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Phil Knight

Philip Hampson Knight (born February 24, 1938) is an American billionaire business magnate who is the co-founder and chairman emeritus of Nike, Inc., a global sports equipment and apparel company. He was previously its chairman and CEO. As of December 2023, Forbes estimated his net worth at $45.0 billion. He is also the owner of the stop motion film production company Laika. Knight is a graduate of the University of Oregon and the Stanford Graduate School of Business. He was part of the track and field club under coach Bill Bowerman at the University of Oregon with whom he would later co-found Nike.

Knight has donated hundreds of millions of dollars to each of his alma maters, as well as Oregon Health & Science University. He has donated over $2 billion to these three institutions.

Early life and education
Phil Hampson Knight was born on February 24, 1938, in Portland, Oregon, to Bill Knight, a lawyer turned newspaper publisher, and his wife, Lota Cloy (née Hatfield) Knight. He grew up in the Portland neighborhood of Eastmoreland, and attended Cleveland High School. According to one source, "When his father refused to give him a summer job at his newspaper [the now defunct Oregon Journal], believing that his son should find work on his own," Knight "went to the rival Oregonian, where he worked the morning shift tabulating sports scores and every morning ran home the full seven miles."

Knight continued his education at the University of Oregon in Eugene, where he ran for the famed Oregon track and field program, was a sports reporter for the Oregon Daily Emerald and was a member of Phi Gamma Delta fraternity. Knight earned a business degree (B.B.A.) in 1959 in just three years. That same year, Knight also received his Army Reserve Commission and was a "Distinguished Military Graduate".

As a middle-distance runner at Oregon, his personal best was 1 mi in 4 minutes, 13 seconds, and he won varsity letters for his track performances in 1957, 1958, and 1959. In 1977, together with Bowerman and Geoff Hollister, Knight founded an American running team called Athletics West.

Early career
Before Blue Ribbon Sports—later Nike—flourished, Knight worked as a Certified Public Accountant, first with Coopers & Lybrand, and then Price Waterhouse. Knight then became an accounting professor at Portland State University.

Nike Inc.
Immediately after graduating from the University of Oregon, Knight enlisted in the Army and served one year on active duty and seven years in the Army Reserve. He next enrolled at Stanford Graduate School of Business, where, for his small business class, Knight produced a paper, "Can Japanese Sports Shoes Do to German Sports Shoes What Japanese Cameras Did to German Cameras?", that essentially foretold his eventual foray into selling running shoes. His ambition was to import high-quality and low-cost running shoes from Japan into the American market. He graduated with a master's degree in business administration from Stanford in 1962.

Knight set out on a trip around the world after graduation, during which he made a stop in Kobe, Japan, in November 1962. It was there that he discovered Tiger brand running shoes, manufactured in Kobe by the Onitsuka Co., now known as Asics. Impressed by the quality and low cost of the shoes, Knight called Mr. Onitsuka, who agreed to meet with him. By the end of the meeting, Knight had secured Tiger distribution rights for the western United States.

The first Tiger samples would take more than a year to be shipped to Knight; during that time he found a job as an accountant in Portland. When Knight finally received the shoe samples, he mailed two pairs to Bowerman at the University of Oregon, hoping to gain both a sale and an influential endorsement. To Knight's surprise, Bowerman not only ordered the Tiger shoes, but also offered to become a partner with Knight and provide product design ideas. The two men agreed to a partnership by handshake on January 25, 1964, the birth date of Blue Ribbon Sports, the company that would later become Nike.

Knight's first sales, were made out of a now storied green Plymouth Valiant automobile at track meets across the Pacific Northwest. By 1969, these early sales allowed Knight to leave his accountant job and work full-time for Blue Ribbon Sports.

Jeff Johnson, Nike's first employee, suggested calling the firm "Nike," named after the Greek winged goddess of victory, and Blue Ribbon Sports was subsequently renamed Nike in 1971.

Nike's "swoosh" logo, now considered one of the most valuable logos in the world, was commissioned for $35 from graphic design student Carolyn Davidson in 1971. According to Nike's website, Knight said at the time: "I don't love it, but it will grow on me." In September 1983, Davidson was given an undisclosed amount of Nike stock for her contribution to the company's brand. On the Oprah television program in April 2011, Knight claimed he gave Davidson "a few hundred shares" when the company went public.

At Nike, Knight developed personal relationships with some of the world's most recognizable athletes, including Michael Jordan and Tiger Woods.

Vinton Studios becomes Laika
Following mainstream success in the late 1990s, Will Vinton Studios animation company sought external investors due to rapid growth. Knight assumed a 15 percent stake in the company in 1998, and his son Travis—who had graduated from Portland State following an unsuccessful attempt at a rap music career—went to work at the studio as an animator.

Citing mismanagement, Knight eventually purchased Will Vinton Studios and assumed control of the company's board with the cooperation of Nike executives. In late 2003, Knight appointed his son to the board and, after Vinton had stepped down—prior to leaving the company with a severance package—Knight rebranded the company Laika. He then invested $180 million into Laika, and the studio released its first feature film, Coraline, in stop motion, in 2009. Coraline was a financial success and Travis Knight was then promoted into the roles of Laika CEO and president.

Death of Matthew Knight
In May 2004, two years after Knight bought Vinton, his son Matthew, aged 34 years, traveled to El Salvador to film a fund-raising video for Christian Children of the World, a Portland nonprofit organization. However, while scuba diving with his colleagues Vincenzo Iannuzzelli and Robert McDonell in Lake Ilopango, near San Salvador, he died from a heart attack 150 ft underwater due to an undetected congenital heart defect. Knight and Travis traveled to El Salvador to return Matthew's body to the US. Laika Studio's 2005 short film Moongirl was dedicated to Matthew's memory.

Knight resigned as Nike CEO on November 18, 2004, several months after Matthew's funeral but retained the position of chairman of the board. Knight's replacement was William Perez, former CEO of S.C. Johnson & Son, Inc., who was eventually replaced by Mark Parker in 2006.

In 2011, the Matthew Knight Arena at the University of Oregon was named in his honor.

Post-Nike CEO role
During the 2009–2010 period, Knight was the largest single contributor to the campaign to defeat Oregon Ballot Measures 66 and 67, which, once passed, increased income tax on some corporations and high-income individuals.

In June 2015, Knight and Nike announced that he would step down as the company's chairman, with president and CEO Mark Parker to succeed him. Knight's retirement from the Nike board took effect at the end of June 2016. In September 2017, Knight decided to come out of retirement to put black back in the UNC jerseys for the Phil Knight Classic in Portland, Oregon.

Memoir
Knight's memoir, Shoe Dog, was released on April 26, 2016, by Simon & Schuster, was rated fifth on The New York Times Best Seller list for business books in July 2018, and details the building of the Nike brand, from importing Japanese shoes to being part of a federal investigation.

Philanthropy
As of July 2021, Knight has a net worth of $60.8 billion. In 1990, Knight founded the Philip H. Knight Charitable Foundation Trust. As of 2016, according to Portland Business Journal, "Knight is the most generous philanthropist in Oregon history. His lifetime gifts now approach $2 billion." In 2023, Knight was America's second largest donor with $1.2 billion in giving.

Stanford University
In 2006, Knight donated US$105 million to the Stanford Graduate School of Business, which, at the time, was the largest ever individual donation to a U.S. business school. The campus was named "The Knight Management Center," in honor of Knight's philanthropic service to the school.

In 2016, it was announced that Knight contributed $400 million to start the Knight-Hennessy Scholars graduate-level education program inspired by the Rhodes Scholarship. Graduates are charged to tackle global challenges, such as climate change and poverty. The first class of 51 scholars from 21 countries was scheduled to arrive at Stanford in the fall of 2018.

In May 2022, it was announced that Phil and Penny Knight gifted Stanford $75 million to establish the Phil and Penny Knight Initiative for Brain Resilience. The initiative will be housed at Stanford's Wu Tsai Neurosciences Institute and is set to study the cognitive decline and degenerative brain diseases such as Parkinson's and Alzheimer's.

University of Oregon
As of 2023, Knight has donated over one billion dollars to the University of Oregon. It is believed that Knight made his first major contribution in the late 1980s. By 2000 Knight had already contributed over $50 million to UO. He was UO's 2018-2019 honorary degree recipient.

UO Academics
Major gifts include funds supporting the renovation of the Knight Library and construction of the Knight Law Center. Contrary to press reports, which claim that Knight financed the whole library renovation project, Knight only financed a portion of the library's renovation. Knight also established endowed chairs across the campus. In the fall of 2016, it was announced that Knight will donate $500 million to UO for a new three-building laboratory and research science complex. This donation was part of a series of large higher-education gifts.

UO Athletic Department (Oregon Ducks)
Knight contributed towards the Moshofsky Center, which opened in 1998.

In August 2007, Knight announced that he and his wife would be donating US$100 million to found the UO Athletics Legacy Fund to help support all athletic programs at the university. In response, athletic director Pat Kilkenny said: "This extraordinary gift will set Oregon athletics on a course toward certain self sufficiency and create the flexibility and financial capacity for the university to move forward with the new athletic arena." At the time, the donation was the largest philanthropic gift in the history of the university.

The 2010 construction of the UO basketball team's Matthew Knight Arena was the result of a partnership between Knight and former Oregon athletic director Pat Kilkenny. Although Knight didn't pay for the project directly, he established a $100 million "Athletic Legacy Fund." The fund supports the athletic department. Named after Knight's deceased son, the venue replaced the McArthur Court building and cost over US$200 million to build. The facility was built using bonds backed by the State of Oregon.

Knight was responsible for financing the UO's US$68 million 145,000 square-foot gridiron football facility that was officially opened in late July 2013. Knight's personal locker in the team's locker room displays the title "Uncle Phil", and other features include a gym with Brazilian hardwood floors, Apple iPhone chargers in each of the player's lockers, various auditoriums and meeting rooms, a games room for the players that includes flat-screen televisions and foosball machines, and a cafeteria.

In November 2015, it was announced that Knight and his wife would be donating $19.2 million towards a new sports complex project at the University of Oregon. The plans for the 29,000 square foot complex was announced in September. Construction started in January 2016 and ended in September 2016. The sports complex was named the Marcus Mariota Sports Performance Center and includes motion capture systems, neurocognitive assessment tools, 40-yard dash track, and steam machines made by Nike to help athletes break into their footwear more quickly.

In 2021, Knight helped to fund the renovation of Hayward Field, a track and field stadium at the university. The project was estimated to cost $270 million, although Knight's total contribution remained private.

Controversy
Knight's contributions to the athletic department at UO have also led to controversy. In April 2000, student leaders began organizing an anti-sweatshop and fair labor practices campaign, and called for Dave Frohnmayer, president of the school, to support the Workers Rights Consortium (WRC). On April 4, 2000, students began a sit-in at Johnson Hall, the UO's administrative center. In early April, an open meeting of students further demanded that the organization Fair Labor Association (FLA) would receive no consideration from the university, as it was perceived as a group founded, funded and backed by Nike and other corporations, and had also been criticized by worker rights advocates as an exercise in dishonest public relations.

University President Dave Frohnmayer subsequently signed a one-year contract with the WRC; Knight then withdrew a US$30 million commitment toward the Autzen Stadium expansion project and offered no further donations to the university. In a public statement, Knight criticized the WRC for having unrealistic provisions and called it misguided, while praising the FLA for being "balanced" in its approach. In the face of ongoing conflict with students, Frohnmayer sided with Knight's assertion that the WRC was providing unbalanced representation, and in October 2000 the Eugene Weekly reported Frohnmayer stating that: ... he would refuse to pay dues to the WRC based on a legal opinion from UO General Counsel Melinda Grier arguing that to do so would be illegal and open the university to liability. Grier claimed the WRC had not yet incorporated, had not yet filed as a non-profit, and served no public purpose justifying a dues payment.

On February 16, 2001, the Oregon University System enacted a mandate that all institutions within the system choose business partners from a politically neutral standpoint, barring all universities in Oregon from joining either the WRC or the FLA. Following the dissolved relationship between the university and the WRC, Knight reinstated the donation and increased the amount to over US$50 million.

Also controversial was Knight's success in lobbying for former insurance executive Pat Kilkenny to be named as athletic director at the university. Kilkenny had neither a college degree nor any prior experience in athletics administration. He attended but did not graduate from UO, as he left the school with several credit hours still owing. Prior to his appointment at UO, Kilkenny had been the chairman and chief executive officer of the San Diego-based Arrowhead General Insurance Agency, and grew the business into a nationwide organization, with written premiums of nearly US$1 billion when he sold the company in 2006.

Other projects
In October 2008, Knight and his wife pledged US$100 million to the OHSU Cancer Institute, the largest gift in the history of Oregon Health & Science University. In recognition, the university renamed the organization the "OHSU Knight Cancer Institute."

In October 2010, Knight donated several million dollars to the Catlin Gabel School to establish a scholarship for incoming freshmen students.

On May 18, 2012, Knight contributed US$65,000 to a higher education Political Action Committee (PAC) formed by Columbia Sportswear CEO Tim Boyle. According to Boyle, the PAC will help facilitate an increase in the autonomy of schools in the Oregon University System.

On September 27, 2013, Knight announced to the audience at the OHSU Knight Cancer Institute's biennial gala, when he announced his intention to donate US$500 million for research if OHSU could match it over the subsequent two years. On June 25, 2015, OHSU met that $500 million goal, and Knight announced his upcoming $500 million donation, to bring the total to $1 billion raised.

Knight and wife Penny also donated to the Marylhurst Knights Opportunity Scholarship Program at Marylhurst University, a private Roman Catholic university in Marylhurst, Oregon; as a result, the university named a lawn on their campus "Knight's Green" in the family's honor.

In December 2016, Knight disclosed that he had donated $112 million in Nike stock to charity.

Accolades
In 2000, Knight was inducted into the Oregon Sports Hall of Fame for his Special Contribution to Sports in Oregon. At the time of his induction, he had contributed approximately US$230 million to UO, the majority of which was for athletics.

On February 24, 2012, Knight was announced as a 2012 inductee of the Naismith Memorial Basketball Hall of Fame, as a contributor. The Hall recognized him as the driving force behind Nike's huge financial support of U.S. basketball and its players. Knight was formally inducted on September 7, 2012.

In 1989, Knight received the Golden Plate Award of the American Academy of Achievement. For his "contributions to business, corporate and philanthropic leadership", Knight was elected to the 2015 American Academy of Arts and Sciences membership class.

In 2020, the university polled alumni and fans on social media, asking them which four UO alumni they would place on a notional Mount Rushmore for the university. Knight was one of the four final choices, along with Ducks track legend Steve Prefontaine; current NFL player Marcus Mariota, the 2014 Heisman Trophy winner; and Sabrina Ionescu, who had just completed an epic college basketball career for the Ducks.

Personal life
Knight met his wife, Penelope "Penny" Parks, while he was working at Portland State University and they were married on September 13, 1968. They own a home in La Quinta, California.

Knight's son, Matthew, died in a scuba diving accident in El Salvador in 2004. Another son of Knight's, Travis Knight, runs the Laika animation studio. Phil Knight serves as chairman.

Knight lists Nike's headquarters in Beaverton, Oregon, as his home address on government records. He resides in a rural area outside Hillsboro, Oregon and owns a personal flight hangar along with two private jets at the Hillsboro Airport.

Ben Affleck and Matt Damon produced a film called Air about Nike's signing of Michael Jordan and ultimately the Air Jordan brand. Affleck plays Knight in the film.

Politics
Knight is a registered Republican and a frequent contributor to Republican political candidates in Oregon. In 2010, he donated $400,000 to Chris Dudley, the then Republican nominee for Governor of Oregon.

Knight occasionally supports Democrats, however; in 1986 he financed Neil Goldschmidt's successful campaign, and in 2014, he donated $250,000 to the re-election campaign of Democrat John Kitzhaber.

He donated $3.5 million to Republican Knute Buehler during the 2018 Oregon gubernatorial election. Regarding this donation, Knight criticized the unfunded Oregon Public Employees Retirement System and the general fiscal state of Oregon, implying Democratic governor Kate Brown was at fault.

Knight was once again a major donor during the 2022 Oregon gubernatorial election. He gave $3.75 million to independent candidate Betsy Johnson until September 2022. After Johnson's support declined in the polls, he switched to backing Republican Christine Drazan, writing her a $1 million check on October 6 of that year. He made a second donation to the Drazan campaign later in October. Also in 2022, Knight donated $2 million to a political action committee seeking to elect Republican state legislators in Oregon. In April 2023, Knight contributed $2 million more to the same conservative PAC.

While Knight is generally reluctant to give interviews, he spoke with the New York Times in October 2022 regarding his support for Republican candidates. He argued Oregon's government had drifted too far to the left and described himself as "more conservative than Nike," responding to criticism of him backing an anti-abortion candidate which apparently contrasted with the company's image of supporting progressive social justice causes. Nike itself donated $75,000 to Drazan and Johnson's Democratic opponent, Tina Kotek, the eventual winner of the election.

title: Phil Knight


text:
Gigabit interface converter

A gigabit interface converter (GBIC) is a standard for transceivers, first defined in 1995 and commonly used with Gigabit Ethernet and Fibre Channel for some time. By offering a standard, hot swappable electrical interface, a single gigabit port can support a wide range of physical media, from copper to long-wave single-mode optical fiber, at lengths of hundreds of kilometers.

A smaller variation of the GBIC called the small form-factor pluggable transceiver (SFP), also known as mini-GBIC, has the same functionality but in a smaller form factor. Announced in 2001, it largely made the GBIC obsolete.

Appeal
The appeal of the GBIC standard (and hot-swappable transceivers in general) in networking equipment, as opposed to fixed physical interface configurations, is its flexibility. Where multiple different optical technologies are in use, an administrator can purchase GBICs as needed, not in advance, and they can be the specific type needed for each link. This lowers the cost of the base system and gives the administrator far more flexibility. On the other hand, if a switch will mostly have one port type (especially if that port type is copper) purchasing a switch with that port type built in will be cheaper and take up less space per port.

Standards
The GBIC standard is non-proprietary and is defined by the Small Form Factor Committee in document number 8053i. The first publication of the proposal was in November 1995. A few corrections and additions were made through September 2000. Robert Snively of Brocade Communications was technical editor. Original contributors were AMP Incorporated, Compaq Computers, Sun Microsystems, and Vixel Corporation.

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

title: Gigabit interface converter


text:
Talk:Second Industrial Revolution

Comments
I believe Wik is correct in this instance about the need for commas. But will the two of you just knock it off? I think both of you can find better use for your energies than edit wars about placement of a comma. Sheesh. -- Infrogmation 02:32, 8 Nov 2003 (UTC)

I see no evidence to indicate that I am wrong. The article on comma appears to include an example, of proper comma usage, which is akin to my usage here. I cannot "knock it off" because Wik follows me from article to article; trying to mess with my punctuation. This issue needs to be addressed. Lirath Q. Pynnor

 * Wik is definitely correct. What's more disturbing, though, is Lir reverting against the grammatical opinion expressed by every other user who has edited the document and commented on this page.  Daniel Quinlan 02:36, Nov 8, 2003 (UTC)

Just because a horde of you say that Im wrong, doesn't mean I am. If you are so confident, why are you unable to explain why Im wrong? I am using the comma to mark off the sentence's seperate elements; what's the problem? Lirath Q. Pynnor

France and the First Industrial Revolution
Ti belkieve that anybody who uses wikipedia should just do the work themselves becuase anybody who relies on the internet that much isnt a real student.........o see that France was an 'industrial power' is simply erroneus. France's economy was still predominately agricultural unlike Britain it did not see mass migration on anywhere near the same scale and its industry relied on small scale Workshop factories also unlike Britain. So the sentence 'Since this period includes the rise of industrial powers other than France and Britain, such as Germany and the USA, it may be used by writers who want to stress the contribution of these countries or relativize the position of the UK.' is only correct in its latter half... France cannot be considered an industrial power at this point... it may have been a world power but not an industrialized nation. —Preceding unsigned comment added by 129.11.198.174 (talk) 10:42, 15 January 2009 (UTC)

comma usage
Well, could you prove that ! I can't put an exclamation point where I just did? It's hard for me to describe as a non-grammarian, but commas definitely don't go where you're putting them. I finally found a formal description in the "comma" section of The Elements of Style:

 * Restrictive clauses, by contrast, are not parenthetic and are not set off by commas. Thus,
 * People who live in glass houses shouldn't throw stones.

Note that is basically what I told you on IRC some minutes ago:

<quinlan> commas are used for parenthetical comments which can be removed, but not just any part of a sentence that may be removed

Daniel Quinlan 02:43, Nov 8, 2003 (UTC)

It should be: "Those who live in glass houses, shouldn't throw stones." In any case, I don't see how it applies here. Lirath Q. Pynnor

I believe I am supported by: http://www.stark.kent.edu/writing/commas.htm
 * The writing-mobile, which has a 600 horse-power engine, is the fastest machine on the road today. Lirath Q. Pynnor
 * Germany spent more money, which was spent on science, than England. Lirath Q. Pynnor

 * No, those are a different case. Specifically, they are non-restrictive clauses and therefore require the offsetting commas.  Daniel Quinlan 03:04, Nov 8, 2003 (UTC)

How is my sentence restrictive? Lirath Q. Pynnor

 * Your sentence is "restrictive" because it defines or limits rather than merely adding something. Let me add two quotes from The Elements of Style:

 * People sitting in the rear couldn't hear. (restrictive)
 * Uncle Bert, being slightly dead, moved forward. (non-restrictive)

 * Daniel Quinlan 03:11, Nov 8, 2003 (UTC)

Not the same at all! Lirath Q. Pynnor

I protected the page because User:Lir started blanking it. I did not protect the page over the comma disagreement since I was involved in trying to convince Lir that his use of commas was ungrammatical, but blanking pages over a grammar dispute is unacceptable, so I protected the most recent unblanked version. Daniel Quinlan 03:41, Nov 8, 2003 (UTC)

Lir says he won't blank it again so I unprotected it. Don't forget to list pages on Protected page if you protect or unprotect them. Angela 04:46, Nov 8, 2003 (UTC)

 * Okay. (There isn't an automatic listing?  How annoying.  ;-)  Daniel Quinlan 05:09, Nov 8, 2003 (UTC)

Ok, agreeing that two commas are unnecessary; how about,

 * In science, the Germans invested more than the British.
 * It is necessary to mention science first because this section is already clearly discussing Germany; so, science should be singled out as the defining part of this sentence.
 * Lirath Q. Pynnor

Pardon me for butting in, but I believe "Those who live in glass houses, shouldn't throw stones" is incorrect because "shouldn't throw stones" is not a clause of any sort, and it is lacking a subject if you put the comma in. "Those who live in glass houses" is similarily missing a verb (as it is not a clause on its own either). The comma strikes me as a hypercorrection. Adam Bishop 05:03, 8 Nov 2003 (UTC)

That is perhaps technically an okay sentence (although the sentences before and after your proposal are not grammatical), but the current version of the page is correct, less awkward, clearer, and more informative. Daniel Quinlan 05:09, Nov 8, 2003 (UTC)

Intro
The introductory paragraph should say what the Second Industrial Revolution actually was. Ben Finn 15:30, 29 December 2005 (UTC)

Also needed are the reasons for the second industrial revolution and a comparison with the first industrial revolution.

other industrialising nations
Shouldn't Belgium be mentioned, as should France and the rest of Europe. Belgium was the first european country to indsustrialize after the UK. 12.220.94.199 22:05, 27 April 2006 (UTC)

Not just Belgium should be mentioned, nor the other European nations, but the United States should also be mentioned, especially that her industrial might just before the first world war was comparable to the the European nations.

I would considere that one as a third industrial revolution.

 * In the English speaking countries (USA, UK and Canada), it is wrongly advocated that the first industrial revolution is related to Watt and his machine and its impact afterwards. Not only the UK entered the IR but all the Breitish Empire since a city like Montreal was heavily industrialized by 1850 and much more than most US ones.

 * At the begining of the XVIII century, the first steam machine has been made by Papin in France and was used on the Fardier of Cugnot (the first automobile in 1770). Papin's machine inspired Newcomen with his single stroke steam machine. Papin's design was used in mines of the Lorraine region during that period. In the XVIII century, the first real industrial revolution truly happened in Lyon with the the advent of the textile industry using successive inventions that led to the Jacquard's loom. I have recently discovered that there existed a textile industry in Lancaster during that period.

 * The US entered into the IR very late: as a matter of fact at the end of the XIX century just like Germany.

 * I even have reasons to beleive that the first industrial revolution happened in Italy but I am still gathering evidence and I am discussing the matter with University professors over there. —The preceding unsigned comment was added by Special:Contributions/ (talk)

What is "Torre Moore" in the germany section suppose to mean? Never heart of it. No entry for that term in wiki and only 1 entry in google as something in a city in Illinois... —Preceding unsigned comment added by 82.83.41.54 (talk) 17:26, 6 April 2008 (UTC)

Was there really a "Second Industrial Revolution"?
This is one of those "categories of history" pages. What is really distinctive about a "second" industrial revolution? Does it make the original one less revolutionary? Have a significant number of major historians really labeled some period as the "second" one and do they agree on the dates at which it occurred? I would say not.

The dates 1871 and 1914 are very arbitrary, though they are of obvious political importance. That doesn't seem to be mentioned here though. It is clear that Industrial Revolution is a very major topic, and that many contemporaries living during the time or shortly thereafter used the term, or very similar terms in other languages, to desribe how the world was changing. I don't see any such agreement on a Second Industrial Revolution.

I don't suggest removing this page, but I believe it should be deprecated from "an era of history" to "a conceptual way of organizing and presenting the history of an era." I have read two of the four books that are "references" for this page (Hobsbawm and Landes), and I cannot recall that either of those authors used the term, Second Industrial Revolution. Who has read the other two. Does either of them use the term?

This page looks too much like a chapter from a high school textbook.

Would whoever champions the idea of this as an important era in history please tell us which historians, journalists, or well-known writers have used this term, or terms like it, to describe this era. --Metzenberg 03:31, 23 July 2006 (UTC)

 * At first, industry was mainly powered by falling water which limited the number and extent of industrial sites. Until electric generators were developed, power could only be distributed by belts over a distance of perhaps 100 meters so transportation networks could not be developed from it.

 * Development of steam engines (initially to pump water out of coal mines) allowed industry to be located away from the finite number of sites with potential water power. It also led to the development of railroads, steamships and promoted understanding of carnot cycles leading to internal combustion engines that led to automobiles and airplanes.  Understanding of connections between magnetism and electricity developed about the same time, making electricity into an industrial commodity that created additional options for industrial locations and transportation infrastructure.  For example it is difficult to imagine subway systems powered by combustion in any form. LADave (talk) 17:20, 2 February 2010 (UTC)

Vaclav Smill called the period 1867-1914 "The Age of Synergy" during which all of the great innovations were developed. Unlike the Fisrt Industrial Revolution, the inventions and innovations were science based. The U.S. had its highest growth during this period with growth three to five times greater than the First Inudstrial Revolution.Phmoreno (talk) 02:40, 1 October 2010 (UTC)

+++++++++++++++++++++++++++++++++

Let us discuss the THIRD Industrial Revolution -- the one thru the foothills of which we are at this time inattentively traveling.

"First Industrial Revolution"?
In a very interesting book in French, Jean Gimpel discusses what he considers to be the "First Industrial Revolution that took place in the 12th Century when the large abbeys were developed as industrial centres using mostly water powered machinery for forging iron, fulling cloth etc. This went on until the Black Death. (see: Gimpel Jean, "la Révolution Industrielle du Moyen Age". Editions Seuil, collection "Points Histoire" 1975 — still available, I believe.)  This would make the subject of the present article the "Third industrial Revolution"(!!?)--John of Paris 12:09, 16 October 2007 (UTC)

 * And Maxine Berg discusses an "industrious revolution" that also occurred before the first industrial revolution. Numbering these things can become slippery.  I think Gimpel's revolution, though, is often referred to as the Medieval Industrial Revolution.  The book is published in English as The Medieval Machine: The Industrial Revolution of the Middle Ages (Holt, Rhinehart, & Winston, 1976; I have a Penguin reprint).  Lynn White, Medieval Technology and Social Change, though might have something else to say about that.--RedJ 17 15:56, 28 October 2007 (UTC)

Agree about the numbering. Even so, it would be interesting to start an aricle on Twelfth Century technology to counter the usual received ideas on the Middle Ages — when life was "short and brutish", as we all know!??--John of Paris 16:53, 30 October 2007 (UTC)

New "Other eras' infobox needs repair
It overlaps (or is overlapped by) the text of the article when viewed using Firefox, if no easy fix, please revert these changes. Thanks, CliffC (talk) 01:19, 26 August 2008 (UTC)

What version are you using? I view it in firefox and it looks fine. J. D. Redding 01:42, 26 August 2008 (UTC)

 * 3.0.1 - just installed it maybe three days ago when it was offered at startup.  Guess that will teach me to be an early adopter. :( ... I have not noticed any other problems, though; and in fact had forgotten I had a new version.  Not a biggie if I'm the only one seeing it.  Best, CliffC (talk) 01:50, 26 August 2008 (UTC)

Agreed.
This article on the Second Industrial Revolution, though better than it had been, is still somewhat problematic. The sections on the various innovations (which I cut down from a considerably larger number) are still confusing and too detailed in all the wrong places. The random information on Germany and the United States either needs to be augmented or deleted so that it could add to the article rather than seem out of place. —Preceding unsigned comment added by WM2011 (talk • contribs) 19:00, 2 April 2009 (UTC)

Citation in lead section
The lead section cites "Atkenson and Kehoe". That would imply that these names appear in the sources section to specify the work, but they don't. What work is it? Hairy Dude (talk) 18:02, 5 April 2009 (UTC)

The graph
What does that graph even mean? I see it says relative levels of industrialization but I can't decipher exactly what it's measuring or what it's supposed to show the reader. 68.237.231.35 (talk) 21:53, 24 July 2010 (UTC)

Overlaps between First and Second Industrial Revolutions
A recent version separated the events taking place between the First and Second Industrial Revolutions by using a "Transition" section. Now the transition period (1830s-1860s) is mixed in and some First Industrial Revolution material has been added. In addition to making the article overly long with too much background detail, it is also confusing and misleading to the typical reader.Phmoreno (talk) 01:29, 10 October 2013 (UTC)

Monopolies
I read through the article and the broad topics that people have elaborated on. I've discovered that one of the main skill technique that many businesses used was not included. I researched more in depth information about monopolies and how they benefited people in the second industrial revolution. New technologies required larger amounts of money, so in order to get the needed capital many entrepreneurs created new ways to organize money. Many owners sold stock to investors, thus making each stockholder an owner of a company. As new directions for businesses flourished many corporations started to move towards monopolies. Many powerful business leaders started to developed monopolies and trusts in hopes to gain more income. Arrogant business leaders destroyed competing companies who stood in the way of their pursuit to profit. In this monopoly some businesses lowered prices to force other competitors out of business. With eliminating their competition they were able to raise their product prices to any level. Many times, a group of competitors formed a cartel, which is an association of fix prices, set production quotas etc. For example in Germany, a cartel fixed prices for 170 coal mines. It also explains how the rise of these businesses created many debates, some felt that businesses destroying each others market would hurt the free enterprise. Some felt that the government in the 1900s would soon move against monopolies and regulate the economy. I specifically went in depth about monopolies among businesses and would be opened to anyone with a better understanding who would want to update this topic. — Preceding unsigned comment added by Morganbhs (talk • contribs) 18:32, 3 January 2014 (UTC)

 * Why is it that articles on technology and economics end up with social commentary? Yes, its true that the Second Industrial Revolution saw the rise of major corporations spread over large geographical areas. This was due to the railroad and telegraph and machinery that gave rise to large scale factories.  The great deflation developed as a result of falling prices and many industries consolidated, creating monopolies.  However, the significant thing was that prices kept falling and standards of living rose at a faster rate than is happening at present.Phmoreno (talk) 15:34, 18 January 2014 (UTC)

Yes i agree that the most important thing was how monopolies effected prices and standards of living. I elaborated more on the start of large corporations and explained why monopolies didn't start solely because of machinery. I didn't intend on this article to end up with any social commentary. — Preceding unsigned comment added by 24.151.103.221 (talk) 00:14, 18 February 2014 (UTC)

 * An excellent reference on monopolies during this period is The Visible Hand by Chandler. Monopolies had great economies of scale, allowing them to lower prices.  One of the biggest monopolies was AT&T, which owned Bell Labs, where the transistor and the laser were developed, two technologies that lowered the price of telecommunications to almost nothing, eventually undermining AT&T's business and starting the computer and internet industries..Phmoreno (talk) 01:28, 18 February 2014 (UTC)

 * First minor point, these were half a century or more later. Second minor point, Laser is right to say, "The question of just how to assign credit for inventing the laser remains unresolved by historians." But still, yes. Jim.henderson (talk) 19:46, 19 February 2014 (UTC)

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Bias towards England
The article is simply unrelenting in identifying almost every English innovator as English, while other engineers, scientists and entrepreneurs from other nationalities are mostly only referred by name. “The English ‘INSERT A NAME’ was the first...” is a common phrase in this article. A lot of Canadians, Americans and Irishmen were described here, but all the concern about nationality of the inventors are absent in this case. Maybe it’s a ‘coincidence’... By the way, the contributions from Thomas Edison are ignored here, in favor of earlier important and less known British contributors (I know you’re now anxious to reply this and say i’m not aware of their seminal works BUT i absolutely think that they were fundamental to this area, but ignoring the importance Edison had to incandescent light sounds strange). Preceding unsigned comment added by Special:Contributions/2804:d55:2a12:571:642f:51aa:38f7:8d34 15:49, 8 June 2021 (UTC)

 * Yeah, the Electrification/Swan centric section is pure nationalist baloney. Needs work. Fountains of Bryn Mawr (talk) 19:11, 8 June 2021 (UTC)

 * You’re saying that. Because you just confirmed what i was saying. I don’t know which part of the:
 * “ I know you’re now anxious to reply this and say i’m not aware of their seminal works BUT i absolutely think that they were fundamental to this area, but ignoring the importance Edison had to incandescent light sounds strange “ you didn’t understood. I wrote that in parenthesis because i was expecting this kind of comment by some individuals. But what you just did is a known argumentative tactic: Plays the “fool”, pretends to misunderstand and make assumptions mostly exaggerating people’s opinions. This isn’t new, the amount of people answering things not said here is appalling. I think it’s becoming popular (I’m unwilling to endorse the “not that smart” individual thesis, which i think is imprecise as an explanation to this acts). Have a good day, I have nothing more to say here, considering the level of the first reply. — Preceding unsigned comment added by 191.219.96.47 (talk) 19:25, 8 June 2021 (UTC)

 * Actually I think I was agreeing with you there, you may want to re-read that. Fountains of Bryn Mawr (talk) 22:15, 8 June 2021 (UTC)
 * It's absolutely right, it presented itself as false sarcasm for which i blame myself only. Probably one more case of cultural schock... too used to sarcasm and irony everywhere.

United State > Employment distribution
An entire section is devoted to a difference of 3% in economic performance between the North and the South. That's tiny!

Thinking of cutting entirely. Uhoj (talk) 19:21, 6 June 2024 (UTC)

title: Talk:Second Industrial Revolution


text:
Cumulant

In probability theory and statistics, the cumulants $κ_{n}$ of a probability distribution are a set of quantities that provide an alternative to the moments of the distribution. Any two probability distributions whose moments are identical will have identical cumulants as well, and vice versa. The first cumulant is the mean, the second cumulant is the variance, and the third cumulant is the same as the third central moment. But fourth and higher-order cumulants are not equal to central moments. In some cases theoretical treatments of problems in terms of cumulants are simpler than those using moments. In particular, when two or more random variables are statistically independent, the $n$th-order cumulant of their sum is equal to the sum of their $n$th-order cumulants. As well, the third and higher-order cumulants of a normal distribution are zero, and it is the only distribution with this property.

Just as for moments, where joint moments are used for collections of random variables, it is possible to define joint cumulants.

Definition
The cumulants of a random variable $X$ are defined using the cumulant-generating function $K(t)$, which is the natural logarithm of the moment-generating function: $$K(t)=\log\operatorname{E}\left[e^{tX}\right].$$

The cumulants $κ_{n}$ are obtained from a power series expansion of the cumulant generating function: $$K(t)=\sum_{n=1}^\infty \kappa_{n} \frac{t^{n}}{n!} =\kappa_1 \frac{t}{1!} + \kappa_2 \frac{t^2}{2!}+ \kappa_3 \frac{t^3}{3!}+ \cdots = \mu t + \sigma^2 \frac{t^2}{2} + \cdots.$$

This expansion is a Maclaurin series, so the $n$th cumulant can be obtained by differentiating the above expansion $n$ times and evaluating the result at zero: $$\kappa_{n} = K^{(n)}(0).$$

If the moment-generating function does not exist, the cumulants can be defined in terms of the relationship between cumulants and moments discussed later.

Alternative definition of the cumulant generating function
Some writers prefer to define the cumulant-generating function as the natural logarithm of the characteristic function, which is sometimes also called the second characteristic function, $$H(t)=\log\operatorname{E} \left[e^{i t X}\right]=\sum_{n=1}^\infty \kappa_n \frac{(it)^n}{n!}=\mu it - \sigma^2 \frac{ t^2}{2} + \cdots$$

An advantage of $H(t)$—in some sense the function $K(t)$ evaluated for purely imaginary arguments—is that $E[e^{itX}]$ is well defined for all real values of $t$ even when $E[e^{tX}]$ is not well defined for all real values of $t$, such as can occur when there is "too much" probability that $X$ has a large magnitude. Although the function $H(t)$ will be well defined, it will nonetheless mimic $K(t)$ in terms of the length of its Maclaurin series, which may not extend beyond (or, rarely, even to) linear order in the argument $t$, and in particular the number of cumulants that are well defined will not change. Nevertheless, even when $H(t)$ does not have a long Maclaurin series, it can be used directly in analyzing and, particularly, adding random variables. Both the Cauchy distribution (also called the Lorentzian) and more generally, stable distributions (related to the Lévy distribution) are examples of distributions for which the power-series expansions of the generating functions have only finitely many well-defined terms.

Some basic properties
The $n$ th cumulant $\kappa_n(X)$ of (the distribution of) a random variable $X$  enjoys the following properties:
 * If $n>1$ and $c$  is constant (i.e. not random) then $ \kappa_n(X+c) = \kappa_n(X),$  i.e. the cumulant is translation invariant. (If $ n=1$  then we have $ \kappa_1(X+c) = \kappa_1(X)+c.) $
 * If $c$ is constant (i.e. not random) then $ \kappa_n(cX) = c^n\kappa_n(X),$  i.e. the $n$ th cumulant is homogeneous of degree $n$.
 * If random variables $X_1,\ldots,X_m$ are independent then $$ \kappa_n(X_1+\cdots+X_m) = \kappa_n(X_1) + \cdots + \kappa_n(X_m)\,. $$ That is, the cumulant is cumulative — hence the name.

The cumulative property follows quickly by considering the cumulant-generating function: $$\begin{align} K_{X_1+\cdots+X_m}(t) & =\log\operatorname{E} \left[e^{t(X_1+\cdots+X_m)}\right] \\[5pt] & = \log \left(\operatorname{E} \left[e^{tX_1}\right] \cdots \operatorname{E} \left[ e^{tX_m} \right] \right) \\[5pt] & = \log\operatorname{E}\left[e^{tX_1}\right] + \cdots + \log \operatorname{E} \left[ e^{tX_m} \right] \\[5pt] &= K_{X_1}(t) + \cdots + K_{X_m}(t), \end{align}$$ so that each cumulant of a sum of independent random variables is the sum of the corresponding cumulants of the addends. That is, when the addends are statistically independent, the mean of the sum is the sum of the means, the variance of the sum is the sum of the variances, the third cumulant (which happens to be the third central moment) of the sum is the sum of the third cumulants, and so on for each order of cumulant.

A distribution with given cumulants $κ_{n}$ can be approximated through an Edgeworth series.

First several cumulants as functions of the moments
All of the higher cumulants are polynomial functions of the central moments, with integer coefficients, but only in degrees 2 and 3 are the cumulants actually central moments.
 * $ \kappa_1(X) = \operatorname E(X)={} $ mean
 * $ \kappa_2(X) = \operatorname{var}(X) = \operatorname E\big((X-\operatorname E(X))^2\big) ={}$ the variance, or second central moment.
 * $ \kappa_3(X) = \operatorname E\big((X-\operatorname E(X))^3\big)={} $ the third central moment.
 * $ \kappa_4(X) = \operatorname E\big((X-\operatorname E(X))^4\big) - 3\left( \operatorname E\big((X-\operatorname E(X))^2\big) \right)^2={} $ the fourth central moment minus three times the square of the second central moment. Thus this is the first case in which cumulants are not simply moments or central moments. The central moments of degree more than 3 lack the cumulative property.
 * $ \kappa_5(X) = \operatorname E\big((X-\operatorname E(X))^5\big) - 10\operatorname E\big((X-\operatorname E(X))^3\big) \operatorname E\big((X-\operatorname E(X))^2\big).$

Cumulants of some discrete probability distributions

 * The constant random variables $X = μ$. The cumulant generating function is $K(t) = μt$. The first cumulant is $κ_{1} = K′(0) = μ$ and the other cumulants are zero, $κ_{2} = κ_{3} = κ_{4} = ⋅⋅⋅ = 0$.
 * The Bernoulli distributions, (number of successes in one trial with probability $p$ of success). The cumulant generating function is $K(t) = log(1 − p + pe^{t})$. The first cumulants are $κ_{1} = K '(0) = p$ and $κ_{2} = K′′(0) = p·(1 − p)$. The cumulants satisfy a recursion formula $$\kappa_{n+1}=p (1-p) \frac{d\kappa_n}{dp}.$$
 * The geometric distributions, (number of failures before one success with probability $p$ of success on each trial). The cumulant generating function is $K(t) = log(p / (1 + (p − 1)e^{t}))$. The first cumulants are $κ_{1} = K′(0) = p^{−1} − 1$, and $κ_{2} = K′′(0) = κ_{1}p^{−1}$. Substituting $p = (μ + 1)^{−1}$ gives $K(t) = −log(1 + μ(1−e^{t}))$ and $κ_{1} = μ$.
 * The Poisson distributions. The cumulant generating function is $K(t) = μ(e^{t} − 1)$. All cumulants are equal to the parameter: $κ_{1} = κ_{2} = κ_{3} = ... = μ$.
 * The binomial distributions, (number of successes in $n$ independent trials with probability $p$ of success on each trial). The special case $n = 1$ is a Bernoulli distribution. Every cumulant is just $n$ times the corresponding cumulant of the corresponding Bernoulli distribution. The cumulant generating function is $K(t) = n log(1 − p + pe^{t})$. The first cumulants are $κ_{1} = K′(0) = np$ and $κ_{2} = K′′(0) = κ_{1}(1 − p)$. Substituting $p = μ·n^{−1}$ gives $K '(t) = ((μ^{−1} − n^{−1})·e^{−t} + n^{−1})^{−1}$ and $κ_{1} = μ$. The limiting case $n^{−1} = 0$ is a Poisson distribution.
 * The negative binomial distributions, (number of failures before $r$ successes with probability $p$ of success on each trial). The special case $r = 1$ is a geometric distribution. Every cumulant is just $r$ times the corresponding cumulant of the corresponding geometric distribution. The derivative of the cumulant generating function is $K′(t) = r·((1 − p)^{−1}·e^{−t}−1)^{−1}$. The first cumulants are $κ_{1} = K′(0) = r·(p^{−1}−1)$, and $κ_{2} = K′′(0) = κ_{1}·p^{−1}$. Substituting $p = (μ·r^{−1}+1)^{−1}$ gives $K′(t) = ((μ^{−1} + r^{−1})e^{−t} − r^{−1})^{−1}$ and $κ_{1} = μ$. Comparing these formulas to those of the binomial distributions explains the name 'negative binomial distribution'. The limiting case $r^{−1} = 0$ is a Poisson distribution.

Introducing the variance-to-mean ratio $$\varepsilon=\mu^{-1}\sigma^2=\kappa_1^{-1}\kappa_2,$$ the above probability distributions get a unified formula for the derivative of the cumulant generating function: $$K'(t)=(1+(e^{-t}-1)\varepsilon)^{-1}\mu$$

The second derivative is $$K''(t)=(\varepsilon-(\varepsilon-1)e^t)^{-2}\mu\varepsilon e^t$$ confirming that the first cumulant is $κ_{1} = K′(0) = μ$ and the second cumulant is $κ_{2} = K′′(0) = με$.

The constant random variables $X = μ$ have $ε = 0$.

The binomial distributions have $ε = 1 − p$ so that $0 < ε < 1$.

The Poisson distributions have $ε = 1$.

The negative binomial distributions have $ε = p^{−1}$ so that $ε > 1$.

Note the analogy to the classification of conic sections by eccentricity: circles $ε = 0$, ellipses $0 < ε < 1$, parabolas $ε = 1$, hyperbolas $ε > 1$.

Cumulants of some continuous probability distributions

 * For the normal distribution with expected value $μ$ and variance $σ^{2}$, the cumulant generating function is $K(t) = μt + σ^{2}t^{2}/2$. The first and second derivatives of the cumulant generating function are $K′(t) = μ + σ^{2}·t$ and $K′′(t) = σ^{2}$. The cumulants are $κ_{1} = μ$, $κ_{2} = σ^{2}$, and $κ_{3} = κ_{4} = ⋅⋅⋅ = 0$. The special case $σ^{2} = 0$ is a constant random variable $X = μ$.
 * The cumulants of the uniform distribution on the interval $[−1, 0]$ are $κ_{n} = B_{n} /n$, where $B_{n}$ is the $n$th Bernoulli number.
 * The cumulants of the exponential distribution with rate parameter $λ$ are $κ_{n} = λ^{−n} (n − 1)!$.

Some properties of the cumulant generating function
The cumulant generating function $K(t)$, if it exists, is infinitely differentiable and convex, and passes through the origin. Its first derivative ranges monotonically in the open interval from the infimum to the supremum of the support of the probability distribution, and its second derivative is strictly positive everywhere it is defined, except for the degenerate distribution of a single point mass. The cumulant-generating function exists if and only if the tails of the distribution are majorized by an exponential decay, that is, (see Big O notation) $$ \begin{align} & \exists c>0,\,\, F(x)=O(e^{cx}), x\to-\infty; \text{ and} \\[4pt] & \exists d>0,\,\, 1-F(x)=O(e^{-dx}),x\to+\infty; \end{align} $$ where $F$ is the cumulative distribution function. The cumulant-generating function will have vertical asymptote(s) at the negative supremum of such $c$, if such a supremum exists, and at the supremum of such $d$, if such a supremum exists, otherwise it will be defined for all real numbers.

If the support of a random variable $X$ has finite upper or lower bounds, then its cumulant-generating function $y = K(t)$, if it exists, approaches asymptote(s) whose slope is equal to the supremum or infimum of the support, $$ \begin{align} y & =(t+1)\inf \operatorname{supp} X-\mu(X), \text{ and} \\[5pt] y & =(t-1)\sup \operatorname{supp}X+\mu(X), \end{align} $$ respectively, lying above both these lines everywhere. (The integrals $$\int_{-\infty}^0 \left[t\inf \operatorname{supp}X-K'(t)\right]\,dt, \qquad \int_{\infty}^0 \left[t\inf \operatorname{supp}X-K'(t) \right]\,dt$$ yield the $y$-intercepts of these asymptotes, since $K(0) = 0$.)

For a shift of the distribution by $c$, $K_{X+c}(t)=K_X(t)+ct.$ For a degenerate point mass at $c$, the cumulant generating function is the straight line $K_c(t)=ct$, and more generally, $K_{X+Y}=K_X+K_Y$  if and only if $X$ and $Y$ are independent and their cumulant generating functions exist; (subindependence and the existence of second moments sufficing to imply independence. )

The natural exponential family of a distribution may be realized by shifting or translating $K(t)$, and adjusting it vertically so that it always passes through the origin: if $f$ is the pdf with cumulant generating function $K(t)=\log M(t),$ and $f|\theta$  is its natural exponential family, then $f(x\mid\theta)=\frac1{M(\theta)}e^{\theta x} f(x),$  and $K(t\mid\theta)=K(t+\theta)-K(\theta).$

If $K(t)$ is finite for a range $t_{1} < Re(t) < t_{2}$ then if $t_{1} < 0 < t_{2}$ then $K(t)$ is analytic and infinitely differentiable for $t_{1} < Re(t) < t_{2}$. Moreover for $t$ real and $t_{1} < t < t_{2} K(t)$ is strictly convex, and $K′(t)$ is strictly increasing.

A negative result
Given the results for the cumulants of the normal distribution, it might be hoped to find families of distributions for which $κ_{m} = κ_{m+1} = ⋯ = 0$ for some $m > 3$, with the lower-order cumulants (orders 3 to $m − 1$) being non-zero. There are no such distributions. The underlying result here is that the cumulant generating function cannot be a finite-order polynomial of degree greater than 2.

Cumulants and moments
The moment generating function is given by: $$M(t) = 1+\sum_{n=1}^\infty \frac{\mu'_n t^n}{n!} = \exp \left(\sum_{n=1}^\infty \frac{\kappa_n t^n}{n!}\right) = \exp(K(t)).$$

So the cumulant generating function is the logarithm of the moment generating function $$K(t) = \log M(t).$$

The first cumulant is the expected value; the second and third cumulants are respectively the second and third central moments (the second central moment is the variance); but the higher cumulants are neither moments nor central moments, but rather more complicated polynomial functions of the moments.

The moments can be recovered in terms of cumulants by evaluating the $n$th derivative of $\exp(K(t))$ at $t=0$, $$\mu'_n = M^{(n)}(0) = \left. \frac{\mathrm{d}^n \exp (K(t))}{\mathrm{d}t^n}\right|_{t=0}. $$

Likewise, the cumulants can be recovered in terms of moments by evaluating the $n$th derivative of $\log M(t)$ at $t=0$, $$\kappa_n = K^{(n)}(0) = \left. \frac{\mathrm{d}^n \log M(t)}{\mathrm{d}t^n} \right|_{t=0}.$$

The explicit expression for the $n$th moment in terms of the first $n$ cumulants, and vice versa, can be obtained by using Faà di Bruno's formula for higher derivatives of composite functions. In general, we have $$\mu'_n = \sum_{k=1}^n B_{n,k}(\kappa_1,\ldots,\kappa_{n-k+1}) $$ $$\kappa_n = \sum_{k=1}^n (-1)^{k-1} (k-1)! B_{n,k}(\mu'_1, \ldots, \mu'_{n-k+1}),$$ where $B_{n,k}$ are incomplete (or partial) Bell polynomials.

In the like manner, if the mean is given by $\mu$, the central moment generating function is given by $$C(t) = \operatorname{E}[e^{t(x-\mu)}] = e^{-\mu t} M(t) = \exp(K(t) - \mu t), $$ and the $n$th central moment is obtained in terms of cumulants as $$\mu_n = C^{(n)}(0) = \left. \frac{\mathrm{d}^n}{\mathrm{d}t^n} \exp (K(t) - \mu t) \right|_{t=0} = \sum_{k=1}^n B_{n,k}(0,\kappa_2,\ldots,\kappa_{n-k+1}).$$

Also, for $n > 1$, the $n$th cumulant in terms of the central moments is $$ \begin{align} \kappa_n & = K^{(n)}(0) = \left. \frac{\mathrm{d}^n}{\mathrm{d}t^n} (\log C(t) + \mu t) \right|_{t=0} \\[4pt] & = \sum_{k=1}^n (-1)^{k-1} (k-1)! B_{n,k}(0,\mu_2,\ldots,\mu_{n-k+1}). \end{align} $$ The $n$th moment $μ′_{n}$ is an $n$th-degree polynomial in the first $n$ cumulants. The first few expressions are:

$$ \begin{align} \mu'_1 = {} & \kappa_1 \\[5pt] \mu'_2 = {} & \kappa_2+\kappa_1^2 \\[5pt] \mu'_3 = {} & \kappa_3+3\kappa_2\kappa_1+\kappa_1^3 \\[5pt] \mu'_4 = {} & \kappa_4 + 4\kappa_3\kappa_1 + 3\kappa_2^2 + 6\kappa_2\kappa_1^2 + \kappa_1^4 \\[5pt] \mu'_5 = {} & \kappa_5+5\kappa_4\kappa_1+10\kappa_3\kappa_2 + 10\kappa_3\kappa_1^2 + 15\kappa_2^2\kappa_1 + 10\kappa_2\kappa_1^3 + \kappa_1^5 \\[5pt] \mu'_6 = {} & \kappa_6 + 6\kappa_5\kappa_1 + 15\kappa_4\kappa_2 + 15\kappa_4\kappa_1^2 + 10\kappa_3^2 + 60\kappa_3\kappa_2\kappa_1 + 20\kappa_3\kappa_1^3 \\ & {} + 15\kappa_2^3 + 45\kappa_2^2\kappa_1^2 + 15\kappa_2\kappa_1^4 + \kappa_1^6. \end{align} $$

The "prime" distinguishes the moments $μ′_{n}$ from the central moments $μ_{n}$. To express the central moments as functions of the cumulants, just drop from these polynomials all terms in which $κ_{1}$ appears as a factor: $$ \begin{align} \mu_1 & =0 \\[4pt] \mu_2 & =\kappa_2 \\[4pt] \mu_3 & =\kappa_3 \\[4pt] \mu_4 & =\kappa_4+3\kappa_2^2 \\[4pt] \mu_5 & =\kappa_5+10\kappa_3\kappa_2 \\[4pt] \mu_6 & =\kappa_6+15\kappa_4\kappa_2+10\kappa_3^2+15\kappa_2^3. \end{align} $$

Similarly, the $n$th cumulant $κ_{n}$ is an $n$th-degree polynomial in the first $n$ non-central moments. The first few expressions are: $$ \begin{align} \kappa_1 = {} & \mu'_1 \\[4pt] \kappa_2 = {} & \mu'_2-{\mu'_1}^2 \\[4pt] \kappa_3 = {} & \mu'_3-3\mu'_2\mu'_1+2{\mu'_1}^3 \\[4pt] \kappa_4 = {} & \mu'_4-4\mu'_3\mu'_1-3{\mu'_2}^2+12\mu'_2{\mu'_1}^2-6{\mu'_1}^4 \\[4pt] \kappa_5 = {} & \mu'_5-5\mu'_4\mu'_1-10\mu'_3\mu'_2 + 20\mu'_3{\mu'_1}^2 + 30{\mu'_2}^2\mu'_1-60\mu'_2{\mu'_1}^3 + 24{\mu'_1}^5 \\[4pt] \kappa_6 = {} & \mu'_6-6\mu'_5\mu'_1-15\mu'_4\mu'_2+30\mu'_4{\mu'_1}^2-10{\mu'_3}^2 + 120\mu'_3\mu'_2\mu'_1 \\ & {} - 120\mu'_3{\mu'_1}^3 + 30{\mu'_2}^3 - 270{\mu'_2}^2 {\mu'_1}^2+360\mu'_2{\mu'_1}^4-120{\mu'_1}^6\,. \end{align} $$

In general, the cumulant is the determinant of a matrix: $$\kappa_l = (-1)^{l+1} \left|\begin{array}{cccccccc} \mu'_1 & 1 & 0 & 0 & 0 & 0 & \ldots & 0 \\ \mu'_2 & \mu'_1 & 1 & 0 & 0 & 0 & \ldots & 0 \\ \mu'_3 & \mu'_2 & \left(\begin{array}{l} 2 \\ 1 \end{array}\right) \mu'_1 & 1 & 0 & 0 & \ldots & 0 \\ \mu'_4 & \mu'_3 & \left(\begin{array}{l} 3 \\ 1 \end{array}\right) \mu'_2 & \left(\begin{array}{l} 3 \\ 2 \end{array}\right) \mu'_1 & 1 & 0 & \ldots & 0 \\ \mu'_5 & \mu'_4 & \left(\begin{array}{l} 4 \\ 1 \end{array}\right) \mu'_3 & \left(\begin{array}{l} 4 \\ 2 \end{array}\right) \mu'_2 & \left(\begin{array}{c} 4 \\ 3 \end{array}\right) \mu'_1 & 1 & \ldots & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \ddots & \ddots & \vdots \\ \mu'_{l-1} & \mu'_{l-2} & \ldots & \ldots & \ldots & \ldots & \ddots & 1 \\ \mu'_l & \mu'_{l-1} & \ldots & \ldots & \ldots & \ldots & \ldots & \left(\begin{array}{l} l-1 \\ l-2 \end{array}\right) \mu'_1 \end{array}\right|$$

To express the cumulants $κ_{n}$ for $n > 1$ as functions of the central moments, drop from these polynomials all terms in which μ'1 appears as a factor: $$\kappa_2=\mu_2\,$$ $$\kappa_3=\mu_3\,$$ $$\kappa_4=\mu_4-3{\mu_2}^2\,$$ $$\kappa_5=\mu_5-10\mu_3\mu_2\,$$ $$\kappa_6=\mu_6-15\mu_4\mu_2-10{\mu_3}^2+30{\mu_2}^3\,.$$

To express the cumulants $κ_{n}$ for $n > 2$ as functions of the standardized central moments $μ″_{n}$, also set $μ'_{2}=1$ in the polynomials: $$\kappa_3=\mu''_3\,$$ $$\kappa_4=\mu''_4-3\,$$ $$\kappa_5=\mu_5-10\mu_3\,$$ $$\kappa_6=\mu_6-15\mu_4-10{\mu''_3}^2+30\,.$$

The cumulants can be related to the moments by differentiating the relationship $log M(t) = K(t)$ with respect to $t$, giving $M′(t) = K′(t) M(t)$, which conveniently contains no exponentials or logarithms. Equating the coefficient of $t^{ n−1} / (n−1)!$ on the left and right sides and using $μ′_{0} = 1$ gives the following formulas for $n ≥ 1$: $$ \begin{align} \mu'_1 = {} & \kappa_1 \\[1pt] \mu'_2 = {} & \kappa_1\mu'_1+\kappa_2 \\[1pt] \mu'_3 = {} & \kappa_1\mu'_2+2\kappa_2\mu'_1+\kappa_3 \\[1pt] \mu'_4 = {} & \kappa_1\mu'_3+3\kappa_2\mu'_2+3\kappa_3\mu'_1+\kappa_4 \\[1pt] \mu'_5 = {} & \kappa_1\mu'_4+4\kappa_2\mu'_3+6\kappa_3\mu'_2+4\kappa_4\mu'_1+\kappa_5 \\[1pt] \mu'_6 = {} & \kappa_1\mu'_5+5\kappa_2\mu'_4+10\kappa_3\mu'_3+10\kappa_4\mu'_2+5\kappa_5\mu'_1+\kappa_6 \\[1pt] \mu'_n = {} & \sum_{m=1}^{n-1}{n-1 \choose m-1}\kappa_m \mu'_{n-m} + \kappa_n\,. \end{align} $$ These allow either $\kappa_n$ or $\mu'_n$  to be computed from the other using knowledge of the lower-order cumulants and moments. The corresponding formulas for the central moments $\mu_n$ for $n \ge 2$  are formed from these formulas by setting $\mu'_1 = \kappa_1 = 0$  and replacing each $\mu'_n$  with $\mu_n$  for $n \ge 2$ : $$ \begin{align} \mu_2 = {} & \kappa_2 \\[1pt] \mu_3 = {} & \kappa_3 \\[1pt] \mu_n = {} & \sum_{m=2}^{n-2}{n-1 \choose m-1}\kappa_m \mu_{n-m} + \kappa_n\,. \end{align} $$

Cumulants and set-partitions
These polynomials have a remarkable combinatorial interpretation: the coefficients count certain partitions of sets. A general form of these polynomials is $$\mu'_n=\sum_{\pi \, \in \, \Pi} \prod_{B \, \in \, \pi} \kappa_{|B|}$$ where
 * $π$ runs through the list of all partitions of a set of size $n$;
 * "$B ∈ \pi$" means $B$ is one of the "blocks" into which the set is partitioned; and
 * $|B|$ is the size of the set $B$.

Thus each monomial is a constant times a product of cumulants in which the sum of the indices is $n$ (e.g., in the term $κ_{3} κ_{2}^{2} κ_{1}$, the sum of the indices is 3 + 2 + 2 + 1 = 8; this appears in the polynomial that expresses the 8th moment as a function of the first eight cumulants). A partition of the integer $n$ corresponds to each term. The coefficient in each term is the number of partitions of a set of $n$ members that collapse to that partition of the integer $n$ when the members of the set become indistinguishable.

Cumulants and combinatorics
Further connection between cumulants and combinatorics can be found in the work of Gian-Carlo Rota, where links to invariant theory, symmetric functions, and binomial sequences are studied via umbral calculus.

Joint cumulants
The joint cumulant $κ$ of several random variables $X_{1}, ..., X_{n}$ is defined as the coefficient $κ_{1,...,1}(X_{1}, ..., X_{n})$ in the Maclaurin series of the multivariate cumulant generating function, see Section 3.1 in, $$G(t_1,\dots,t_n)=\log \mathrm{E}(\mathrm{e}^{\sum_{j=1}^n t_j X_j}) =\sum_{k_1,\ldots,k_n} \kappa_{k_1,\ldots,k_n} \frac{t_1^{k_1} \cdots t_n^{k_n}}{k_1! \cdots k_n!} \,.$$ Note that $$\kappa_{k_1,\dots,k_n} = \left.\left(\frac{\mathrm{d}}{\mathrm{d} t_1}\right)^{k_1} \cdots \left(\frac{\mathrm{d}}{\mathrm{d} t_n}\right)^{k_n} G(t_1,\dots,t_n) \right|_{t_1 = \dots = t_n = 0}\,,$$ and, in particular $$\kappa(X_1,\ldots,X_n) = \left. \frac{\mathrm{d}^n}{\mathrm{d} t_1 \cdots \mathrm{d} t_n} G(t_1,\dots,t_n) \right|_{t_1 = \dots = t_n = 0}\,.$$ As with a single variable, the generating function and cumulant can instead be defined via $$H(t_1,\dots,t_n) =\log \mathrm{E}(\mathrm{e}^{\sum_{j=1}^n i t_j X_j}) =\sum_{k_1,\ldots,k_n} \kappa_{k_1,\ldots,k_n} i^{k_1+\cdots+k_n} \frac{t_1^{k_1} \cdots t_n^{k_n}}{k_1! \cdots k_n!}\,,$$ in which case $$\kappa_{k_1,\dots,k_n} = (-i)^{k_1+\cdots+k_n} \left.\left(\frac{\mathrm{d}}{\mathrm{d} t_1}\right)^{k_1} \cdots \left(\frac{\mathrm{d}}{\mathrm{d} t_n}\right)^{k_n} H(t_1,\dots,t_n) \right|_{t_1 = \dots = t_n = 0}\,,$$ and $$\kappa(X_1,\ldots,X_n) = \left. (-i)^{n} \frac{\mathrm{d}^n}{\mathrm{d} t_1 \cdots \mathrm{d} t_n} H(t_1,\dots,t_n) \right|_{t_1 = \dots = t_n = 0}\,.$$

Repeated random variables and relation between the coefficients κk 1, ..., kn
Observe that $\kappa_{k_1,\dots,k_n} (X_1,\ldots,X_n)$  can also be written as $$\kappa_{k_1,\dots,k_n} = \left. \frac{\mathrm{d}^{k_1}}{\mathrm{d} t_{1,1} \cdots \mathrm{d} t_{1,k_1}} \cdots \frac{\mathrm{d}^{k_n}}{\mathrm{d} t_{n,1} \cdots \mathrm{d} t_{n,k_n}} G\left(\sum_{j=1}^{k_1}t_{1,j},\dots,\sum_{j=1}^{k_n}t_{n,j}\right) \right|_{t_{i,j}=0},$$ from which we conclude that $$\kappa_{k_1,\dots,k_n} (X_1,\ldots,X_n) = \kappa_{1,\ldots,1} ( \underbrace{X_1,\dots,X_1}_{k_1}, \ldots, \underbrace{X_n,\dots,X_n}_{k_n} ) .$$ For example $$\kappa_{2,0,1}(X,Y,Z) = \kappa(X,X,Z),\,$$ and $$\kappa_{0,0,n,0}(X,Y,Z,T) = \kappa_{n}(Z) = \kappa(\underbrace{Z,\dots,Z}_{n}) .\,$$ In particular, the last equality shows that the cumulants of a single random variable are the joint cumulants of multiple copies of that random variable.

Relation with mixed moments
The joint cumulant or random variables can be expressed as an alternate sum of products of their mixed moments, see Equation (3.2.7) in, $$\kappa(X_1,\dots,X_n)=\sum_\pi (|\pi|-1)!(-1)^{|\pi|-1}\prod_{B\in\pi}E\left(\prod_{i\in B}X_i\right)$$ where $\pi$ runs through the list of all partitions of $\{1, ..., n\}$; where $B$ runs through the list of all blocks of the partition π; and where $|\pi|$ is the number of parts in the partition.

For example, $$\kappa(X)=\operatorname E(X),$$ is the expected value of $X$ , $$\kappa(X,Y)=\operatorname E(XY) - \operatorname E(X) \operatorname E(Y),$$ is the covariance of $X$  and $Y$, and $$\kappa(X,Y,Z)=\operatorname E(XYZ) - \operatorname E(XY) \operatorname E(Z) - \operatorname E(XZ) \operatorname E(Y) - \operatorname E(YZ) \operatorname E(X) + 2\operatorname E(X)\operatorname E(Y)\operatorname E(Z).\,$$

For zero-mean random variables $X_1,\ldots,X_n$, any mixed moment of the form $\prod_{B\in\pi} E\left(\prod_{i\in B} X_i\right)$ vanishes if $\pi$  is a partition of $\{ 1,\ldots,n \}$  which contains a singleton $B=\{k\}$. Hence, the expression of their joint cumulant in terms of mixed moments simplifies. For example, if X,Y,Z,W are zero mean random variables, we have $$\kappa(X,Y,Z) = \operatorname E(XYZ).\,$$ $$\kappa(X,Y,Z,W) = \operatorname E(XYZW) - \operatorname E(XY) \operatorname E(ZW) - \operatorname E(XZ) \operatorname E(YW) - \operatorname E(XW) \operatorname E(YZ).\,$$

More generally, any coefficient of the Maclaurin series can also be expressed in terms of mixed moments, although there are no concise formulae. Indeed, as noted above, one can write it as a joint cumulant by repeating random variables appropriately, and then apply the above formula to express it in terms of mixed moments. For example $$\kappa_{201}(X,Y,Z) = \kappa(X,X,Z)=\operatorname E(X^2Z) -2\operatorname E(XZ)\operatorname E(X) - \operatorname E(X^2)\operatorname E(Z) + 2\operatorname E(X)^2\operatorname E(Z).\,$$

If some of the random variables are independent of all of the others, then any cumulant involving two (or more) independent random variables is zero.

The combinatorial meaning of the expression of mixed moments in terms of cumulants is easier to understand than that of cumulants in terms of mixed moments, see Equation (3.2.6) in: $$\operatorname E(X_1\cdots X_n)=\sum_\pi\prod_{B\in\pi}\kappa(X_i : i \in B). $$

For example: $$\operatorname E(XYZ) = \kappa(X,Y,Z) + \kappa(X,Y)\kappa(Z) + \kappa(X,Z)\kappa(Y) + \kappa(Y,Z)\kappa(X) + \kappa(X)\kappa(Y)\kappa(Z).\,$$

Further properties
Another important property of joint cumulants is multilinearity: $$\kappa(X+Y,Z_1,Z_2,\dots) = \kappa(X,Z_1,Z_2,\ldots) + \kappa(Y,Z_1,Z_2,\ldots).\,$$

Just as the second cumulant is the variance, the joint cumulant of just two random variables is the covariance. The familiar identity $$\operatorname{var}(X+Y) = \operatorname{var}(X) + 2\operatorname{cov}(X,Y) + \operatorname{var}(Y)\,$$ generalizes to cumulants: $$\kappa_n(X+Y)=\sum_{j=0}^n {n \choose j} \kappa( \, \underbrace{X,\dots,X}_j, \underbrace{Y,\dots,Y}_{n-j}\,).\,$$

Conditional cumulants and the law of total cumulance
The law of total expectation and the law of total variance generalize naturally to conditional cumulants. The case $n = 3$, expressed in the language of (central) moments rather than that of cumulants, says $$\mu_3(X) = \operatorname E(\mu_3(X\mid Y)) + \mu_3(\operatorname E(X\mid Y)) + 3 \operatorname{cov}(\operatorname E(X\mid Y), \operatorname{var} (X\mid Y)).$$

In general, $$\kappa(X_1,\dots,X_n)=\sum_\pi \kappa(\kappa(X_{\pi_1}\mid Y), \dots, \kappa(X_{\pi_b}\mid Y))$$ where
 * the sum is over all partitions π of the set $\{1, ..., n\}$ of indices, and
 * π1, ..., πb are all of the "blocks" of the partition π; the expression $κ(X_{\pi_{m}})|undefined$ indicates that the joint cumulant of the random variables whose indices are in that block of the partition.

Conditional cumulants and the conditional expectation
For certain settings, a derivative identity can be established between the conditional cumulant and the conditional expectation. For example, suppose that $Y = X + Z$ where $Z$ is standard normal independent of $X$, then for any $X$ it holds that $$\kappa_{n+1}(X\mid Y=y) = \frac{ \mathrm{d}^n}{ \mathrm{d} y^n}\operatorname E(X\mid Y = y), \, n \in \mathbb{N}, \, y \in \mathbb{R}.$$ The results can also be texted to the exponential family.

Relation to statistical physics
In statistical physics many extensive quantities – that is quantities that are proportional to the volume or size of a given system – are related to cumulants of random variables. The deep connection is that in a large system an extensive quantity like the energy or number of particles can be thought of as the sum of (say) the energy associated with a number of nearly independent regions. The fact that the cumulants of these nearly independent random variables will (nearly) add make it reasonable that extensive quantities should be expected to be related to cumulants.

A system in equilibrium with a thermal bath at temperature $T$ have a fluctuating internal energy $E$, which can be considered a random variable drawn from a distribution $ E\sim p(E)$. The partition function of the system is $$Z(\beta) = \langle\exp(-\beta E)\rangle,$$ where $β$ = $1/(kT)$ and $k$ is the Boltzmann constant and the notation $\langle A \rangle$ has been used rather than $\operatorname{E}[A]$  for the expectation value to avoid confusion with the energy, $E$. Hence the first and second cumulant for the energy $E$ give the average energy and heat capacity. $$ \begin{align} \langle E \rangle_c & = \frac{\partial \log Z}{\partial (-\beta)} = \langle E \rangle \\[6pt] \langle E^2 \rangle_c & = \frac{\partial\langle E\rangle_c}{\partial (-\beta)} = k T^2 \frac{\partial \langle E\rangle}{\partial T} = kT^2C \end{align} $$

The Helmholtz free energy expressed in terms of $$F(\beta) = -\beta^{-1}\log Z(\beta) \, $$ further connects thermodynamic quantities with cumulant generating function for the energy. Thermodynamics properties that are derivatives of the free energy, such as its internal energy, entropy, and specific heat capacity, all can be readily expressed in terms of these cumulants. Other free energy can be a function of other variables such as the magnetic field or chemical potential $\mu$, e.g. $$\Omega=-\beta^{-1}\log(\langle \exp(-\beta E -\beta\mu N) \rangle),\,$$ where $N$ is the number of particles and $\Omega$ is the grand potential. Again the close relationship between the definition of the free energy and the cumulant generating function implies that various derivatives of this free energy can be written in terms of joint cumulants of $E$ and $N$.

History
The history of cumulants is discussed by Anders Hald.

Cumulants were first introduced by Thorvald N. Thiele, in 1889, who called them semi-invariants. They were first called cumulants in a 1932 paper by Ronald Fisher and John Wishart. Fisher was publicly reminded of Thiele's work by Neyman, who also notes previous published citations of Thiele brought to Fisher's attention. Stephen Stigler has said that the name cumulant was suggested to Fisher in a letter from Harold Hotelling. In a paper published in 1929, Fisher had called them cumulative moment functions.

The partition function in statistical physics was introduced by Josiah Willard Gibbs in 1901. The free energy is often called Gibbs free energy. In statistical mechanics, cumulants are also known as Ursell functions relating to a publication in 1927.

Formal cumulants
More generally, the cumulants of a sequence ${ m_{n} : n = 1, 2, 3, ... }$, not necessarily the moments of any probability distribution, are, by definition, $$1+\sum_{n=1}^\infty \frac{m_n t^n}{n!} = \exp \left( \sum_{n=1}^\infty \frac{\kappa_n t^n}{n!} \right) ,$$ where the values of $κ_{n}$ for $n = 1, 2, 3, ...$ are found formally, i.e., by algebra alone, in disregard of questions of whether any series converges. All of the difficulties of the "problem of cumulants" are absent when one works formally. The simplest example is that the second cumulant of a probability distribution must always be nonnegative, and is zero only if all of the higher cumulants are zero. Formal cumulants are subject to no such constraints.

Bell numbers
In combinatorics, the $n$th Bell number is the number of partitions of a set of size $n$. All of the cumulants of the sequence of Bell numbers are equal to 1. The Bell numbers are the moments of the Poisson distribution with expected value 1.

Cumulants of a polynomial sequence of binomial type
For any sequence $\{ κ_{n} : n = 1, 2, 3, ... \}$ of scalars in a field of characteristic zero, being considered formal cumulants, there is a corresponding sequence $\{ μ′ : n = 1, 2, 3, ... \}$ of formal moments, given by the polynomials above. For those polynomials, construct a polynomial sequence in the following way. Out of the polynomial $$ \begin{align} \mu'_6 = {} & \kappa_6 + 6\kappa_5\kappa_1 + 15\kappa_4\kappa_2 + 15\kappa_4\kappa_1^2 + 10\kappa_3^2+60\kappa_3\kappa_2\kappa_1 + 20\kappa_3\kappa_1^3 \\ & {} + 15\kappa_2^3 + 45\kappa_2^2\kappa_1^2 + 15\kappa_2\kappa_1^4 + \kappa_1^6 \end{align} $$ make a new polynomial in these plus one additional variable $x$: $$ \begin{align} p_6(x) = {} & \kappa_6 \,x + (6\kappa_5\kappa_1 + 15\kappa_4\kappa_2 + 10\kappa_3^2)\,x^2 + (15\kappa_4\kappa_1^2 + 60\kappa_3\kappa_2\kappa_1 + 15\kappa_2^3)\,x^3 \\ & {} + (45\kappa_2^2\kappa_1^2)\,x^4+(15\kappa_2\kappa_1^4)\,x^5 +(\kappa_1^6)\,x^6, \end{align} $$ and then generalize the pattern. The pattern is that the numbers of blocks in the aforementioned partitions are the exponents on $x$. Each coefficient is a polynomial in the cumulants; these are the Bell polynomials, named after Eric Temple Bell.

This sequence of polynomials is of binomial type. In fact, no other sequences of binomial type exist; every polynomial sequence of binomial type is completely determined by its sequence of formal cumulants.

Free cumulants
In the above moment-cumulant formula\ $$\operatorname E(X_1\cdots X_n)=\sum_\pi\prod_{B\,\in\,\pi}\kappa(X_i : i\in B)$$ for joint cumulants, one sums over all partitions of the set $\{ 1, ..., n \}$. If instead, one sums only over the noncrossing partitions, then, by solving these formulae for the $\kappa$ in terms of the moments, one gets free cumulants rather than conventional cumulants treated above. These free cumulants were introduced by Roland Speicher and play a central role in free probability theory. In that theory, rather than considering independence of random variables, defined in terms of tensor products of algebras of random variables, one considers instead free independence of random variables, defined in terms of free products of algebras.

The ordinary cumulants of degree higher than 2 of the normal distribution are zero. The free cumulants of degree higher than 2 of the Wigner semicircle distribution are zero. This is one respect in which the role of the Wigner distribution in free probability theory is analogous to that of the normal distribution in conventional probability theory.

title: Cumulant