobel laureate in physics for 1923 (his official title was "Chairman of the Executive Council")
 * Lee A. DuBridge (1946–1969), experimental physicist (first to officially hold the title of President)
 * Harold Brown (1969–1977), physicist and public servant (left Caltech to serve as United States Secretary of Defense in the administration of Jimmy Carter)
 * Robert F. Christy (1977–1978), astrophysicist (acting president)
 * Marvin L. Goldberger (1978–1987), theoretical physicist (left to serve as Director of Institute for Advanced Study)
 * Thomas E. Everhart (1987–1997), experimental physicist
 * David Baltimore (1997–2006), molecular biologist, Nobel laureate in Physiology or Medicine for 1975
 * Jean-Lou Chameau (2006–2013), civil engineer and educational administrator (left to serve as president of King Abdullah University of Science and Technology)
 * Thomas F. Rosenbaum (2014–), condensed matter physicist and administrator

Caltech startups
Over the years Caltech has actively promoted the commercialization of technologies developed within its walls. Through its Office of Technology Transfer & Corporate Partnerships, scientific breakthroughs have led to the transfer of numerous technologies in a wide variety of scientific-related fields such as photovoltaic, radio-frequency identification (RFID), semiconductors, hyperspectral imaging, electronic devices, protein design, solid state amplifiers and many more. Companies such as Quora, Contour Energy Systems, Impinj, Fulcrum Microsystems, Nanosys, Inc., Photon etc., Xencor, and Wavestream Wireless have emerged from Caltech.

In media and popular culture
Caltech has appeared in many works of popular culture, both as itself and in disguised form. On television, it played a prominent role and was the workplace of all four male lead characters and one female lead character in the sitcom The Big Bang Theory. Caltech is also the inspiration, and frequent film location, for the California Institute of Science in Numb3rs. On film, the Pacific Tech of The War of the Worlds and Real Genius is based on Caltech.

In nonfiction, two 2007 documentaries examine aspects of Caltech: Curious, its researchers, and Quantum Hoops, its men's basketball team.

Caltech is also prominently featured in many comics and television series by Marvel Entertainment. In Marvel Comics, the university serves as the alma mater of Hulk, Mister Fantastic, Bill Foster (Black Goliath), and Madman. In the Marvel Cinematic Universe, Bruno Carrelli (Kamala Khan's best friend and love interest) attends Caltech in the miniseries Ms. Marvel.

Given its Los Angeles-area location, the grounds of the Institute are often host to short scenes in movies and television. The Athenaeum dining club appears in the Beverly Hills Cop series, The X-Files, True Romance, and The West Wing.

title: California Institute of Technology


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Talk:Cretaceous–Paleogene extinction event

Why did only ectothermic large tetrapods survive the extinction event?
The second sentence of the article is interesting but puzzling. Why would ectothermic species be advantaged rather than disadvantaged by their lack of ability to regulate their own heat?

Not sure, and it the distinction only seemed to matter for animals over 55 pounds. I assume it's because endotherms have higher food and water requirements, which would make it much harder for large ones to survive under such conditions. Their habitats would have been destroyed, and with them would have gone their food and water. If they didn't die immediately, they would have starved or been poisoned by food, water and air tainted with sulfur and other nasty things by the impact. Sumanuil (talk) 22:12, 27 January 2019 (UTC)

 * Was the sea turtle not the only tetrapod over 55 pounds to survive? With only one example, it's impossible to say anything about any particular quality such as ecto- or endothermy. Firejuggler86 (talk) 02:22, 24 June 2021 (UTC)

Notes to update Cretaceous stage
Prior to the 2021, that's results of end Cretaceous when asteroid hits the Mexico did happen to 66 million years ago, but according to 2021's paper by Bland et al., they gave it to update the Cretaceous stage was 146-67 million years ago (Early Cretaceous from 146-101 Ma; Cenomanian from 101-94 Ma; Coniacian from 90-86 Ma; Campanian from 84-75 Ma, Maastrichtian from 75-67 Ma).
 * You have to provide a reference, not just change the text so that it no longer reflects the source cited. It would be best to give the source on this talk page so that it can be discussed. It will probably take more than one paper to gain general acceptance of revision of the dating of the International Commission on Stratigraphy. Dudley Miles (talk) 16:02, 13 July 2021 (UTC)
 * It would also be great if that sentence would be rewritten, right now it's an unintelligible blob that says nothing. What's the "that's results of end Cretaceous when asteroid hits the Mexico did happen to 66 million years ago, but according to 2021's paper by Bland et al."? What should it mean? Artem.G (talk) 15:37, 21 August 2021 (UTC)
 * This paper is by "Chris Bland the Whale King"? The same barely coherent vandal who, until he was finally banned, kept vandalizing pages with blatant errors and nonsensical original research?--Mr Fink (talk) 18:37, 30 August 2021 (UTC)

Citation 5
Citation 5 is a photography book, and so doesn't really seem relevant to the statements connected to it, 'With the exception of some ectothermic species such as sea turtles and crocodilians, no tetrapods weighing more than 25 kilograms (55 pounds) survived' I suspect theres an error here — Preceding unsigned comment added by 80.4.214.138 (talk) 11:53, 10 November 2021 (UTC)

Please add info on decline in dinosaur biodiversity millions of years before & timeline items
I think at the least a brief mention is warranted. It's featured in 2022 in science like so:

"A study indicates a substantial decline in dinosaur biodiversity millions of years before the Cretaceous-Paleogene extinction event."

There may also be more info in the latest section of Timeline of Cretaceous–Paleogene extinction event research not yet included in the article. Please check that too.

Prototyperspective (talk) 21:13, 28 November 2022 (UTC)

 * This issue is discussed in the article, which states that there have been some studies showing a decline in diversity in the late Cretaceous and others no decline. This is one more regional study and does not seem significant enough to warrant mentioning. Dudley Miles (talk) 23:01, 28 November 2022 (UTC)

Massively glitched page--huge amount of content potentially lost
I just finished a large edit in visual editing mode and added dozens of new papers like "Deccan volcanism caused coupled pCO2 and terrestrial temperature rises, and pre-impact extinctions in northern China" and "No post-Cretaceous ecosystem depression in European forests? Rich insect-feeding damage on diverse middle Palaeocene plants, Menat, France". After I submitted my edit, I looked at the Citations section and the papers I added were nowhere to be found. After several hours of work. What the **** happened?! How did this glitch happen??? I remember copying and pasting every link to single reference I added into the citation tool!!! Is there any way to recover my work so I do not have to do it all over again???!!! Anteosaurus magnificus (talk) 01:18, 19 November 2023 (UTC)
 * I do not know about visual editing mode but my method in standard mode is to click preview first to check and paste and copy the revised article into Word if there looks like there might be a problem. This usually works to provide a backup, but not always. Dudley Miles (talk) 08:54, 19 November 2023 (UTC)

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/


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Carlo Goldoni

Carlo Osvaldo Goldoni (, also, ; 25 February 1707 – 6 February 1793) was an Italian playwright and librettist from the Republic of Venice. His works include some of Italy's most famous and best-loved plays. Audiences have admired the plays of Goldoni for their ingenious mix of wit and honesty. His plays offered his contemporaries images of themselves, often dramatizing the lives, values, and conflicts of the emerging middle classes. Though he wrote in French and Italian, his plays make rich use of the Venetian language, regional vernacular, and colloquialisms. Goldoni also wrote under the pen name and title Polisseno Fegeio, Pastor Arcade, which he claimed in his memoirs the "Arcadians of Rome" bestowed on him.

Memoirs
There is an abundance of autobiographical information on Goldoni, most of which comes from the introductions to his plays and from his Memoirs. However, these memoirs are known to contain many errors of fact, especially about his earlier years.

In these memoirs, he paints himself as a born comedian, careless, light-hearted and with a happy temperament, proof against all strokes of fate, yet thoroughly respectable and honourable.

Early life and studies
Goldoni was born in Venice in 1707, the son of Margherita Salvioni (or Saioni) and Giulio Goldoni. In his memoirs, Goldoni describes his father as a physician, and claims that he was introduced to theatre by his grandfather Carlo Alessandro Goldoni. In reality, it seems that Giulio was an apothecary; as for the grandfather, he had died four years before Carlo's birth. In any case, Goldoni was deeply interested in theatre from his earliest years, and all attempts to direct his activity into other channels were of no avail; his toys were puppets, and his books were plays.

His father placed him under the care of the philosopher Caldini at Rimini but the youth soon ran away with a company of strolling players and returned to Venice. In 1723 his father matriculated him into the stern Collegio Ghislieri in Pavia, which imposed the tonsure and monastic habits on its students. However, he relates in his Memoirs that a considerable part of his time was spent reading Greek and Latin comedies. He had already begun writing at this time and, in his third year, he composed a libellous poem (Il colosso) in which he ridiculed the daughters of certain Pavian families. As a result of that incident (and/or of a visit with some schoolmates to a local brothel), he was expelled from the school and had to leave the city (1725). He studied law at Udine, and eventually took his degree at University of Modena. He was employed as a law clerk at Chioggia and Feltre, after which he returned to his native city and began practising. Educated as a lawyer, and holding lucrative positions as secretary and counsellor, he seemed, indeed, at one time to have settled down to the practice of law, but following an unexpected summons to Venice, after an absence of several years, he changed his career, and thenceforth he devoted himself to writing plays and managing theatres. His father died in 1731. In 1732, to avoid an unwanted marriage, he left the town for Milan and then for Verona where the theatre manager Giuseppe Imer helped him on his way to becoming a comical poet as well as introducing him to his future wife, Nicoletta Conio. Goldoni returned with her to Venice, where he stayed until 1743.

Theatrical career

Goldoni entered the Italian theatre scene with a tragedy, Amalasunta, produced in Milan. The play was a critical and financial failure.

Submitting it to Count Prata, director of the opera, he was told that his piece "was composed with due regard for the rules of Aristotle and Horace, but not according to those laid down for the Italian drama." "In France", continued the count, "you can try to please the public, but here in Italy it is the actors and actresses whom you must consult, as well as the composer of the music and the stage decorators. Everything must be done according to a certain form which I will explain to you."

Goldoni thanked his critic, went back to his inn and ordered a fire, into which he threw the manuscript of his Amalasunta.

His next play, Belisario, written in 1734, was more successful, though of its success he afterwards professed himself ashamed.

During this period he also wrote librettos for opera seria and served for a time as literary director of the San Giovanni Grisostomo, Venice's most distinguished opera house.

He wrote other tragedies for a time, but he was not long in discovering that his bent was for comedy. He had come to realize that the Italian stage needed reforming; adopting Molière as his model, he went to work in earnest and in 1738 produced his first real comedy, L'uomo di mondo ("The Man of the World"). During his many wanderings and adventures in Italy, he was constantly at work and when, at Livorno, he became acquainted with the manager Medebac, he determined to pursue the profession of playwriting in order to make a living. He was employed by Medebac to write plays for his theatre in Venice. He worked for other managers and produced during his stay in that city some of his most characteristic works. He also wrote Momolo Cortesan in 1738. By 1743, he had perfected his hybrid style of playwriting (combining the model of Molière with the strengths of Commedia dell'arte and his own wit and sincerity). This style was typified in La Donna di garbo, the first Italian comedy of its kind.

After 1748, Goldoni collaborated with the composer Baldassare Galuppi, making significant contributions to the new form of 'opera buffa'. Galuppi composed the score for more than twenty of Goldoni's librettos. As with his comedies, Goldoni's opera buffa integrates elements of the Commedia dell'arte with recognisable local and middle-class realities. His operatic works include two of the most successful musical comedies of the eighteenth century, Il filosofo di campagna (The Country Philosopher), set by Galuppi (1752) and La buona figliuola (The Good Girl), set by Niccolò Piccinni (1760).

In 1753, following his return from Bologna, he defected to the Teatro San Luca of the Vendramin family, where he performed most of his plays to 1762.

Move to France and death

In 1757, he engaged in a bitter dispute with playwright Carlo Gozzi, which left him utterly disgusted with the tastes of his countrymen; so much so that in 1761 he moved to Paris, where he received a position at court and was put in charge of the Théâtre-Italien. He spent the rest of his life in France, composing most of his plays in French and writing his memoirs in that language.

Among the plays which he wrote in French, the most successful was Le bourru bienfaisant, dedicated to Marie Adélaïde, a daughter of Louis XV and aunt to the dauphin, the future Louis XVI of France. It premiered on 4 February 1771, almost nine months after the dauphin's marriage to Marie Antoinette. Goldoni enjoyed considerable popularity in France; in 1769, when he retired to Versailles, the King gave him a pension. He lost this pension after the French Revolution. The Convention eventually voted to restore his pension the day after his death. It was restored to his widow, at the pleading of the poet André Chénier; "She is old", he urged, "she is seventy-six, and her husband has left her no heritage save his illustrious name, his virtues and his poverty."

Goldoni's impact on Italian theatre
In his Memoirs Goldoni amply discusses the state of Italian comedy when he began writing. At that time, Italian comedy revolved around the conventionality of the Commedia dell'arte, or improvised comedy. Goldoni took to himself the task of superseding the comedy of masks and the comedy of intrigue with representations of actual life and manners through the characters and their behaviours. He maintained that Italian life and manners were susceptible of artistic treatment such as had not been given them before.

His works are a lasting monument to the changes that he initiated: a dramatic revolution that had been attempted but not achieved before. Goldoni's importance lies in providing good examples rather than precepts. Goldoni says that he took for his models the plays of Molière and that whenever a piece of his own succeeded he whispered to himself: "Good, but not yet Molière". Goldoni's plays are gentler and more optimistic in tone than Molière's.

It was this very success that was the object of harsh critiques by Carlo Gozzi, who accused Goldoni of having deprived the Italian theatre of the charms of poetry and imagination. The great success of Gozzi's fairy dramas so irritated Goldoni that it led to his self-exile to France.

Goldoni gave to his country a classical form, which, though it has since been cultivated, has yet to be cultivated by a master.

Themes
Goldoni's plays that were written while he was still in Italy ignore religious and ecclesiastical subjects. This may be surprising, considering his staunch Catholic upbringing. No thoughts are expressed about death or repentance in his memoirs or in his comedies. After his move to France, his position became clearer, as his plays took on a clear anti-clerical tone and often satirized the hypocrisy of monks and of the Church.

Goldoni was inspired by his love of humanity and the admiration he had for his fellow men. He wrote, and was obsessed with, the relationships that humans establish with one another, their cities and homes, the Humanist movement, and the study of philosophy. The moral and civil values that Goldoni promotes in his plays are those of rationality, civility, humanism, the importance of the rising middle class, a progressive stance on state affairs, honour and honesty. Goldoni had a dislike for arrogance, intolerance and the abuse of power.

Goldoni's main characters are no abstract examples of human virtue, nor monstrous examples of human vice. They occupy the middle ground of human temperament. Goldoni maintains an acute sensibility for the differences in social classes between his characters as well as environmental and generational changes. Goldoni pokes fun at the arrogant nobility and the pauper who lacks dignity.

Venetian and Tuscan
As in other theatrical works of the time and place, the characters in Goldoni's Italian comedies spoke originally either the literary Tuscan variety (which became modern Italian) or the Venetian dialect, depending on their station in life. However, in some printed editions of his plays, he often turned the Venetian texts into Tuscan, too.

Goldoni in popular culture
One of his best-known works is the comic play Servant of Two Masters, which has been translated and adapted internationally numerous times. In 1966 it was adapted into an opera buffa by the American composer Vittorio Giannini. In 2011, Richard Bean adapted the play for the National Theatre of Great Britain as One Man, Two Guvnors. Its popularity led to a transfer to the West End and in 2012 to Broadway.

The film Carlo Goldoni – Venice, Grand Theatre of the World, directed by Alessandro Bettero, was released in 2007 and is available in English, Italian, French, and Japanese.

Selected works
The following is a small sampling of Goldoni's enormous output.

Tragedies

 * Rosmonda (1734)
 * Griselda (1734)

Tragicomedies

 * Belisario (1734)
 * Don Giovanni Tenorio o sia Il dissoluto, "The Dissolute" (1735)
 * Rinaldo di Montalbano (1736)

Comedies

 * Il servitore di due padroni, (1745) "The Servant of Two Masters" (now often retitled Arlecchino servitore di due padroni "Harlequin Servant of two Masters")
 * I due gemelli veneziani, "The Two Venetian Twins" (1747)
 * La vedova scaltra, "The Shrewd Widow" (1748)
 * La putta onorata, "The Honorable Maid" (1749)
 * Il cavaliere e la dama, "The Gentleman and the Lady" (1749)
 * La famiglia dell'antiquario, "The Antiquarian's Family" (1750)
 * Il teatro comico, "The Comical Theatre" (1750–1751)
 * Il bugiardo, "The Liar" (1750–1751)
 * Il vero amico, "The True Friend" (1750) translated by Anna Cuffaro
 * I pettegolezzi delle donne, "Women's Gossip" (1750–1751)
 * La locandiera, "The Mistress of the Inn" (1751)
 * Il feudatario "The Feudal Lord" (1752)
 * Gl'innamorati, "The Lovers" (1759)
 * I rusteghi, "The Boors" (1760)
 * Le baruffe chiozzotte, "The Chioggia Scuffles" (1762)
 * Gli amori di Zelinda e Lindoro, "The Love of Zelinda and Lindoro" (1764)

Opera seria libretti

 * Amalasunta (1732)
 * Gustavo primo, re di Svezia (c. 1738)
 * Oronte, re de' Sciti (1740)
 * Statira (c. 1740)

Opera buffa libretti

 * La contessina (The Young Countess) by Maccari (1743)
 * L'Arcadia in Brenta (The Arcadia in Brenta) by Galuppi (1749)
 * Il mondo della luna (The World on the Moon), set to music by Galuppi (1750), Haydn (1777), Paisiello (1782) and other composers.
 * Il filosofo di campagna (The Country Philosopher) by Galuppi (1754)
 * Il mercato di Malmantile (The Malmantile Market) by Fischietti (1757)
 * Buovo d'Antona, set to music by Tommaso Traetta (1758, incorrectly recorded as 1750 in Zatta's edition)
 * La buona figliuola (The Good Girl) by Niccolò Piccinni (1760)
 * Lo speziale (The Apothecary) by Joseph Haydn (1768)
 * La finta semplice (The Fake Innocent) by Wolfgang Amadeus Mozart (1769)
 * Le pescatrici (The Fisherwomen) by Haydn (1770), Florian Leopold Gassmann (1771)

Intermezzo libretti

 * Le donne vendicate, "The Revenge of the Women" (1751)

Cantatas and serenades

 * La ninfa saggia, "The Wise Nymph" (17??)
 * Gli amanti felici, "The Happy Lovers" (17??)

Poetry

 * Il colosso, a satire against Pavia girls which led to Goldoni being expelled from Collegio Ghislieri (1725)
 * Il quaresimale in epilogo (1725–1726)

Books

 * Nuovo teatro comico, "New Comic Theater", plays. Pitteri, Venice (1757)
 * Mémoires, "Memoirs". Paris (1787)
 * Goldoni's collected works. Zalta, Venice (1788–1795)

Selected translations of Goldoni's works

 * Il vero amico, "The True Friend" translated by Anna Cuffaro. Publisher: Sparkling Books.
 * Archifanfaro translated by W. H. Auden with an introduction by Michael Andre in Unmuzzled OX.

title: Carlo Goldoni


text:
Cumulative distribution function

In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable $$X$$, or just distribution function of $$X$$, evaluated at $$x$$, is the probability that $$X$$ will take a value less than or equal to $$x$$.

Every probability distribution supported on the real numbers, discrete or "mixed" as well as continuous, is uniquely identified by a right-continuous monotone increasing function (a càdlàg function) $$F \colon \mathbb R \rightarrow [0,1]$$ satisfying $$\lim_{x\rightarrow-\infty}F(x)=0$$ and $$\lim_{x\rightarrow\infty}F(x)=1$$.

In the case of a scalar continuous distribution, it gives the area under the probability density function from negative infinity to $$x$$. Cumulative distribution functions are also used to specify the distribution of multivariate random variables.

Definition
The cumulative distribution function of a real-valued random variable $$X$$ is the function given by

where the right-hand side represents the probability that the random variable $$X$$ takes on a value less than or equal to $$x$$.

The probability that $$X$$ lies in the semi-closed interval $$(a,b]$$, where $$a < b$$, is therefore

In the definition above, the "less than or equal to" sign, "≤", is a convention, not a universally used one (e.g. Hungarian literature uses "<"), but the distinction is important for discrete distributions. The proper use of tables of the binomial and Poisson distributions depends upon this convention. Moreover, important formulas like Paul Lévy's inversion formula for the characteristic function also rely on the "less than or equal" formulation.

If treating several random variables $$X, Y, \ldots$$ etc. the corresponding letters are used as subscripts while, if treating only one, the subscript is usually omitted. It is conventional to use a capital $$F$$ for a cumulative distribution function, in contrast to the lower-case $$f$$ used for probability density functions and probability mass functions. This applies when discussing general distributions: some specific distributions have their own conventional notation, for example the normal distribution uses $$\Phi$$ and $$\phi$$ instead of $$F$$ and $$f$$, respectively.

The probability density function of a continuous random variable can be determined from the cumulative distribution function by differentiating using the Fundamental Theorem of Calculus; i.e. given $$F(x)$$, $$f(x) = \frac{dF(x)}{dx}$$ as long as the derivative exists.

The CDF of a continuous random variable $$X$$ can be expressed as the integral of its probability density function $$f_X$$ as follows: $$F_X(x) = \int_{-\infty}^x f_X(t) \, dt.$$

In the case of a random variable $$X$$ which has distribution having a discrete component at a value $$b$$, $$\operatorname{P}(X=b) = F_X(b) - \lim_{x \to b^-} F_X(x).$$

If $$F_X$$ is continuous at $$b$$, this equals zero and there is no discrete component at $$b$$.

Properties

Every cumulative distribution function $$F_X$$ is non-decreasing and right-continuous,  which makes it a càdlàg function. Furthermore, $$\lim_{x \to -\infty} F_X(x) = 0, \quad \lim_{x \to +\infty} F_X(x) = 1.$$

Every function with these three properties is a CDF, i.e., for every such function, a random variable can be defined such that the function is the cumulative distribution function of that random variable.

If $$X$$ is a purely discrete random variable, then it attains values $$x_1,x_2,\ldots$$ with probability $$p_i = p(x_i)$$, and the CDF of $$X$$ will be discontinuous at the points $$x_i$$: $$F_X(x) = \operatorname{P}(X\leq x) = \sum_{x_i \leq x} \operatorname{P}(X = x_i) = \sum_{x_i \leq x} p(x_i).$$

If the CDF $$F_X$$ of a real valued random variable $$X$$ is continuous, then $$X$$ is a continuous random variable; if furthermore $$F_X$$ is absolutely continuous, then there exists a Lebesgue-integrable function $$f_X(x)$$ such that $$F_X(b)-F_X(a) = \operatorname{P}(a< X\leq b) = \int_a^b f_X(x)\,dx$$ for all real numbers $$a$$ and $$b$$. The function $$f_X$$ is equal to the derivative of $$F_X$$ almost everywhere, and it is called the probability density function of the distribution of $$X$$.

If $$X$$ has finite L1-norm, that is, the expectation of $$|X|$$ is finite, then the expectation is given by the Riemann–Stieltjes integral $$\mathbb E[X] = \int_{-\infty}^\infty t dF_X(t)$$and for any $$x \geq 0$$, $$\begin{align} x (1-F_X(x)) & \leq \int_x^{\infty} t dF_X(t) \\ x F_X(-x)   & \leq \int_{-\infty}^{-x} (-t) dF_X(t) \end{align}$$as shown in the diagram.

In particular, we have $$\lim_{x \to -\infty} x F(x) = 0, \quad \lim_{x \to +\infty} x (1-F(x)) = 0. $$

Examples
As an example, suppose $$X$$ is uniformly distributed on the unit interval $$[0,1]$$.

Then the CDF of $$X$$ is given by $$F_X(x) = \begin{cases} 0 &:\ x < 0\\ x &:\ 0 \le x \le 1\\ 1 &:\ x > 1 \end{cases}$$

Suppose instead that $$X$$ takes only the discrete values 0 and 1, with equal probability.

Then the CDF of $$X$$ is given by $$F_X(x) = \begin{cases} 0 &:\ x < 0\\ 1/2 &:\ 0 \le x < 1\\ 1 &:\ x \ge 1 \end{cases}$$

Suppose $$X$$ is exponential distributed. Then the CDF of $$X$$ is given by $$F_X(x;\lambda) = \begin{cases} 1-e^{-\lambda x} & x \ge 0, \\ 0 & x < 0. \end{cases}$$

Here λ > 0 is the parameter of the distribution, often called the rate parameter.

Suppose $$X$$ is normal distributed. Then the CDF of $$X$$ is given by $$F(x;\mu,\sigma) = \frac{1}{\sigma\sqrt{2\pi}} \int_{-\infty}^x \exp \left( -\frac{(t - \mu)^2}{2\sigma^2} \right)\, dt. $$

Here the parameter $$\mu$$ is the mean or expectation of the distribution; and $$\sigma$$ is its standard deviation.

A table of the CDF of the standard normal distribution is often used in statistical applications, where it is named the standard normal table, the unit normal table, or the Z table.

Suppose $$X$$ is binomial distributed. Then the CDF of $$X$$ is given by $$F(k;n,p) = \Pr(X\leq k) = \sum _{i=0}^{\lfloor k\rfloor }{n \choose i} p^{i} (1-p)^{n-i}$$

Here $$p$$ is the probability of success and the function denotes the discrete probability distribution of the number of successes in a sequence of $$n$$ independent experiments, and $$\lfloor k\rfloor$$ is the "floor" under $$k$$, i.e. the greatest integer less than or equal to $$k$$.

Complementary cumulative distribution function (tail distribution)
Sometimes, it is useful to study the opposite question and ask how often the random variable is above a particular level. This is called the ' (') or simply the ' or ', and is defined as $$\bar F_X(x) = \operatorname{P}(X > x) = 1 - F_X(x).$$

This has applications in statistical hypothesis testing, for example, because the one-sided p-value is the probability of observing a test statistic at least as extreme as the one observed. Thus, provided that the test statistic, T, has a continuous distribution, the one-sided p-value is simply given by the ccdf: for an observed value $$t$$ of the test statistic $$p= \operatorname{P}(T \ge t) = \operatorname{P}(T > t) = 1 - F_T(t).$$

In survival analysis, $$\bar F_X(x)$$ is called the survival function and denoted $$S(x)$$, while the term reliability function is common in engineering.

Proof: Assuming $$X$$ has a density function $$f_X$$, for any $$c > 0$$ $$ \operatorname{E}(X) = \int_0^\infty x f_X(x) \, dx \geq \int_0^c x f_X(x) \, dx + c\int_c^\infty f_X(x) \, dx $$ Then, on recognizing $$\bar F_X(c) = \int_c^\infty f_X(x) \, dx$$ and rearranging terms, $$ 0 \leq c\bar F_X(c) \leq \operatorname{E}(X) - \int_0^c x f_X(x) \, dx \to 0 \text{ as } c \to \infty $$ as claimed. If the random variable can only take non-negative integer values, this is equivalent to $$\operatorname{E}(X) = \sum_{n=0}^\infty \bar F_X(n).$$
 * Properties
 * For a non-negative continuous random variable having an expectation, Markov's inequality states that $$\bar F_X(x) \leq \frac{\operatorname{E}(X)}{x} .$$
 * As $$x \to \infty, \bar F_X(x) \to 0$$, and in fact $$\bar F_X(x) = o(1/x)$$ provided that $$\operatorname{E}(X)$$ is finite.
 * For a random variable having an expectation, $$\operatorname{E}(X) = \int_0^\infty \bar F_X(x) \, dx - \int_{-\infty}^0 F_X(x) \, dx$$ and for a non-negative random variable the second term is 0.

Folded cumulative distribution
While the plot of a cumulative distribution $$F$$ often has an S-like shape, an alternative illustration is the folded cumulative distribution or mountain plot, which folds the top half of the graph over, that is
 * $$F_\text{fold}(x)=F(x)1_{\{F(x)\leq 0.5\}}+(1-F(x))1_{\{F(x)>0.5\}}$$

where $$1_{\{A\}}$$ denotes the indicator function and the second summand is the survivor function, thus using two scales, one for the upslope and another for the downslope. This form of illustration emphasises the median, dispersion (specifically, the mean absolute deviation from the median ) and skewness of the distribution or of the empirical results.

Inverse distribution function (quantile function)
If the CDF F is strictly increasing and continuous then $$ F^{-1}( p ), p \in [0,1], $$ is the unique real number $$ x $$ such that $$ F(x) = p $$. This defines the inverse distribution function or quantile function.

Some distributions do not have a unique inverse (for example if $$f_X(x)=0$$ for all $$a<x<b$$, causing $$F_X$$ to be constant). In this case, one may use the generalized inverse distribution function, which is defined as

F^{-1}(p) = \inf \{x \in \mathbb{R}: F(x) \geq p \}, \quad \forall p \in [0,1]. $$
 * Example 1: The median is $$F^{-1}( 0.5 )$$.
 * Example 2: Put $$ \tau = F^{-1}( 0.95 ) $$. Then we call $$ \tau $$ the 95th percentile.

Some useful properties of the inverse cdf (which are also preserved in the definition of the generalized inverse distribution function) are:

 * 1) $$F^{-1}$$ is nondecreasing
 * 2) $$F^{-1}(F(x)) \leq x$$
 * 3) $$F(F^{-1}(p)) \geq p$$
 * 4) $$F^{-1}(p) \leq x$$ if and only if $$p \leq F(x)$$
 * 5) If $$Y$$ has a $$U[0, 1]$$ distribution then $$F^{-1}(Y)$$ is distributed as $$F$$. This is used in random number generation using the inverse transform sampling-method.
 * 6) If $$\{X_\alpha\}$$ is a collection of independent $$F$$-distributed random variables defined on the same sample space, then there exist random variables $$Y_\alpha$$ such that $$Y_\alpha$$ is distributed as $$U[0,1]$$ and $$F^{-1}(Y_\alpha) = X_\alpha$$ with probability 1 for all $$\alpha$$.

The inverse of the cdf can be used to translate results obtained for the uniform distribution to other distributions.

Empirical distribution function
The empirical distribution function is an estimate of the cumulative distribution function that generated the points in the sample. It converges with probability 1 to that underlying distribution. A number of results exist to quantify the rate of convergence of the empirical distribution function to the underlying cumulative distribution function.

Definition for two random variables
When dealing simultaneously with more than one random variable the joint cumulative distribution function can also be defined. For example, for a pair of random variables $$X,Y$$, the joint CDF $$F_{XY}$$ is given by

where the right-hand side represents the probability that the random variable $$X$$ takes on a value less than or equal to $$x$$ and that $$Y$$ takes on a value less than or equal to $$y$$.

Example of joint cumulative distribution function:

For two continuous variables X and Y: $$ \Pr(a < X < b \text{ and } c < Y < d) = \int_a^b \int_c^d f(x,y) \, dy \, dx;$$

For two discrete random variables, it is beneficial to generate a table of probabilities and address the cumulative probability for each potential range of X and Y, and here is the example:

given the joint probability mass function in tabular form, determine the joint cumulative distribution function. Solution: using the given table of probabilities for each potential range of X and Y, the joint cumulative distribution function may be constructed in tabular form:

Definition for more than two random variables
For $$N$$ random variables $$X_1,\ldots,X_N$$, the joint CDF $$F_{X_1,\ldots,X_N}$$ is given by

Interpreting the $$N$$ random variables as a random vector $$\mathbf{X} = (X_1, \ldots, X_N)^T$$ yields a shorter notation: $$F_{\mathbf{X}}(\mathbf{x}) = \operatorname{P}(X_1 \leq x_1,\ldots,X_N \leq x_N)$$

Properties
Every multivariate CDF is:
 * 1) Monotonically non-decreasing for each of its variables,
 * 2) Right-continuous in each of its variables,
 * 3) $$0\leq F_{X_1 \ldots X_n}(x_1,\ldots,x_n)\leq 1,$$
 * 4) $$\lim_{x_1,\ldots,x_n \rightarrow+\infty}F_{X_1 \ldots X_n}(x_1,\ldots,x_n)=1 \text{ and } \lim_{x_i\rightarrow-\infty}F_{X_1 \ldots X_n}(x_1,\ldots,x_n)=0, \text{for all } i.$$

Not every function satisfying the above four properties is a multivariate CDF, unlike in the single dimension case. For example, let $$F(x,y)=0$$ for $$x<0$$ or $$x+y<1$$ or $$y<0$$ and let $$F(x,y)=1$$ otherwise. It is easy to see that the above conditions are met, and yet $$F$$ is not a CDF since if it was, then $\operatorname{P}\left(\frac{1}{3} < X \leq 1, \frac{1}{3} < Y \leq 1\right)=-1$ as explained below.

The probability that a point belongs to a hyperrectangle is analogous to the 1-dimensional case: $$F_{X_1,X_2}(a, c) + F_{X_1,X_2}(b, d) - F_{X_1,X_2}(a, d) - F_{X_1,X_2}(b, c) = \operatorname{P}(a < X_1 \leq b, c < X_2 \leq d) = \int ...$$

Complex random variable
The generalization of the cumulative distribution function from real to complex random variables is not obvious because expressions of the form $$ P(Z \leq 1+2i) $$ make no sense. However expressions of the form $$ P(\Re{(Z)} \leq 1, \Im{(Z)} \leq 3) $$ make sense. Therefore, we define the cumulative distribution of a complex random variables via the joint distribution of their real and imaginary parts: $$ F_Z(z) = F_{\Re{(Z)},\Im{(Z)}}(\Re{(z)},\Im{(z)}) = P(\Re{(Z)} \leq \Re{(z)}, \Im{(Z)} \leq \Im{(z)}). $$

Complex random vector
Generalization of $$ yields $$F_{\mathbf{Z}}(\mathbf{z}) = F_{\Re{(Z_1)},\Im{(Z_1)}, \ldots, \Re{(Z_n)},\Im{(Z_n)}}(\Re{(z_1)}, \Im{(z_1)},\ldots,\Re{(z_n)}, \Im{(z_n)}) = \operatorname{P}(\Re{(Z_1)} \leq \Re{(z_1)},\Im{(Z_1)} \leq \Im{(z_1)},\ldots,\Re{(Z_n)} \leq \Re{(z_n)},\Im{(Z_n)} \leq \Im{(z_n)})$$ as definition for the CDS of a complex random vector $$\mathbf{Z} = (Z_1,\ldots,Z_N)^T$$.

Use in statistical analysis
The concept of the cumulative distribution function makes an explicit appearance in statistical analysis in two (similar) ways. Cumulative frequency analysis is the analysis of the frequency of occurrence of values of a phenomenon less than a reference value. The empirical distribution function is a formal direct estimate of the cumulative distribution function for which simple statistical properties can be derived and which can form the basis of various statistical hypothesis tests. Such tests can assess whether there is evidence against a sample of data having arisen from a given distribution, or evidence against two samples of data having arisen from the same (unknown) population distribution.

Kolmogorov–Smirnov and Kuiper's tests
The Kolmogorov–Smirnov test is based on cumulative distribution functions and can be used to test to see whether two empirical distributions are different or whether an empirical distribution is different from an ideal distribution. The closely related Kuiper's test is useful if the domain of the distribution is cyclic as in day of the week. For instance Kuiper's test might be used to see if the number of tornadoes varies during the year or if sales of a product vary by day of the week or day of the month.

title: Cumulative distribution function


text:
Central tendency

In statistics, a central tendency (or measure of central tendency) is a central or typical value for a probability distribution.

Colloquially, measures of central tendency are often called averages. The term central tendency dates from the late 1920s.

The most common measures of central tendency are the arithmetic mean, the median, and the mode. A middle tendency can be calculated for either a finite set of values or for a theoretical distribution, such as the normal distribution. Occasionally authors use central tendency to denote "the tendency of quantitative data to cluster around some central value."

The central tendency of a distribution is typically contrasted with its dispersion or variability; dispersion and central tendency are the often characterized properties of distributions. Analysis may judge whether data has a strong or a weak central tendency based on its dispersion.

Measures
The following may be applied to one-dimensional data. Depending on the circumstances, it may be appropriate to transform the data before calculating a central tendency. Examples are squaring the values or taking logarithms. Whether a transformation is appropriate and what it should be, depend heavily on the data being analyzed.

 * Arithmetic mean or simply, mean: the sum of all measurements divided by the number of observations in the data set.
 * Median: the middle value that separates the higher half from the lower half of the data set. The median and the mode are the only measures of central tendency that can be used for ordinal data, in which values are ranked relative to each other but are not measured absolutely.
 * Mode: the most frequent value in the data set. This is the only central tendency measure that can be used with nominal data, which have purely qualitative category assignments.
 * Generalized mean: A generalization of the Pythagorean means, specified by an exponent.
 * Geometric mean: the nth root of the product of the data values, where there are n of these. This measure is valid only for data that are measured on a strictly positive scale.
 * Harmonic mean: the reciprocal of the arithmetic mean of the reciprocals of the data values. This measure is valid only for data that are measured either on a strictly positive or a strictly negative scale.
 * Weighted arithmetic mean: an arithmetic mean that incorporates weighting to certain data elements.
 * Truncated mean or trimmed mean: the arithmetic mean of data values after a certain number or proportion of the highest and lowest data values have been discarded.
 * Interquartile mean: a truncated mean based on data within the interquartile range.
 * Midrange: the arithmetic mean of the maximum and minimum values of a data set.
 * Midhinge: the arithmetic mean of the first and third quartiles.
 * Quasi-arithmetic mean: A generalization of the generalized mean, specified by a continuous injective function.
 * Trimean: the weighted arithmetic mean of the median and two quartiles.
 * Winsorized mean: an arithmetic mean in which extreme values are replaced by values closer to the median.

Any of the above may be applied to each dimension of multi-dimensional data, but the results may not be invariant to rotations of the multi-dimensional space.

 * Geometric median: the point minimizing the sum of distances to a set of sample points. This is the same as the median when applied to one-dimensional data, but it is not the same as taking the median of each dimension independently. It is not invariant to different rescaling of the different dimensions.
 * Quadratic mean (often known as the root mean square) : useful in engineering, but not often used in statistics. This is because it is not a good indicator of the center of the distribution when the distribution includes negative values.
 * Simplicial depth: the probability that a randomly chosen simplex with vertices from the given distribution will contain the given center
 * Tukey median: a point with the property that every halfspace containing it also contains many sample points

Solutions to variational problems
Several measures of central tendency can be characterized as solving a variational problem, in the sense of the calculus of variations, namely minimizing variation from the center. That is, given a measure of statistical dispersion, one asks for a measure of central tendency that minimizes variation: such that variation from the center is minimal among all choices of center. In a quip, "dispersion precedes location". These measures are initially defined in one dimension, but can be generalized to multiple dimensions. This center may or may not be unique. In the sense of $L^{p}$ spaces, the correspondence is:

The associated functions are called $L^{p}$-norms: respectively 0-"norm", 1-norm, 2-norm, and ∞-norm. The function corresponding to the L0 space is not a norm, and is thus often referred to in quotes: 0-"norm".

In equations, for a given (finite) data set $L^{0}$, thought of as a vector $L^{1}$, the dispersion about a point $L^{2}$ is the "distance" from $L^{∞}$ to the constant vector $p$ in the p-norm (normalized by the number of points n):

 * $$f_p(c) = \left\| \mathbf{x} - \mathbf{c} \right\|_p := \bigg( \frac{1}{n} \sum_{i=1}^n \left| x_i - c\right| ^p \bigg) ^{1/p}$$

For $X$ and $\mathbf{x} = (x1,…,xn)$ these functions are defined by taking limits, respectively as $\mathbf{c}$ and $\mathbf{x}$. For $\mathbf{c} = (c,…,c)$ the limiting values are $p = 0$ and $p = ∞$ or $p → 0$, so the difference becomes simply equality, so the 0-norm counts the number of unequal points. For $p → ∞$ the largest number dominates, and thus the ∞-norm is the maximum difference.

Uniqueness
The mean (L2 center) and midrange (L∞ center) are unique (when they exist), while the median (L1 center) and mode (L0 center) are not in general unique. This can be understood in terms of convexity of the associated functions (coercive functions).

The 2-norm and ∞-norm are strictly convex, and thus (by convex optimization) the minimizer is unique (if it exists), and exists for bounded distributions. Thus standard deviation about the mean is lower than standard deviation about any other point, and the maximum deviation about the midrange is lower than the maximum deviation about any other point.

The 1-norm is not strictly convex, whereas strict convexity is needed to ensure uniqueness of the minimizer. Correspondingly, the median (in this sense of minimizing) is not in general unique, and in fact any point between the two central points of a discrete distribution minimizes average absolute deviation.

The 0-"norm" is not convex (hence not a norm). Correspondingly, the mode is not unique – for example, in a uniform distribution any point is the mode.

Clustering
Instead of a single central point, one can ask for multiple points such that the variation from these points is minimized. This leads to cluster analysis, where each point in the data set is clustered with the nearest "center". Most commonly, using the 2-norm generalizes the mean to k-means clustering, while using the 1-norm generalizes the (geometric) median to k-medians clustering. Using the 0-norm simply generalizes the mode (most common value) to using the k most common values as centers.

Unlike the single-center statistics, this multi-center clustering cannot in general be computed in a closed-form expression, and instead must be computed or approximated by an iterative method; one general approach is expectation–maximization algorithms.

Information geometry
The notion of a "center" as minimizing variation can be generalized in information geometry as a distribution that minimizes divergence (a generalized distance) from a data set. The most common case is maximum likelihood estimation, where the maximum likelihood estimate (MLE) maximizes likelihood (minimizes expected surprisal), which can be interpreted geometrically by using entropy to measure variation: the MLE minimizes cross-entropy (equivalently, relative entropy, Kullback–Leibler divergence).

A simple example of this is for the center of nominal data: instead of using the mode (the only single-valued "center"), one often uses the empirical measure (the frequency distribution divided by the sample size) as a "center". For example, given binary data, say heads or tails, if a data set consists of 2 heads and 1 tails, then the mode is "heads", but the empirical measure is 2/3 heads, 1/3 tails, which minimizes the cross-entropy (total surprisal) from the data set. This perspective is also used in regression analysis, where least squares finds the solution that minimizes the distances from it, and analogously in logistic regression, a maximum likelihood estimate minimizes the surprisal (information distance).

Relationships between the mean, median and mode
For unimodal distributions the following bounds are known and are sharp:

 * $$ \frac{| \theta - \mu |}{ \sigma } \le \sqrt{ 3 } ,$$

 * $$ \frac{| \nu - \mu |}{ \sigma } \le \sqrt{ 0.6 } ,$$

 * $$ \frac{| \theta - \nu |}{ \sigma } \le \sqrt{ 3 } ,$$

where μ is the mean, ν is the median, θ is the mode, and σ is the standard deviation.

For every distribution,

 * $$ \frac{| \nu - \mu |}{ \sigma } \le 1.$$

title: Central tendency