text:
Expected value

In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of the possible values a random variable can take, weighted by the probability of those outcomes. Since it is obtained through arithmetic, the expected value sometimes may not even be included in the sample data set; it is not the value you would "expect" to get in reality.

The expected value of a random variable with a finite number of outcomes is a weighted average of all possible outcomes. In the case of a continuum of possible outcomes, the expectation is defined by integration. In the axiomatic foundation for probability provided by measure theory, the expectation is given by Lebesgue integration.

The expected value of a random variable $X$ is often denoted by $E(X)$, $E[X]$, or $EX$, with $E$ also often stylized as $$\mathbb{E}$$ or $E$.

History
The idea of the expected value originated in the middle of the 17th century from the study of the so-called problem of points, which seeks to divide the stakes in a fair way between two players, who have to end their game before it is properly finished. This problem had been debated for centuries. Many conflicting proposals and solutions had been suggested over the years when it was posed to Blaise Pascal by French writer and amateur mathematician Chevalier de Méré in 1654. Méré claimed that this problem could not be solved and that it showed just how flawed mathematics was when it came to its application to the real world. Pascal, being a mathematician, was provoked and determined to solve the problem once and for all.

He began to discuss the problem in the famous series of letters to Pierre de Fermat. Soon enough, they both independently came up with a solution. They solved the problem in different computational ways, but their results were identical because their computations were based on the same fundamental principle. The principle is that the value of a future gain should be directly proportional to the chance of getting it. This principle seemed to have come naturally to both of them. They were very pleased by the fact that they had found essentially the same solution, and this in turn made them absolutely convinced that they had solved the problem conclusively; however, they did not publish their findings. They only informed a small circle of mutual scientific friends in Paris about it.

In Dutch mathematician Christiaan Huygens' book, he considered the problem of points, and presented a solution based on the same principle as the solutions of Pascal and Fermat. Huygens published his treatise in 1657, (see Huygens (1657)) "De ratiociniis in ludo aleæ" on probability theory just after visiting Paris. The book extended the concept of expectation by adding rules for how to calculate expectations in more complicated situations than the original problem (e.g., for three or more players), and can be seen as the first successful attempt at laying down the foundations of the theory of probability.

In the foreword to his treatise, Huygens wrote:

"It should be said, also, that for some time some of the best mathematicians of France have occupied themselves with this kind of calculus so that no one should attribute to me the honour of the first invention. This does not belong to me. But these savants, although they put each other to the test by proposing to each other many questions difficult to solve, have hidden their methods. I have had therefore to examine and go deeply for myself into this matter by beginning with the elements, and it is impossible for me for this reason to affirm that I have even started from the same principle. But finally I have found that my answers in many cases do not differ from theirs."

In the mid-nineteenth century, Pafnuty Chebyshev became the first person to think systematically in terms of the expectations of random variables.

Etymology
Neither Pascal nor Huygens used the term "expectation" in its modern sense. In particular, Huygens writes:

"That any one Chance or Expectation to win any thing is worth just such a Sum, as wou'd procure in the same Chance and Expectation at a fair Lay. ... If I expect a or b, and have an equal chance of gaining them, my Expectation is worth (a+b)/2."

More than a hundred years later, in 1814, Pierre-Simon Laplace published his tract "Théorie analytique des probabilités", where the concept of expected value was defined explicitly:

"... this advantage in the theory of chance is the product of the sum hoped for by the probability of obtaining it; it is the partial sum which ought to result when we do not wish to run the risks of the event in supposing that the division is made proportional to the probabilities. This division is the only equitable one when all strange circumstances are eliminated; because an equal degree of probability gives an equal right for the sum hoped for. We will call this advantage mathematical hope."

Notations
The use of the letter $E$ to denote "expected value" goes back to W. A. Whitworth in 1901. The symbol has since become popular for English writers. In German, $E$ stands for Erwartungswert, in Spanish for esperanza matemática, and in French for espérance mathématique.

When "E" is used to denote "expected value", authors use a variety of stylizations: the expectation operator can be stylized as $E$ (upright), $E$ (italic), or $$\mathbb{E}$$ (in blackboard bold), while a variety of bracket notations (such as $E(X)$, $E[X]$, and $EX$) are all used.

Another popular notation is $μ_{X}$, whereas $⟨X⟩$, $⟨X⟩_{av}$, and $$\overline{X}$$ are commonly used in physics, and $M(X)$ in Russian-language literature.

Definition
As discussed above, there are several context-dependent ways of defining the expected value. The simplest and original definition deals with the case of finitely many possible outcomes, such as in the flip of a coin. With the theory of infinite series, this can be extended to the case of countably many possible outcomes. It is also very common to consider the distinct case of random variables dictated by (piecewise-)continuous probability density functions, as these arise in many natural contexts. All of these specific definitions may be viewed as special cases of the general definition based upon the mathematical tools of measure theory and Lebesgue integration, which provide these different contexts with an axiomatic foundation and common language.

Any definition of expected value may be extended to define an expected value of a multidimensional random variable, i.e. a random vector $X$. It is defined component by component, as $E[X]_{i} = E[X_{i}]$. Similarly, one may define the expected value of a random matrix $X$ with components $X_{ij}$ by $E[X]_{ij} = E[X_{ij}]$.

Random variables with finitely many outcomes
Consider a random variable $X$ with a finite list $x_{1}, ..., x_{k}$ of possible outcomes, each of which (respectively) has probability $p_{1}, ..., p_{k}$ of occurring. The expectation of $X$ is defined as $$\operatorname{E}[X] =x_1p_1 + x_2p_2 + \cdots + x_kp_k.$$

Since the probabilities must satisfy $p_{1} + ⋅⋅⋅ + p_{k} = 1$, it is natural to interpret $E[X]$ as a weighted average of the $x_{i}$ values, with weights given by their probabilities $p_{i}$.

In the special case that all possible outcomes are equiprobable (that is, $p_{1} = ⋅⋅⋅ = p_{k}$), the weighted average is given by the standard average. In the general case, the expected value takes into account the fact that some outcomes are more likely than others.

Examples

 * Let $$X$$ represent the outcome of a roll of a fair six-sided . More specifically, $$X$$ will be the number of pips showing on the top face of the after the toss. The possible values for $$X$$ are 1, 2, 3, 4, 5, and 6, all of which are equally likely with a probability of . The expectation of $$X$$ is $$ \operatorname{E}[X] = 1 \cdot \frac{1}{6} + 2 \cdot \frac{1}{6} + 3\cdot\frac{1}{6} + 4\cdot\frac{1}{6} + 5\cdot\frac{1}{6} + 6\cdot\frac{1}{6} = 3.5.$$ If one rolls the  $$n$$ times and computes the average (arithmetic mean) of the results, then as $$n$$ grows, the average will almost surely converge to the expected value, a fact known as the strong law of large numbers.
 * The roulette game consists of a small ball and a wheel with 38 numbered pockets around the edge. As the wheel is spun, the ball bounces around randomly until it settles down in one of the pockets. Suppose random variable $$X$$ represents the (monetary) outcome of a $1 bet on a single number ("straight up" bet). If the bet wins (which happens with probability in American roulette), the payoff is $35; otherwise the player loses the bet. The expected profit from such a bet will be $$ \operatorname{E}[\,\text{gain from }\$1\text{ bet}\,] = -\$1 \cdot \frac{37}{38} + \$35 \cdot \frac{1}{38} = -\$\frac{1}{19}.$$ That is, the expected value to be won from a $1 bet is −$. Thus, in 190 bets, the net loss will probably be about $10.

Random variables with countably infinitely many outcomes
Informally, the expectation of a random variable with a countably infinite set of possible outcomes is defined analogously as the weighted average of all possible outcomes, where the weights are given by the probabilities of realizing each given value. This is to say that $$\operatorname{E}[X] = \sum_{i=1}^\infty x_i\, p_i,$$ where $x_{1}, x_{2}, ...$ are the possible outcomes of the random variable $X$ and $p_{1}, p_{2}, ...$ are their corresponding probabilities. In many non-mathematical textbooks, this is presented as the full definition of expected values in this context.

However, there are some subtleties with infinite summation, so the above formula is not suitable as a mathematical definition. In particular, the Riemann series theorem of mathematical analysis illustrates that the value of certain infinite sums involving positive and negative summands depends on the order in which the summands are given. Since the outcomes of a random variable have no naturally given order, this creates a difficulty in defining expected value precisely.

For this reason, many mathematical textbooks only consider the case that the infinite sum given above converges absolutely, which implies that the infinite sum is a finite number independent of the ordering of summands. In the alternative case that the infinite sum does not converge absolutely, one says the random variable does not have finite expectation.

Examples
+ 2(\tfrac{c}{8}) + 3 (\tfrac{c}{24}) + \cdots \,= \, \tfrac{c}{2} + \tfrac{c}{4} + \tfrac{c}{8} + \cdots \,=\, c \,=\, \tfrac{1}{\ln 2}.$$
 * Suppose $$x_i = i$$ and $$p_i = \tfrac{c}{i \cdot 2^i}$$ for $$i = 1, 2, 3, \ldots,$$ where $$c = \tfrac{1}{\ln 2}$$ is the scaling factor which makes the probabilities sum to 1. Then we have $$\operatorname{E}[X] \,= \sum_i x_i p_i = 1(\tfrac{c}{2})

Random variables with density
Now consider a random variable $X$ which has a probability density function given by a function $f$ on the real number line. This means that the probability of $X$ taking on a value in any given open interval is given by the integral of $f$ over that interval. The expectation of $X$ is then given by the integral $$\operatorname{E}[X] = \int_{-\infty}^\infty x f(x)\, dx.$$ A general and mathematically precise formulation of this definition uses measure theory and Lebesgue integration, and the corresponding theory of absolutely continuous random variables is described in the next section. The density functions of many common distributions are piecewise continuous, and as such the theory is often developed in this restricted setting. For such functions, it is sufficient to only consider the standard Riemann integration. Sometimes continuous random variables are defined as those corresponding to this special class of densities, although the term is used differently by various authors.

Analogously to the countably-infinite case above, there are subtleties with this expression due to the infinite region of integration. Such subtleties can be seen concretely if the distribution of $X$ is given by the Cauchy distribution $Cauchy(0, π)$, so that $f(x) = (x^{2} + π^{2})^{−1}$. It is straightforward to compute in this case that $$\int_a^b xf(x)\,dx=\int_a^b \frac{x}{x^2+\pi^2}\,dx=\frac{1}{2}\ln\frac{b^2+\pi^2}{a^2+\pi^2}.$$ The limit of this expression as $a → −∞$ and $b → ∞$ does not exist: if the limits are taken so that $a = −b$, then the limit is zero, while if the constraint $2a = −b$ is taken, then the limit is $ln(2)$.

To avoid such ambiguities, in mathematical textbooks it is common to require that the given integral converges absolutely, with $E[X]$ left undefined otherwise. However, measure-theoretic notions as given below can be used to give a systematic definition of $E[X]$ for more general random variables $X$.

Arbitrary real-valued random variables
All definitions of the expected value may be expressed in the language of measure theory. In general, if $X$ is a real-valued random variable defined on a probability space $(Ω, Σ, P)$, then the expected value of $X$, denoted by $E[X]$, is defined as the Lebesgue integral $$\operatorname{E} [X] = \int_\Omega X\,d\operatorname{P}.$$ Despite the newly abstract situation, this definition is extremely similar in nature to the very simplest definition of expected values, given above, as certain weighted averages. This is because, in measure theory, the value of the Lebesgue integral of $X$ is defined via weighted averages of approximations of $X$ which take on finitely many values. Moreover, if given a random variable with finitely or countably many possible values, the Lebesgue theory of expectation is identical to the summation formulas given above. However, the Lebesgue theory clarifies the scope of the theory of probability density functions. A random variable $X$ is said to be absolutely continuous if any of the following conditions are satisfied: These conditions are all equivalent, although this is nontrivial to establish. In this definition, $f$ is called the probability density function of $A$ (relative to Lebesgue measure). According to the change-of-variables formula for Lebesgue integration, combined with the law of the unconscious statistician, it follows that $$\operatorname{E}[X] \equiv \int_\Omega X\,d\operatorname{P} = \int_\Reals x f(x)\, dx$$ for any absolutely continuous random variable $X$. The above discussion of continuous random variables is thus a special case of the general Lebesgue theory, due to the fact that every piecewise-continuous function is measurable.
 * there is a nonnegative measurable function $A$ on the real line such that $$\operatorname{P}(X \in A) = \int_A f(x) \, dx,$$ for any Borel set $X$, in which the integral is Lebesgue.
 * the cumulative distribution function of $A$ is absolutely continuous.
 * for any Borel set $A$ of real numbers with Lebesgue measure equal to zero, the probability of $X$ being valued in $A$ is also equal to zero
 * for any positive number $ε$ there is a positive number $δ$ such that: if $f$ is a Borel set with Lebesgue measure less than $δ$, then the probability of $X$ being valued in $X$ is less than $ε$.

The expected value of any real-valued random variable $$X$$ can also be defined on the graph of its cumulative distribution function $$F$$ by a nearby equality of areas. In fact, $$\operatorname{E}[X] = \mu$$ with a real number $$\mu$$ if and only if the two surfaces in the $$x$$-$$y$$-plane, described by $$ x \le \mu, \;\, 0\le y \le F(x) \quad\text{or}\quad x \ge \mu, \;\, F(x) \le y \le 1 $$ respectively, have the same finite area, i.e. if $$ \int_{-\infty}^\mu F(x)\,dx = \int_\mu^\infty \big(1 - F(x)\big)\,dx $$ and both improper Riemann integrals converge. Finally, this is equivalent to the representation

$$ \operatorname{E}[X] = \int_0^\infty \bigl(1 - F(x)\bigr) \, dx - \int_{-\infty}^0 F(x) \, dx, $$ also with convergent integrals.

Infinite expected values
Expected values as defined above are automatically finite numbers. However, in many cases it is fundamental to be able to consider expected values of $±∞$. This is intuitive, for example, in the case of the St. Petersburg paradox, in which one considers a random variable with possible outcomes $x_{i} = 2^{i}$, with associated probabilities $p_{i} = 2^{−i}$, for $μ$ ranging over all positive integers. According to the summation formula in the case of random variables with countably many outcomes, one has $$ \operatorname{E}[X]= \sum_{i=1}^\infty x_i\,p_i = 2\cdot \frac{1}{2}+4\cdot\frac{1}{4} + 8\cdot\frac{1}{8}+ 16\cdot\frac{1}{16}+ \cdots = 1 + 1 + 1 + 1 + \cdots.$$ It is natural to say that the expected value equals $+∞$.

There is a rigorous mathematical theory underlying such ideas, which is often taken as part of the definition of the Lebesgue integral. The first fundamental observation is that, whichever of the above definitions are followed, any nonnegative random variable whatsoever can be given an unambiguous expected value; whenever absolute convergence fails, then the expected value can be defined as $+∞$. The second fundamental observation is that any random variable can be written as the difference of two nonnegative random variables. Given a random variable $𝑚$, one defines the positive and negative parts by $X^{ +} = max(X, 0)$ and $X^{ −} = −min(X, 0)$. These are nonnegative random variables, and it can be directly checked that $X = X^{ +} − X^{ −}$. Since $E[X^{ +}]$ and $E[X^{ −}]$ are both then defined as either nonnegative numbers or $+∞$, it is then natural to define: $$ \operatorname{E}[X] = \begin{cases} \operatorname{E}[X^+] - \operatorname{E}[X^-] & \text{if } \operatorname{E}[X^+] < \infty \text{ and } \operatorname{E}[X^-] < \infty;\\ +\infty & \text{if } \operatorname{E}[X^+] = \infty \text{ and } \operatorname{E}[X^-] < \infty;\\ -\infty & \text{if } \operatorname{E}[X^+] < \infty \text{ and } \operatorname{E}[X^-] = \infty;\\ \text{undefined} & \text{if } \operatorname{E}[X^+] = \infty \text{ and } \operatorname{E}[X^-] = \infty. \end{cases} $$

According to this definition, $E[X]$ exists and is finite if and only if $E[X^{ +}]$ and $E[X^{ −}]$ are both finite. Due to the formula $|X| = X^{ +} + X^{ −}$, this is the case if and only if $E|X|$ is finite, and this is equivalent to the absolute convergence conditions in the definitions above. As such, the present considerations do not define finite expected values in any cases not previously considered; they are only useful for infinite expectations.
 * In the case of the St. Petersburg paradox, one has $X^{ −} = 0$ and so $E[X] = +∞$ as desired.
 * Suppose the random variable $i$ takes values $1, −2,3, −4, ...$ with respective probabilities $6π^{−2}, 6(2π)^{−2}, 6(3π)^{−2}, 6(4π)^{−2}, ...$. Then it follows that $X^{ +}$ takes value $2k−1$ with probability $6((2k−1)π)^{−2}$ for each positive integer $X$, and takes value $0$ with remaining probability. Similarly, $X^{ −}$ takes value $2k$ with probability $6(2kπ)^{−2}$ for each positive integer $X$ and takes value $0$ with remaining probability. Using the definition for non-negative random variables, one can show that both $E[X^{ +}] = ∞$ and $E[X^{ −}] = ∞$ (see Harmonic series). Hence, in this case the expectation of $k$ is undefined.
 * Similarly, the Cauchy distribution, as discussed above, has undefined expectation.

Expected values of common distributions
The following table gives the expected values of some commonly occurring probability distributions. The third column gives the expected values both in the form immediately given by the definition, as well as in the simplified form obtained by computation therefrom. The details of these computations, which are not always straightforward, can be found in the indicated references.

Properties
The basic properties below (and their names in bold) replicate or follow immediately from those of Lebesgue integral. Note that the letters "a.s." stand for "almost surely"—a central property of the Lebesgue integral. Basically, one says that an inequality like $$X \geq 0$$ is true almost surely, when the probability measure attributes zero-mass to the complementary event $$\left\{ X < 0 \right\}.$$ \operatorname{E}[X + Y] &=  \operatorname{E}[X] + \operatorname{E}[Y], \\ \operatorname{E}[aX]   &= a \operatorname{E}[X], \end{align} $$ whenever the right-hand side is well-defined. By induction, this means that the expected value of the sum of any finite number of random variables is the sum of the expected values of the individual random variables, and the expected value scales linearly with a multiplicative constant. Symbolically, for $$N$$ random variables $$X_{i}$$ and constants $$a_{i} (1\leq i \leq N),$$ we have $ \operatorname{E}\left[\sum_{i=1}^{N}a_{i}X_{i}\right] = \sum_{i=1}^{N}a_{i}\operatorname{E}[X_{i}].$ If we think of the set of random variables with finite expected value as forming a vector space, then the linearity of expectation implies that the expected value is a linear form on this vector space.
 * Non-negativity: If $$X \geq 0$$ (a.s.), then $$\operatorname{E}[X] \geq 0.$$
 * Linearity of expectation: The expected value operator (or expectation operator) $$\operatorname{E}[\cdot]$$ is linear in the sense that, for any random variables $$X$$ and $$Y,$$ and a constant $$a,$$ $$\begin{align}
 * Monotonicity: If $$X\leq Y$$ (a.s.), and both $$\operatorname{E}[X]$$ and $$\operatorname{E}[Y]$$ exist, then $$\operatorname{E}[X]\leq\operatorname{E}[Y].$$ Proof follows from the linearity and the non-negativity property for $$Z=Y-X,$$ since $$Z\geq 0$$ (a.s.).
 * Non-degeneracy: If $$\operatorname{E}[|X|]=0,$$ then $$X=0$$ (a.s.).
 * If $$X = Y$$ (a.s.), then $$\operatorname{E}[X] = \operatorname{E}[ Y].$$ In other words, if X and Y are random variables that take different values with probability zero, then the expectation of X will equal the expectation of Y.
 * If $$X = c$$ (a.s.) for some real number $k$, then $$\operatorname{E}[X] = c.$$ In particular, for a random variable $$X$$ with well-defined expectation, $$\operatorname{E}[\operatorname{E}[X]] = \operatorname{E}[X].$$ A well defined expectation implies that there is one number, or rather, one constant that defines the expected value. Thus follows that the expectation of this constant is just the original expected value.
 * As a consequence of the formula $|X| = X + X$ as discussed above, together with the triangle inequality, it follows that for any random variable $$X$$ with well-defined expectation, one has $$|\operatorname{E}[X]| \leq \operatorname{E}|X|.$$
 * Let $1_{A}$ denote the indicator function of an event $X$, then $E[1_{A}]$ is given by the probability of $c$. This is nothing but a different way of stating the expectation of a Bernoulli random variable, as calculated in the table above.
 * Formulas in terms of CDF: If $$F(x)$$ is the cumulative distribution function of a random variable $A$, then where the values on both sides are well defined or not well defined simultaneously, and the integral is taken in the sense of Lebesgue-Stieltjes. As a consequence of integration by parts as applied to this representation of $E[X]$, it can be proved that $$ \operatorname{E}[X] = \int_0^\infty (1-F(x))\,dx - \int^0_{-\infty} F(x)\,dx,$$ with the integrals taken in the sense of Lebesgue. As a special case, for any random variable $A$ valued in the nonnegative integers ${0, 1, 2, 3, ...}$, one has $$ \operatorname{E}[X] = \sum _{n=0}^\infty \Pr(X>n), $$ where $X$ denotes the underlying probability measure.
 * Non-multiplicativity: In general, the expected value is not multiplicative, i.e. $$\operatorname{E}[XY]$$ is not necessarily equal to $$\operatorname{E}[X]\cdot \operatorname{E}[Y].$$ If $$X$$ and $$Y$$ are independent, then one can show that $$\operatorname{E}[XY]=\operatorname{E}[X] \operatorname{E}[Y].$$ If the random variables are dependent, then generally $$\operatorname{E}[XY] \neq \operatorname{E}[X] \operatorname{E}[Y],$$ although in special cases of dependency the equality may hold.
 * Law of the unconscious statistician: The expected value of a measurable function of $$X,$$ $$g(X),$$ given that $$X$$ has a probability density function $$f(x),$$ is given by the inner product of $$f$$ and $$g$$: $$\operatorname{E}[g(X)] = \int_{\R} g(x) f(x)\, dx .$$ This formula also holds in multidimensional case, when $$g$$ is a function of several random variables, and $$f$$ is their joint density.

Inequalities
Concentration inequalities control the likelihood of a random variable taking on large values. Markov's inequality is among the best-known and simplest to prove: for a nonnegative random variable $X$ and any positive number $P$, it states that $$ \operatorname{P}(X\geq a)\leq\frac{\operatorname{E}[X]}{a}. $$

If $X$ is any random variable with finite expectation, then Markov's inequality may be applied to the random variable $|X−E[X]|^{2}$ to obtain Chebyshev's inequality $$ \operatorname{P}(|X-\text{E}[X]|\geq a)\leq\frac{\operatorname{Var}[X]}{a^2}, $$ where $Var$ is the variance. These inequalities are significant for their nearly complete lack of conditional assumptions. For example, for any random variable with finite expectation, the Chebyshev inequality implies that there is at least a 75% probability of an outcome being within two standard deviations of the expected value. However, in special cases the Markov and Chebyshev inequalities often give much weaker information than is otherwise available. For example, in the case of an unweighted dice, Chebyshev's inequality says that odds of rolling between 1 and 6 is at least 53%; in reality, the odds are of course 100%. The Kolmogorov inequality extends the Chebyshev inequality to the context of sums of random variables.

The following three inequalities are of fundamental importance in the field of mathematical analysis and its applications to probability theory. f(\operatorname{E}(X)) \leq \operatorname{E} (f(X)). $$ Part of the assertion is that the negative part of $f: R → R$ has finite expectation, so that the right-hand side is well-defined (possibly infinite). Convexity of $a$ can be phrased as saying that the output of the weighted average of two inputs under-estimates the same weighted average of the two outputs; Jensen's inequality extends this to the setting of completely general weighted averages, as represented by the expectation. In the special case that $f(X)$ for positive numbers $f(x) = |x|^{t/s}$, one obtains the Lyapunov inequality $$ \left(\operatorname{E}|X|^s\right)^{1/s} \leq \left(\operatorname{E}|X|^t\right)^{1/t}. $$ This can also be proved by the Hölder inequality. In measure theory, this is particularly notable for proving the inclusion $s < t$ of $L^{s} ⊂ L^{t}$, in the special case of probability spaces. \operatorname{E}|XY|\leq(\operatorname{E}|X|^p)^{1/p}(\operatorname{E}|Y|^q)^{1/q}. $$ for any random variables $X$ and $X$. The special case of $L^{p} spaces$ is called the Cauchy–Schwarz inequality, and is particularly well-known. \Bigl(\operatorname{E}|X+Y|^p\Bigr)^{1/p}\leq\Bigl(\operatorname{E}|X|^p\Bigr)^{1/p}+\Bigl(\operatorname{E}|Y|^p\Bigr)^{1/p}. $$ The Hölder and Minkowski inequalities can be extended to general measure spaces, and are often given in that context. By contrast, the Jensen inequality is special to the case of probability spaces.
 * Jensen's inequality: Let $p > 1$ be a convex function and $f$ a random variable with finite expectation. Then $$
 * Hölder's inequality: if $q > 1$ and $p^{ −1} + q^{ −1} = 1$ are numbers satisfying $p = q = 2$, then $$
 * Minkowski inequality: given any number $p ≥ 1$, for any random variables $X$ and $Y$ with $E|X|^{p}$ and $E|Y|^{p}$ both finite, it follows that $E|X + Y|^{p}$ is also finite and $$

Expectations under convergence of random variables
In general, it is not the case that $$\operatorname{E}[X_n] \to \operatorname{E}[X]$$ even if $$X_n\to X$$ pointwise. Thus, one cannot interchange limits and expectation, without additional conditions on the random variables. To see this, let $$U$$ be a random variable distributed uniformly on $$[0,1].$$ For $$n\geq 1,$$ define a sequence of random variables $$X_n = n \cdot \mathbf{1}\left\{ U \in \left(0,\tfrac{1}{n}\right)\right\},$$ with $$\mathbf{1}\{A\}$$ being the indicator function of the event $$A.$$ Then, it follows that $$X_n \to 0$$ pointwise. But, $$\operatorname{E}[X_n] = n \cdot \Pr\left(U \in \left[ 0, \tfrac{1}{n}\right] \right) = n \cdot \tfrac{1}{n} = 1$$ for each $$n.$$ Hence, $$\lim_{n \to \infty} \operatorname{E}[X_n] = 1 \neq 0 = \operatorname{E}\left[ \lim_{n \to \infty} X_n \right].$$

Analogously, for general sequence of random variables $$\{ Y_n : n \geq 0\},$$ the expected value operator is not $$\sigma$$-additive, i.e. $$\operatorname{E}\left[\sum^\infty_{n=0} Y_n\right] \neq \sum^\infty_{n=0}\operatorname{E}[Y_n].$$

An example is easily obtained by setting $$Y_0 = X_1$$ and $$Y_n = X_{n+1} - X_n$$ for $$n \geq 1,$$ where $$X_n$$ is as in the previous example.

A number of convergence results specify exact conditions which allow one to interchange limits and expectations, as specified below. \operatorname{E}\left[\sum^\infty_{i=0}X_i\right] = \sum^\infty_{i=0}\operatorname{E}[X_i]. $$
 * Monotone convergence theorem: Let $$\{X_n : n \geq 0\}$$ be a sequence of random variables, with $$0 \leq X_n \leq X_{n+1}$$ (a.s) for each $$n \geq 0.$$ Furthermore, let $$X_n \to X$$ pointwise. Then, the monotone convergence theorem states that $$\lim_n\operatorname{E}[X_n]=\operatorname{E}[X].$$ Using the monotone convergence theorem, one can show that expectation indeed satisfies countable additivity for non-negative random variables. In particular, let $$\{X_i\}_{i=0}^\infty$$ be non-negative random variables. It follows from the monotone convergence theorem that $$
 * Fatou's lemma: Let $$\{ X_n \geq 0 : n \geq 0\}$$ be a sequence of non-negative random variables. Fatou's lemma states that $$\operatorname{E}[\liminf_n X_n] \leq \liminf_n \operatorname{E}[X_n].$$ Corollary. Let $$X_n \geq 0$$ with $$\operatorname{E}[X_n] \leq C$$ for all $$n \geq 0.$$ If $$X_n \to X$$ (a.s), then $$\operatorname{E}[X] \leq C.$$  Proof is by observing that $ X = \liminf_n X_n$  (a.s.) and applying Fatou's lemma.
 * Dominated convergence theorem: Let $$\{X_n : n \geq 0 \}$$ be a sequence of random variables. If $$X_n\to X$$ pointwise (a.s.), $$|X_n|\leq Y \leq +\infty$$ (a.s.), and $$\operatorname{E}[Y]<\infty.$$ Then, according to the dominated convergence theorem,
 * $$\operatorname{E}|X| \leq \operatorname{E}[Y] <\infty$$;
 * $$\lim_n\operatorname{E}[X_n]=\operatorname{E}[X]$$
 * $$\lim_n\operatorname{E}|X_n - X| = 0.$$
 * Uniform integrability: In some cases, the equality $$\lim_n\operatorname{E}[X_n]=\operatorname{E}[\lim_n X_n]$$ holds when the sequence $$\{X_n\}$$ is uniformly integrable.

Relationship with characteristic function
The probability density function $$f_X$$ of a scalar random variable $$X$$ is related to its characteristic function $$\varphi_X$$ by the inversion formula: $$f_X(x) = \frac{1}{2\pi}\int_{\mathbb{R}} e^{-itx}\varphi_X(t) \, dt.$$

For the expected value of $$g(X)$$ (where $$g:{\mathbb R}\to{\mathbb R}$$ is a Borel function), we can use this inversion formula to obtain $$\operatorname{E}[g(X)] = \frac{1}{2\pi} \int_\Reals g(x) \left[ \int_\Reals e^{-itx}\varphi_X(t) \, dt \right] dx.$$

If $$\operatorname{E}[g(X)]$$ is finite, changing the order of integration, we get, in accordance with Fubini–Tonelli theorem, $$\operatorname{E}[g(X)] = \frac{1}{2\pi} \int_\Reals G(t) \varphi_X(t) \, dt,$$ where $$G(t) = \int_\Reals g(x) e^{-itx} \, dx$$ is the Fourier transform of $$g(x).$$ The expression for $$\operatorname{E}[g(X)]$$ also follows directly from the Plancherel theorem.

Uses and applications
The expectation of a random variable plays an important role in a variety of contexts.

In statistics, where one seeks estimates for unknown parameters based on available data gained from samples, the sample mean serves as an estimate for the expectation, and is itself a random variable. In such settings, the sample mean is considered to meet the desirable criterion for a "good" estimator in being unbiased; that is, the expected value of the estimate is equal to the true value of the underlying parameter.

For a different example, in decision theory, an agent making an optimal choice in the context of incomplete information is often assumed to maximize the expected value of their utility function.

It is possible to construct an expected value equal to the probability of an event by taking the expectation of an indicator function that is one if the event has occurred and zero otherwise. This relationship can be used to translate properties of expected values into properties of probabilities, e.g. using the law of large numbers to justify estimating probabilities by frequencies.

The expected values of the powers of X are called the moments of X; the moments about the mean of X are expected values of powers of $X − E[X]$. The moments of some random variables can be used to specify their distributions, via their moment generating functions.

To empirically estimate the expected value of a random variable, one repeatedly measures observations of the variable and computes the arithmetic mean of the results. If the expected value exists, this procedure estimates the true expected value in an unbiased manner and has the property of minimizing the sum of the squares of the residuals (the sum of the squared differences between the observations and the estimate). The law of large numbers demonstrates (under fairly mild conditions) that, as the size of the sample gets larger, the variance of this estimate gets smaller.

This property is often exploited in a wide variety of applications, including general problems of statistical estimation and machine learning, to estimate (probabilistic) quantities of interest via Monte Carlo methods, since most quantities of interest can be written in terms of expectation, e.g. $$\operatorname{P}({X \in \mathcal{A}}) = \operatorname{E}[{\mathbf 1}_{\mathcal{A}}],$$ where $${\mathbf 1}_{\mathcal{A}}$$ is the indicator function of the set $$\mathcal{A}.$$

In classical mechanics, the center of mass is an analogous concept to expectation. For example, suppose X is a discrete random variable with values xi and corresponding probabilities pi. Now consider a weightless rod on which are placed weights, at locations xi along the rod and having masses pi (whose sum is one). The point at which the rod balances is E[X].

Expected values can also be used to compute the variance, by means of the computational formula for the variance $$\operatorname{Var}(X)= \operatorname{E}[X^2] - (\operatorname{E}[X])^2.$$

A very important application of the expectation value is in the field of quantum mechanics. The expectation value of a quantum mechanical operator $$\hat{A}$$ operating on a quantum state vector $$|\psi\rangle$$ is written as $$\langle\hat{A}\rangle = \langle\psi|\hat{A}|\psi\rangle.$$ The uncertainty in $$\hat{A}$$ can be calculated by the formula $$(\Delta A)^2 = \langle\hat{A}^2\rangle - \langle \hat{A} \rangle^2$$.

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

title: Expected value


text:
Talk:Euthanasia

Semi-protected edit request on 19 June 2023
Kitluvsmangos4ever (talk) 21:54, 19 June 2023 (UTC)

It is important to recognize the complexity of bioethical subjects in Christianity because although there is simply one sacred text that encapsulates the Christian ideology, there is no distinct set of rules that every Christian could agree upon. Rather, different groups or denominations may interpret their faith in dissimilar manners, thus creating the broad and varied perspectives of ethical issues in Christianity.1

As God is the creator of life in Christian belief, it is believed that the followers of Christianity are to “submit to what God had ordained” or willed.2 Therefore, one’s illness or death is viewed to be brought upon by God in accordance with his plan. Consequently, by interfering with the end of life, doctors were once believed to be defying God’s will. However, with time, this belief has evolved and doctors are now viewed as fulfilling God’s will by healing the sick.2

Based on research, there is no direct discussion about the topics of euthanasia or physician assisted suicide in the Christian bible. Rather, the bible indirectly refers such topics through the discussion of creation, human value, and suffering/death.3 Creation- As God is the sole creator of the universe, “any human interference with hastening the end of life” would be considered a violation of God’s will. Human value- According to the bible, those who do not value God’s creation of human life will face consequences upon God’s judgement. Therefore, killing and the taking of life is a deliberate action that “should only occur at God’s command”. Suffering/Death- Suffering is seen as consequence from God for turning away from him. On the other hand, the bible describes suffering to sometimes be inexplainable. Therefore, the bible explains that life, along with suffering and death, “occurs along the path to a better future of eternal life without suffering”.

Furthermore, the rationale behind opposition to euthanasia slightly differs between various Christian denominations: Roman Catholic Perspectives- The Roman Catholic Church condemns euthanasia and assisted suicide as morally wrong. As paragraph 2324 of the Catechism of the Catholic Church states, "Intentional euthanasia, whatever its forms or motives, is murder. It is gravely contrary to the dignity of the human person and to the respect due to the living God, his Creator". Because of this, per the Declaration on Euthanasia and the Second Vatican Council, the practice of euthanasia is unacceptable within the Church.[73] Eastern Orthodox Perspectives- The Orthodox Church in America, along with other Eastern Orthodox Churches, also opposes euthanasia stating that "euthanasia is the deliberate cessation of human life, and, as such, must be condemned as murder."[74] Therefore, this denomination believes that accepting illness according to God’s plan allows for spiritual growth to take place.3 Protestant Perspectives- Among Protestant denominations, the Episcopal Church passed a resolution in 1991 opposing euthanasia and assisted suicide stating that the act is “mistaken, deceptive, and evil” and that it is "morally wrong and unacceptable to take a human life to relieve the suffering caused by incurable illnesses."3 [74] Other non-Catholic churches which oppose euthanasia include: •	Assemblies of God[75] •	The Church of Jesus Christ of Latter-day Saints[76] •	Church of the Nazarene[77] •	Evangelical Lutheran Church in America[78] •	Presbyterian Church in America[79] •	Lutheran Church–Missouri Synod[80] •	Reformed Church in America[81] •	Salvation Army[82] •	Seventh-day Adventist Church[75] •	Southern Baptist Convention[83] •	United Methodist Church[74]

Partially in favor of The Church of England accepts passive euthanasia under some circumstances, but is strongly against active euthanasia, and has led to the opposition against recent attempts to legalize it.[84] The United Church of Canada accepts passive euthanasia under some circumstances, but is in general against active euthanasia, with growing acceptance now that active euthanasia has been partly legalized in Canada.[85] The Waldensians take a liberal stance on Euthanasia and allow the decision to lie with individuals.[86][87]

Works Cited 1.	Widdows, H (2007) Christian approaches to bioethics. In Richard E. Ashcroft, Angus Dawson, Heather Draper, and John R. McMillan (eds.) Principles of Heath Care Ethics Second Edition. Chichester, West Sussex: Wiley and Sons. pp. 99 - 107. 2.	Badham, P. (n.d.). Guidance on conducting a systematic literature review - sage journals. SHOULD CHRISTIANS ACCEPT THE VALIDITY OF VOLUNTARY EUTHANASIA? https://journals.sagepub.com/doi/pdf/10.1177/0739456X17723971 3.	Grove, G., Lovell, M., & Best, M. (2022). Perspectives of Major World Religions regarding Euthanasia and Assisted Suicide: A Comparative Analysis. Journal of religion and health, 61(6), 4758–4782. https://doi.org/10.1007/s10943-022-01498-5Karma paper
 * Red question icon with gradient background.svg Not done: it's not clear what changes you want to be made. Please mention the specific changes in a "change X to Y" format and provide a reliable source if appropriate. M.Bitton (talk) 23:09, 20 June 2023 (UTC)

 * Change this Section:
 * Christianity Broadly against The Roman Catholic Church condemns euthanasia and assisted suicide as morally wrong. As paragraph 2324 of the Catechism of the Catholic Church states, "Intentional euthanasia, whatever its forms or motives, is murder. It is gravely contrary to the dignity of the human person and to the respect due to the living God, his Creator". Because of this, per the Declaration on Euthanasia, the practice is unacceptable within the Church.[73] The Orthodox Church in America, along with other Eastern Orthodox Churches, also opposes euthanasia stating that "euthanasia is the deliberate cessation of human life, and, as such, must be condemned as murder."[74]
 * Many non-Catholic churches in the United States take a stance against euthanasia. Among Protestant denominations, the Episcopal Church passed a resolution in 1991 opposing euthanasia and assisted suicide stating that it is "morally wrong and unacceptable to take a human life to relieve the suffering caused by incurable illnesses."[74] Protestant and other non-Catholic churches which oppose euthanasia include:
 * Assemblies of God[75] The Church of Jesus Christ of Latter-day Saints[76] Church of the Nazarene[77] Evangelical Lutheran Church in America[78] Presbyterian Church in America[79] Lutheran Church–Missouri Synod[80] Reformed Church in America[81] Salvation Army[82] Seventh-day Adventist Church[75] Southern Baptist Convention[83] United Methodist Church[74]
 * To:
 * Christianity It is important to recognize the complexity of bioethical subjects in Christianity because although there is simply one sacred text that encapsulates the Christian ideology, there is no distinct set of rules that every Christian could agree upon. Rather, different groups or denominations may interpret their faith in dissimilar manners, thus creating the broad and varied perspectives of ethical issues in Christianity.1
 * As God is the creator of life in Christian belief, it is believed that the followers of Christianity are to “submit to what God had ordained” or willed.2 Therefore, one’s illness or death is viewed to be brought upon by God in accordance with his plan. Consequently, by interfering with the end of life, doctors were once believed to be defying God’s will. However, with time, this belief has evolved and doctors are now viewed as fulfilling God’s will by healing the sick.2
 * Based on research, there is no direct discussion about the topics of euthanasia or physician assisted suicide in the Christian bible. Rather, the bible indirectly refers such topics through the discussion of creation, human value, and suffering/death.3 Creation- As God is the sole creator of the universe, “any human interference with hastening the end of life” would be considered a violation of God’s will. Human value- According to the bible, those who do not value God’s creation of human life will face consequences upon God’s judgement. Therefore, killing and the taking of life is a deliberate action that “should only occur at God’s command”. Suffering/Death- Suffering is seen as consequence from God for turning away from him. On the other hand, the bible describes suffering to sometimes be inexplainable. Therefore, the bible explains that life, along with suffering and death, “occurs along the path to a better future of eternal life without suffering”.
 * Furthermore, the rationale behind opposition to euthanasia slightly differs between various Christian denominations: Roman Catholic Perspectives- The Roman Catholic Church condemns euthanasia and assisted suicide as morally wrong. As paragraph 2324 of the Catechism of the Catholic Church states, "Intentional euthanasia, whatever its forms or motives, is murder. It is gravely contrary to the dignity of the human person and to the respect due to the living God, his Creator". Because of this, per the Declaration on Euthanasia and the Second Vatican Council, the practice of euthanasia is unacceptable within the Church.[73] Eastern Orthodox Perspectives- The Orthodox Church in America, along with other Eastern Orthodox Churches, also opposes euthanasia stating that "euthanasia is the deliberate cessation of human life, and, as such, must be condemned as murder."[74] Therefore, this denomination believes that accepting illness according to God’s plan allows for spiritual growth to take place.3 Protestant Perspectives- Among Protestant denominations, the Episcopal Church passed a resolution in 1991 opposing euthanasia and assisted suicide stating that the act is “mistaken, deceptive, and evil” and that it is "morally wrong and unacceptable to take a human life to relieve the suffering caused by incurable illnesses."3 [74] Other non-Catholic churches which oppose euthanasia include: • Assemblies of God[75] • The Church of Jesus Christ of Latter-day Saints[76] • Church of the Nazarene[77] • Evangelical Lutheran Church in America[78] • Presbyterian Church in America[79] • Lutheran Church–Missouri Synod[80] • Reformed Church in America[81] • Salvation Army[82] • Seventh-day Adventist Church[75] • Southern Baptist Convention[83] • United Methodist Church[74]
 * Partially in favor of The Church of England accepts passive euthanasia under some circumstances, but is strongly against active euthanasia, and has led to the opposition against recent attempts to legalize it.[84] The United Church of Canada accepts passive euthanasia under some circumstances, but is in general against active euthanasia, with growing acceptance now that active euthanasia has been partly legalized in Canada.[85] The Waldensians take a liberal stance on Euthanasia and allow the decision to lie with individuals.[86][87]
 * Works Cited 1. Widdows, H (2007) Christian approaches to bioethics. In Richard E. Ashcroft, Angus Dawson, Heather Draper, and John R. McMillan (eds.) Principles of Heath Care Ethics Second Edition. Chichester, West Sussex: Wiley and Sons. pp. 99 - 107. 2. Badham, P. (n.d.). Guidance on conducting a systematic literature review - sage journals. SHOULD CHRISTIANS ACCEPT THE VALIDITY OF VOLUNTARY EUTHANASIA? https://journals.sagepub.com/doi/pdf/10.1177/0739456X17723971 3. Grove, G., Lovell, M., & Best, M. (2022). Perspectives of Major World Religions regarding Euthanasia and Assisted Suicide: A Comparative Analysis. Journal of religion and health, 61(6), 4758–4782. https://doi.org/10.1007/s10943-022-01498-5Karma paper Kitluvsmangos4ever (talk) 14:36, 20 June 2023 (UTC) 2600:1007:A000:654B:9D04:7C1:9DC5:2576 (talk) 13:27, 23 June 2023 (UTC)

Semi-protected edit request on 20 June 2023
Change this Section:

Christianity Broadly against The Roman Catholic Church condemns euthanasia and assisted suicide as morally wrong. As paragraph 2324 of the Catechism of the Catholic Church states, "Intentional euthanasia, whatever its forms or motives, is murder. It is gravely contrary to the dignity of the human person and to the respect due to the living God, his Creator". Because of this, per the Declaration on Euthanasia, the practice is unacceptable within the Church.[73] The Orthodox Church in America, along with other Eastern Orthodox Churches, also opposes euthanasia stating that "euthanasia is the deliberate cessation of human life, and, as such, must be condemned as murder."[74]

Many non-Catholic churches in the United States take a stance against euthanasia. Among Protestant denominations, the Episcopal Church passed a resolution in 1991 opposing euthanasia and assisted suicide stating that it is "morally wrong and unacceptable to take a human life to relieve the suffering caused by incurable illnesses."[74] Protestant and other non-Catholic churches which oppose euthanasia include:

Assemblies of God[75] The Church of Jesus Christ of Latter-day Saints[76] Church of the Nazarene[77] Evangelical Lutheran Church in America[78] Presbyterian Church in America[79] Lutheran Church–Missouri Synod[80] Reformed Church in America[81] Salvation Army[82] Seventh-day Adventist Church[75] Southern Baptist Convention[83] United Methodist Church[74]

To:

Christianity It is important to recognize the complexity of bioethical subjects in Christianity because although there is simply one sacred text that encapsulates the Christian ideology, there is no distinct set of rules that every Christian could agree upon. Rather, different groups or denominations may interpret their faith in dissimilar manners, thus creating the broad and varied perspectives of ethical issues in Christianity.1

As God is the creator of life in Christian belief, it is believed that the followers of Christianity are to “submit to what God had ordained” or willed.2 Therefore, one’s illness or death is viewed to be brought upon by God in accordance with his plan. Consequently, by interfering with the end of life, doctors were once believed to be defying God’s will. However, with time, this belief has evolved and doctors are now viewed as fulfilling God’s will by healing the sick.2

Based on research, there is no direct discussion about the topics of euthanasia or physician assisted suicide in the Christian bible. Rather, the bible indirectly refers such topics through the discussion of creation, human value, and suffering/death.3 Creation- As God is the sole creator of the universe, “any human interference with hastening the end of life” would be considered a violation of God’s will. Human value- According to the bible, those who do not value God’s creation of human life will face consequences upon God’s judgement. Therefore, killing and the taking of life is a deliberate action that “should only occur at God’s command”. Suffering/Death- Suffering is seen as consequence from God for turning away from him. On the other hand, the bible describes suffering to sometimes be inexplainable. Therefore, the bible explains that life, along with suffering and death, “occurs along the path to a better future of eternal life without suffering”.

Furthermore, the rationale behind opposition to euthanasia slightly differs between various Christian denominations: Roman Catholic Perspectives- The Roman Catholic Church condemns euthanasia and assisted suicide as morally wrong. As paragraph 2324 of the Catechism of the Catholic Church states, "Intentional euthanasia, whatever its forms or motives, is murder. It is gravely contrary to the dignity of the human person and to the respect due to the living God, his Creator". Because of this, per the Declaration on Euthanasia and the Second Vatican Council, the practice of euthanasia is unacceptable within the Church.[73] Eastern Orthodox Perspectives- The Orthodox Church in America, along with other Eastern Orthodox Churches, also opposes euthanasia stating that "euthanasia is the deliberate cessation of human life, and, as such, must be condemned as murder."[74] Therefore, this denomination believes that accepting illness according to God’s plan allows for spiritual growth to take place.3 Protestant Perspectives- Among Protestant denominations, the Episcopal Church passed a resolution in 1991 opposing euthanasia and assisted suicide stating that the act is “mistaken, deceptive, and evil” and that it is "morally wrong and unacceptable to take a human life to relieve the suffering caused by incurable illnesses."3 [74] Other non-Catholic churches which oppose euthanasia include: • Assemblies of God[75] • The Church of Jesus Christ of Latter-day Saints[76] • Church of the Nazarene[77] • Evangelical Lutheran Church in America[78] • Presbyterian Church in America[79] • Lutheran Church–Missouri Synod[80] • Reformed Church in America[81] • Salvation Army[82] • Seventh-day Adventist Church[75] • Southern Baptist Convention[83] • United Methodist Church[74]

Partially in favor of The Church of England accepts passive euthanasia under some circumstances, but is strongly against active euthanasia, and has led to the opposition against recent attempts to legalize it.[84] The United Church of Canada accepts passive euthanasia under some circumstances, but is in general against active euthanasia, with growing acceptance now that active euthanasia has been partly legalized in Canada.[85] The Waldensians take a liberal stance on Euthanasia and allow the decision to lie with individuals.[86][87]

Works Cited 1. Widdows, H (2007) Christian approaches to bioethics. In Richard E. Ashcroft, Angus Dawson, Heather Draper, and John R. McMillan (eds.) Principles of Heath Care Ethics Second Edition. Chichester, West Sussex: Wiley and Sons. pp. 99 - 107. 2. Badham, P. (n.d.). Guidance on conducting a systematic literature review - sage journals. SHOULD CHRISTIANS ACCEPT THE VALIDITY OF VOLUNTARY EUTHANASIA? https://journals.sagepub.com/doi/pdf/10.1177/0739456X17723971 3. Grove, G., Lovell, M., & Best, M. (2022). Perspectives of Major World Religions regarding Euthanasia and Assisted Suicide: A Comparative Analysis. Journal of religion and health, 61(6), 4758–4782. https://doi.org/10.1007/s10943-022-01498-5Karma paper Kitluvsmangos4ever (talk) 14:36, 20 June 2023 (UTC)

 * Object This is not the place to have an in-depth discussion about bioethics and the church(es). Beside that, it is not neutral. The Banner  talk 10:29, 24 June 2023 (UTC)
 * Red information icon with gradient background.svg Not done for now: please establish a consensus for this alteration before using the template. Lightoil (talk) 12:26, 25 June 2023 (UTC)

Buddhism
"Buddhism

Since enlightenment is a major goal in the Buddhist religion, human existence is highly valued and therefore protected. In the Buddhist belief of rebirth and karma, one has infinite lifetimes until the process is completed through enlightenment.[90] Here, it is taught that one must experience one’s own karma as failing to do so could lead to other consequences.

In Buddhism, all disorders originate from the mental environment due to negative states of mind. This is because illness and other inevitable realities are linked to one’s karma.[90]

The decision made by a physician to ease the pain and suffering of terminally ill patients is viewed negatively in Buddhism. Rather, they view this act carried out by physicians as “harm intended as help”.[90] Moreover, if one’s karma (suffering) were to be intentionally disrupted, that suffering would then be experienced again in another life. Providing large doses of narcotics to a terminally-ill patient is heavily looked down upon as it does not allow for the patient to “leave in a peaceful manner”. Simply put, Buddhism advocates for “hospice care, not euthanasia”.[90] Ultimately, Buddhism teaches its followers to “accept and work with all aspects of life, including death”.[90]

Buddhism also teaches that physicians perform the act because the patient’s suffering and pain disturb the doctor’s mind. The physician then projects this disturbance onto the patient themselves through the act of euthanasia.[91] Therefore, for the benefit of both the patient and the physician, euthanasia is an unacceptable deed in Buddhism.

These Buddhist ethics are put in place to not only respect the tenants and virtues of Buddhism but also to prevent any selfish desires that either the family or physician may have.[91] The goal here is to protect both the patient as well as their enlightenment, and rejecting euthanasia adheres to this Buddhist goal."

Citations [90]. Euthanasia: A Buddhist perspective - JSTOR. Euthanasia: A Buddhist Perspective. (n.d.). https://www.jstor.org/stable/27505857 [91]. Chaicharoen, P., & Ratanakul, P. (n.d.). Letting-Go or Killing: Thai Buddhist Perspectives on Euthanasia. Eubios Ethics Institute -. https://www.eubios.info/EJ82/ej82c.htm Kitluvsmangos4ever (talk) 22:07, 8 August 2023 (UTC)

Lead sentence does not accurately define euthanasia
Someone who can edit this locked page, please adjust the lead sentence to reflect that both the term and concept of "euthanasia" is not strictly proscribed to "ending pain and suffering" (which ultimately is a subjective matter). It also fundamentally includes involuntary euthanasia (as introduced, belatedly, in paragraph three), in which a subject's life is ended without consent (effectively to relieve the "pain and suffering" of whomever is responsible for it, whether individually, collectively, or by government policy.

This fundamental distinction needs to be worked into the defining term used in the lead sentence, which otherwise, simply, misleads (by offering an over-narrow description of the term without indicating a very different interpretation of the term exists). Different enough that it is construed in many instances as murder, and in others a matter of government policy (in one form or another) interpreted by others outside it variously as murder on the small scale and "genocide" or related concepts on the large. 2601:196:180:DC0:566:1463:F27E:721F (talk) 04:44, 19 January 2024 (UTC)
 * Do you have any sources to back up your claim? The Banner  talk 09:09, 19 January 2024 (UTC)

title: Talk:Euthanasia


text:
Electric light

An electric light, lamp, or light bulb is an electrical component that produces light. It is the most common form of artificial lighting. Lamps usually have a base made of ceramic, metal, glass, or plastic which secures the lamp in the socket of a light fixture, which is often called a "lamp" as well. The electrical connection to the socket may be made with a screw-thread base, two metal pins, two metal caps or a bayonet mount.

The three main categories of electric lights are incandescent lamps, which produce light by a filament heated white-hot by electric current, gas-discharge lamps, which produce light by means of an electric arc through a gas, such as fluorescent lamps, and LED lamps, which produce light by a flow of electrons across a band gap in a semiconductor.

The energy efficiency of electric lighting has increased radically since the first demonstration of arc lamps and the incandescent light bulb of the 19th century. Modern electric light sources come in a profusion of types and sizes adapted to many applications. Most modern electric lighting is powered by centrally generated electric power, but lighting may also be powered by mobile or standby electric generators or battery systems. Battery-powered light is often reserved for when and where stationary lights fail, often in the form of flashlights or electric lanterns, as well as in vehicles.

History
Before electric lighting became common in the early 20th century, people used candles, gas lights, oil lamps, and fires. In 1799–1800, Alessandro Volta created the voltaic pile, the first electric battery. Current from these batteries could heat copper wire to incandescence. Vasily Vladimirovich Petrov developed the first persistent electric arc in 1802, and English chemist Humphry Davy gave a practical demonstration of an arc light in 1806. It took more than a century of continuous and incremental improvement, including numerous designs, patents, and resulting intellectual property disputes, to get from these early experiments to commercially produced incandescent light bulbs in the 1920s.

In 1840, Warren de la Rue enclosed a platinum coil in a vacuum tube and passed an electric current through it, thus creating one of the world's first electric light bulbs. The design was based on the concept that the high melting point of platinum would allow it to operate at high temperatures and that the evacuated chamber would contain fewer gas molecules to react with the platinum, improving its longevity. Although it was an efficient design, the cost of the platinum made it impractical for commercial use.

William Greener, an English inventor, made significant contributions to early electric lighting with his lamp in 1846 (patent specification 11076), laying the groundwork for future innovations such as those by Thomas Edison.

The late 1870s and 1880s were marked by intense competition and innovation, with inventors like Joseph Swan in the UK and Thomas Edison in the US independently developing functional incandescent lamps. Swan's bulbs, based on designs by William Staite, were successful, but the filaments were too thick. Edison worked to create bulbs with thinner filaments, leading to a better design. The rivalry between Swan and Edison eventually led to a merger, forming the Edison and Swan Electric Light Company. By the early twentieth century these had completely replaced arc lamps.

The turn of the century saw further improvements in bulb longevity and efficiency, notably with the introduction of the tungsten filament by William D. Coolidge, who applied for a patent in 1912. This innovation became a standard for incandescent bulbs for many years.

In 1910, Georges Claude introduced the first neon light, paving the way for neon signs which would become ubiquitous in advertising.

In 1934, Arthur Compton, a renowned physicist and GE consultant, reported to the GE lamp department on successful experiments with fluorescent lighting at General Electric Co., Ltd. in Great Britain (unrelated to General Electric in the United States). Stimulated by this report, and with all of the key elements available, a team led by George E. Inman built a prototype fluorescent lamp in 1934 at General Electric's Nela Park (Ohio) engineering laboratory. This was not a trivial exercise; as noted by Arthur A. Bright, "A great deal of experimentation had to be done on lamp sizes and shapes, cathode construction, gas pressures of both argon and mercury vapor, colors of fluorescent powders, methods of attaching them to the inside of the tube, and other details of the lamp and its auxiliaries before the new device was ready for the public."

The first practical LED arrived in 1962.

U.S. transition to LED bulbs
In the United States, incandescent light bulbs including halogen bulbs stopped being sold as of August 1, 2023, because they do not meet minimum lumens per watt performance metrics established by the U.S. Department of Energy. Compact fluorescent bulbs are also banned despite their lumens per watt performance, because of their toxic mercury that can be released into the home if broken and widespread problems with proper disposal of mercury-containing bulbs.

Incandescent
In its modern form, the incandescent light bulb consists of a coiled filament of tungsten sealed in a globular glass chamber, either a vacuum or full of an inert gas such as argon. When an electric current is connected, the tungsten is heated to 2,000 to 3,300 K and glows, emitting light that approximates a continuous spectrum.

Incandescent bulbs are highly inefficient, in that just 2–5% of the energy consumed is emitted as visible, usable light. The remaining 95% is lost as heat. In warmer climates, the emitted heat must then be removed, putting additional pressure on ventilation or air conditioning systems. In colder weather, the heat byproduct has some value, and has been successfully harnessed for warming in devices such as heat lamps. Incandescent bulbs are nonetheless being phased out in favor of technologies like CFLs and LED bulbs in many countries due to their low energy efficiency. The European Commission estimated in 2012 that a complete ban on incandescent bulbs would contribute 5 to 10 billion euros to the economy and save 15 billion metric tonnes of carbon dioxide emissions.

Halogen
Halogen lamps are usually much smaller than standard incandescent lamps, because for successful operation a bulb temperature over 200 °C is generally necessary. For this reason, most have a bulb of fused silica (quartz) or aluminosilicate glass. This is often sealed inside an additional layer of glass. The outer glass is a safety precaution, to reduce ultraviolet emission and to contain hot glass shards should the inner envelope explode during operation. Oily residue from fingerprints may cause a hot quartz envelope to shatter due to excessive heat buildup at the contamination site. The risk of burns or fire is also greater with bare bulbs, leading to their prohibition in some places, unless enclosed by the luminaire.

Those designed for 12- or 24-volt operation have compact filaments, useful for good optical control. Also, they have higher efficacies (lumens per watt) and longer lives than non-halogen types. The light output remains almost constant throughout their life.

Fluorescent
Fluorescent lamps consist of a glass tube that contains mercury vapour or argon under low pressure. Electricity flowing through the tube causes the gases to give off ultraviolet energy. The inside of the tubes are coated with phosphors that give off visible light when struck by ultraviolet photons. They have much higher efficiency than incandescent lamps. For the same amount of light generated, they typically use around one-quarter to one-third the power of an incandescent. The typical luminous efficacy of fluorescent lighting systems is 50–100 lumens per watt, several times the efficacy of incandescent bulbs with comparable light output. Fluorescent lamp fixtures are more costly than incandescent lamps, because they require a ballast to regulate the current through the lamp, but the lower energy cost typically offsets the higher initial cost. Compact fluorescent lamps are available in the same popular sizes as incandescent lamps and are used as an energy-saving alternative in homes. Because they contain mercury, many fluorescent lamps are classified as hazardous waste. The United States Environmental Protection Agency recommends that fluorescent lamps be segregated from general waste for recycling or safe disposal, and some jurisdictions require recycling of them.

LED
The solid-state light-emitting diode (LED) has been popular as an indicator light in consumer electronics and professional audio gear since the 1970s. In the 2000s, efficacy and output have risen to the point where LEDs are now being used in lighting applications such as car headlights and brake lights, in flashlights and bicycle lights, as well as in decorative applications, such as holiday lighting. Indicator LEDs are known for their extremely long life, up to 100,000 hours, but lighting LEDs are operated much less conservatively, and consequently have shorter lives. LED technology is useful for lighting designers, because of its low power consumption, low heat generation, instantaneous on/off control, and in the case of single color LEDs, continuity of color throughout the life of the diode and relatively low cost of manufacture. LED lifetime depends strongly on the temperature of the diode. Operating an LED lamp in conditions that increase the internal temperature can greatly shorten the lamp's life. Some lasers have been adapted as an alternative to LEDs to provide highly focused illumination.

Carbon arc
Carbon arc lamps consist of two carbon rod electrodes in open air, supplied by a current-limiting ballast. The electric arc is struck by touching the rod tips then separating them. The ensuing arc produces a white-hot plasma between the rod tips. These lamps have higher efficacy than filament lamps, but the carbon rods are short-lived and require constant adjustment in use, as the intense heat of the arc erodes them. The lamps produce significant ultraviolet output, they require ventilation when used indoors, and due to their intensity they need protection from direct sight.

Invented by Humphry Davy around 1805, the carbon arc was the first practical electric light. It was used commercially beginning in the 1870s for large building and street lighting until it was superseded in the early 20th century by the incandescent light. Carbon arc lamps operate at high power and produce high intensity white light. They also are a point source of light. They remained in use in limited applications that required these properties, such as movie projectors, stage lighting, and searchlights, until after World War II.

Discharge
A discharge lamp has a glass or silica envelope containing two metal electrodes separated by a gas. Gases used include, neon, argon, xenon, sodium, metal halides, and mercury. The core operating principle is much the same as the carbon arc lamp, but the term "arc lamp" normally refers to carbon arc lamps, with more modern types of gas discharge lamp normally called discharge lamps. With some discharge lamps, very high voltage is used to strike the arc. This requires an electrical circuit called an igniter, which is part of the electrical ballast circuitry. After the arc is struck, the internal resistance of the lamp drops to a low level, and the ballast limits the current to the operating current. Without a ballast, excess current would flow, causing rapid destruction of the lamp.

Some lamp types contain a small amount of neon, which permits striking at normal running voltage with no external ignition circuitry. Low-pressure sodium lamps operate this way. The simplest ballasts are just an inductor, and are chosen where cost is the deciding factor, such as street lighting. More advanced electronic ballasts may be designed to maintain constant light output over the life of the lamp, may drive the lamp with a square wave to maintain completely flicker-free output, and shut down in the event of certain faults.

The most efficient source of electric light is the low-pressure sodium lamp. It produces, for all practical purposes, a monochromatic orange-yellow light, which gives a similarly monochromatic perception of any illuminated scene. For this reason, it is generally reserved for outdoor public lighting applications. Low-pressure sodium lights are favoured for public lighting by astronomers, since the light pollution that they generate can be easily filtered, contrary to broadband or continuous spectra.

Form factor
Many lamp units, or light bulbs, are specified in standardized shape codes and socket names. Incandescent bulbs and their retrofit replacements are often specified as "A19/A60 E26/E27", a common size for those kinds of light bulbs. In this example, the "A" parameters describe the bulb size and shape within the A-series light bulb while the "E" parameters describe the Edison screw base size and thread characteristics.

Comparison parameters
Common comparison parameters include: Less common parameters include color rendering index (CRI).
 * Luminous flux (in lumens)
 * Energy consumption (in watts)
 * Luminous efficacy (in lumens per watt)
 * Color temperature (in kelvins)

Life expectancy
Life expectancy for many types of lamp is defined as the number of hours of operation at which 50% of them fail, that is the median life of the lamps. Production tolerances as low as 1% can create a variance of 25% in lamp life, so in general some lamps will fail well before the rated life expectancy, and some will last much longer. For LEDs, lamp life is defined as the operation time at which 50% of lamps have experienced a 70% decrease in light output. In the 1900s the Phoebus cartel formed in an attempt to reduce the life of electric light bulbs, an example of planned obsolescence.

Some types of lamp are also sensitive to switching cycles. Rooms with frequent switching, such as bathrooms, can expect much shorter lamp life than what is printed on the box. Compact fluorescent lamps are particularly sensitive to switching cycles.

Uses
The total amount of artificial light (especially from street light) is sufficient for cities to be easily visible at night from the air, and from space. External lighting grew at a rate of 3–6 percent for the later half of the 20th century and is the major source of light pollution that burdens astronomers and others with 80% of the world's population living in areas with night time light pollution. Light pollution has been shown to have a negative effect on some wildlife.

Electric lamps can be used as heat sources, for example in incubators, as infrared lamps in fast food restaurants and toys such as the Kenner Easy-Bake Oven.

Lamps can also be used for light therapy to deal with such issues as vitamin D deficiency, skin conditions such as acne and dermatitis, skin cancers, and seasonal affective disorder. Lamps which emit a specific frequency of blue light are also used to treat neonatal jaundice with the treatment which was initially undertaken in hospitals being able to be conducted at home.

Electric lamps can also be used as a grow light to aid in plant growth especially in indoor hydroponics and aquatic plants with recent research into the most effective types of light for plant growth.

Due to their nonlinear resistance characteristics, tungsten filament lamps have long been used as fast-acting thermistors in electronic circuits. Popular uses have included:
 * Stabilization of sine wave oscillators
 * Protection of tweeters in loudspeaker enclosures; excess current that is too high for the tweeter illuminates the light rather than destroying the tweeter.
 * Automatic volume control in telephones

Cultural symbolism
In Western culture, a lightbulb — in particular, the appearance of an illuminated lightbulb above a person's head — signifies sudden inspiration.

In the Middle East, a light bulb symbol has a sexual connotation.

A stylized depiction of a light bulb features as the logo of the Turkish AK Party.

title: Electric light


text:
Edgar Rice Burroughs

Edgar Rice Burroughs (September 1, 1875 – March 19, 1950) was an American writer, best known for his prolific output in the adventure, science fiction, and fantasy genres. Best known for creating the characters Tarzan and John Carter, he also wrote the Pellucidar series, the Amtor series, and the Caspak trilogy.

Tarzan was immediately popular, and Burroughs capitalized on it in every possible way, including a syndicated Tarzan comic strip, films, and merchandise. Tarzan remains one of the most successful fictional characters to this day and is a cultural icon. Burroughs's California ranch is now the center of the Tarzana neighborhood in Los Angeles, named after the character. Burroughs was an explicit supporter of eugenics and scientific racism in both his fiction and nonfiction; Tarzan was meant to reflect these concepts.

Early life and family
Burroughs was born on September 1, 1875, in Chicago (he later lived for many years in the suburb of Oak Park), the fourth son of Major George Tyler Burroughs, a businessman and Civil War veteran, and his wife, Mary Evaline (Zieger) Burroughs. His middle name is from his paternal grandmother, Mary Coleman Rice Burroughs.

Burroughs was of almost entirely English ancestry, with a family line that had been in North America since the Colonial era.

Through his Rice grandmother, Burroughs was descended from settler Edmund Rice, one of the English Puritans who moved to Massachusetts Bay Colony in the early 17th century. He once remarked: "I can trace my ancestry back to Deacon Edmund Rice."

The Burroughs side of the family was also of English origin, having emigrated to Massachusetts around the same time. Many of his ancestors fought in the American Revolution. Some of his ancestors settled in Virginia during the colonial period, and Burroughs often emphasized his connection with that side of his family, seeing it as romantic and warlike.

Burroughs was educated at a number of local schools. He then attended Phillips Academy, in Andover, Massachusetts, and then the Michigan Military Academy. Graduating in 1895, but failing the entrance exam for the United States Military Academy at West Point, he instead became an enlisted soldier with the 7th U.S. Cavalry in Fort Grant, Arizona Territory. After being diagnosed with a heart problem and thus ineligible to serve, he was discharged in 1897.

After his discharge, Burroughs worked at a number of different jobs. During the Chicago influenza epidemic of 1891, he spent half a year at his brother's ranch on the Raft River in Idaho, as a cowboy, drifted somewhat afterward, then worked at his father's Chicago battery factory in 1899, marrying his childhood sweetheart, Emma Hulbert (1876–1944), in January 1900.

In 1903, Burroughs joined his brothers, Yale graduates George and Harry, who were, by then, prominent Pocatello area ranchers in southern Idaho, and partners in the Sweetser-Burroughs Mining Company, where he took on managing their ill-fated Snake River gold dredge, a classic bucket-line dredge. The Burroughs brothers were also the sixth cousins, once removed, of famed miner Kate Rice who, in 1914, became the first female prospector in the Canadian North. Journalist and publisher C. Allen Thorndike Rice was also his third cousin.

When the new mine proved unsuccessful, the brothers secured for Burroughs a position with the Oregon Short Line Railroad in Salt Lake City. Burroughs resigned from the railroad in October 1904.

Later life
By 1911, around age 36, after seven years of low wages as a pencil-sharpener wholesaler, Burroughs began to write fiction. By this time, Emma and he had two children, Joan (1908–1972), and Hulbert (1909–1991). During this period, he had copious spare time and began reading pulp-fiction magazines. In 1929, he recalled thinking that:

""[...] if people were paid for writing rot such as I read in some of those magazines, that I could write stories just as rotten. As a matter of fact, although I had never written a story, I knew absolutely that I could write stories just as entertaining and probably a whole lot more so than any I chanced to read in those magazines.""

In 1913, Burroughs and Emma had their third and last child, John Coleman Burroughs (1913–1979), later known for his illustrations of his father's books.

In the 1920s, Burroughs became a pilot, purchased a Security Airster S-1, and encouraged his family to learn to fly.

Daughter Joan married Tarzan film actor James Pierce. She starred with her husband as the voice of Jane, during 1932–1934 for the Tarzan radio series.

Burroughs divorced Emma in 1934, and, in 1935, married the former actress Florence Gilbert Dearholt, who was the former wife of his friend (who was then himself remarrying), Ashton Dearholt, with whom he had co-founded Burroughs-Tarzan Enterprises while filming The New Adventures of Tarzan. Burroughs adopted the Dearholts' two children. He and Florence divorced in 1942.

Burroughs was in his late 60s and was in Honolulu at the time of the Japanese attack on Pearl Harbor. Despite his age, he applied for and received permission to become a war correspondent, becoming one of the oldest U.S. war correspondents during World War II. This period of his life is mentioned in William Brinkley's bestselling novel Don't Go Near the Water.

Death
After the war ended, Burroughs moved back to Encino, California, where after many health problems, he died of a heart attack on March 19, 1950, having written almost 80 novels. He is buried in Tarzana, California, US.

At the time of his death he was believed to have been the writer who had made the most from films, earning over US$2 million in royalties from 27 Tarzan pictures.

The Science Fiction Hall of Fame inducted Burroughs in 2003.

Literary career
Aiming his work at the pulps—under the name "Norman Bean" to protect his reputation—Burroughs had his first story, Under the Moons of Mars, serialized by Frank Munsey in the February to July 1912 issues of The All-Story. Under the Moons of Mars inaugurated the Barsoom series, introduced John Carter, and earned Burroughs US$400 ($11,922 today). It was first published as a book by A. C. McClurg of Chicago in 1917, entitled A Princess of Mars, after three Barsoom sequels had appeared as serials and McClurg had published the first four serial Tarzan novels as books.

Burroughs soon took up writing full-time, and by the time the run of Under the Moons of Mars had finished, he had completed two novels, including Tarzan of the Apes, published from October 1912 and one of his most successful series.

Burroughs also wrote popular science fiction and fantasy stories involving adventurers from Earth transported to various planets (notably Barsoom, Burroughs's fictional name for Mars, and Amtor, his fictional name for Venus), lost islands (Caspak), and into the interior of the Hollow Earth in his Pellucidar stories. He also wrote Westerns and historical romances. Besides those published in All-Story, many of his stories were published in The Argosy magazine.

Tarzan was a cultural sensation when introduced. Burroughs was determined to capitalize on Tarzan's popularity in every way possible. He planned to exploit Tarzan through several different media including a syndicated Tarzan comic strip, movies, and merchandise. Experts in the field advised against this course of action, stating that the different media would just end up competing against each other. Burroughs went ahead, however, and proved the experts wrong – the public wanted Tarzan in whatever fashion he was offered. Tarzan remains one of the most successful fictional characters to this day and is a cultural icon.

In either 1915 or 1919, Burroughs purchased a large ranch north of Los Angeles, California, which he named "Tarzana". The citizens of the community that sprang up around the ranch voted to adopt that name when their community, Tarzana, California, was formed in 1927. Also, the unincorporated community of Tarzan, Texas, was formally named in 1927 when the US Postal Service accepted the name, reputedly coming from the popularity of the first (silent) Tarzan of the Apes film, starring Elmo Lincoln, and an early "Tarzan" comic strip.

In 1923, Burroughs set up his own company, Edgar Rice Burroughs, Inc., and began printing his own books through the 1930s.

Reception and criticism
Because of the part Burroughs's science fiction played in inspiring real exploration of Mars, an impact crater on Mars was named in his honor after his death. In a Paris Review interview, Ray Bradbury said of Burroughs: ""Edgar Rice Burroughs never would have looked upon himself as a social mover and shaker with social obligations. But as it turns out – and I love to say it because it upsets everyone terribly – Burroughs is probably the most influential writer in the entire history of the world. By giving romance and adventure to a whole generation of boys, Burroughs caused them to go out and decide to become special.""

In Something of Myself (published posthumously in 1937) Rudyard Kipling wrote: "My Jungle Books begat Zoos of [imitators] . But the genius of all the genii was one who wrote a series called Tarzan of the Apes. I read it, but regret I never saw it on the films, where it rages most successfully. He had 'jazzed' the motif of the Jungle Books and, I imagine, had thoroughly enjoyed himself. He was reported to have said that he wanted to find out how bad a book he could write and 'get away with', which is a legitimate ambition."

By 1963, Floyd C. Gale of Galaxy Science Fiction wrote when discussing reprints of several Burroughs novels by Ace Books, "an entire generation has grown up inexplicably Burroughs-less". He stated that most of the author's books had been out of print for years and that only the "occasional laughable Tarzan film" reminded public of his fiction. Gale reported his surprise that after two decades his books were again available, with Canaveral Press, Dover Publications, and Ballantine Books also reprinting them.

Few critical books have been written about Burroughs. From an academic standpoint, the most helpful are Erling Holtsmark's two books: Tarzan and Tradition and Edgar Rice Burroughs; Stan Galloway's The Teenage Tarzan: A Literary Analysis of Edgar Rice Burroughs' Jungle Tales of Tarzan; and Richard Lupoff's two books: Master of Adventure: Edgar Rice Burroughs and Barsoom: Edgar Rice Burroughs and the Martian Vision. Galloway was identified by James Edwin Gunn as "one of the half-dozen finest Burroughs scholars in the world"; Galloway called Holtsmark his "most important predecessor".

Burroughs strongly supported eugenics and scientific racism. His views held that English nobles made up a particular heritable elite among Anglo-Saxons. Tarzan was meant to reflect this, with him being born to English nobles and then adopted by talking apes (the Mangani). They express eugenicist views themselves, but Tarzan is permitted to live despite being deemed "unfit" in comparison, and grows up to surpass not only them but black Africans, whom Burroughs clearly presents as inherently inferior, even not wholly human. In one Tarzan story, he finds an ancient civilization where eugenics has been practiced for over 2,000 years, with the result that it is free of all crime. Criminal behavior is held to be entirely hereditary, with the solution having been to kill not only criminals but also their families. Lost on Venus, a later novel, presents a similar utopia where forced sterilization is practiced and the "unfit" are killed. Burroughs explicitly supported such ideas in his unpublished nonfiction essay I See A New Race. Additionally, his Pirate Blood, which is not speculative fiction and remained unpublished after his death, portrayed the characters as victims of their hereditary criminal traits (one a descendant of the corsair Jean Lafitte, another from the Jukes family). These views have been compared with Nazi eugenics (though noting that they were popular and common at the time), with his Lost on Venus being released the same year the Nazis took power (in 1933).

In 2003, Burroughs was inducted into the Science Fiction and Fantasy Hall of Fame.

Barsoom series (aka Martian series)

 * 1) A Princess of Mars (1912)
 * 2) The Gods of Mars (1913)
 * 3) The Warlord of Mars (1914)
 * 4) Thuvia, Maid of Mars (1916)
 * 5) The Chessmen of Mars (1922)
 * 6) The Master Mind of Mars (1927)
 * 7) A Fighting Man of Mars (1930)
 * 8) Swords of Mars (1934)
 * 9) Synthetic Men of Mars (1939)
 * 10) Llana of Gathol (1941)
 * 11) John Carter of Mars (1964, two stories from 1940 and 1943)

Tarzan series

 * 1) Tarzan of the Apes (1912)
 * 2) The Return of Tarzan (1913)
 * 3) The Beasts of Tarzan (1914)
 * 4) The Son of Tarzan (1915)
 * 5) Tarzan and the Jewels of Opar (1916)
 * 6) Jungle Tales of Tarzan (stories 1916–1917)
 * 7) Tarzan the Untamed (1919)
 * 8) Tarzan the Terrible (1921)
 * 9) Tarzan and the Golden Lion (1922)
 * 10) Tarzan and the Ant Men (1924)
 * 11) Tarzan, Lord of the Jungle (1927)
 * 12) Tarzan and the Lost Empire (1928)
 * 13) Tarzan at the Earth's Core (1929)
 * 14) Tarzan the Invincible (1930)
 * 15) Tarzan Triumphant (1931)
 * 16) Tarzan and the City of Gold (1932)
 * 17) Tarzan and the Lion Man (1933)
 * 18) Tarzan and the Leopard Men (1932)
 * 19) Tarzan's Quest (1935)
 * 20) Tarzan the Magnificent (1936)
 * 21) Tarzan and the Forbidden City (1938)
 * 22) Tarzan and the Foreign Legion (1947, written in 1944)
 * 23) Tarzan and the Tarzan Twins (1963, collects 1927 and 1936 children's books)
 * 24) Tarzan and the Madman (1964, written in 1940)
 * 25) Tarzan and the Castaways (1965, stories from 1940 to 1941)
 * 26) Tarzan: The Lost Adventure (1995, rewritten version of 1946 fragment, completed by Joe R. Lansdale)

Pellucidar series

 * 1) At the Earth's Core (1914)
 * 2) Pellucidar (1915)
 * 3) Tanar of Pellucidar (1929)
 * 4) Tarzan at the Earth's Core (1929)
 * 5) Back to the Stone Age (1937)
 * 6) Land of Terror (1944, written in 1939)
 * 7) Savage Pellucidar (1963, stories from 1942)

Venus series

 * 1) Pirates of Venus (1932)
 * 2) Lost on Venus (1933)
 * 3) Carson of Venus (1938)
 * 4) Escape on Venus (1946, stories from 1941 to 1942)
 * 5) The Wizard of Venus (1970, written in 1941)

Caspak series

 * 1) The Land That Time Forgot (1918)
 * 2) The People That Time Forgot (1918)
 * 3) Out of Time's Abyss (1918)

Moon series
These three texts have been published by various houses in one or two volumes. Adding to the confusion, some editions have the original (significantly longer) introduction to Part I from the first publication as a magazine serial, and others have the shorter version from the first book publication, which included all three parts under the title The Moon Maid.
 * Part I: The Moon Maid (1923, serialized in Argosy, May 5 – June 2, 1923)
 * Part II: The Moon Men (1925, serialized in Argosy, February 21 – March 14, 1925)
 * Part III: The Red Hawk (1925 serialized in Argosy, September 5–19, 1925)

Mucker series

 * The Mucker (1914)
 * The Return of the Mucker (1916)
 * The Oakdale Affair (1918)

Other science fiction

 * The Monster Men (1913)
 * The Lost Continent (1916; a.k.a. Beyond Thirty)
 * The Resurrection of Jimber-Jaw (1937)
 * Beyond the Farthest Star (1942)

Jungle adventure novels

 * The Cave Girl (1913, revised 1917)
 * The Eternal Lover (1914, rev. 1915; A.K.A. The Eternal Savage)
 * The Man-Eater (1915)
 * The Lad and the Lion (1917)
 * Jungle Girl (1931; A.K.A. The Land of Hidden Men)

Western novels

 * The Bandit of Hell's Bend (1924)
 * The War Chief (1927)
 * Apache Devil (1933)
 * The Deputy Sheriff of Comanche County (1940)

Historical novels

 * The Outlaw of Torn (1914)
 * I am a Barbarian (1967; written in 1941)

Other works

 * Minidoka: 937th Earl of One Mile Series M (1998; written in 1903)
 * The Mad King (1914, rev. 1915)
 * The Girl from Farris's (1916)
 * The Rider (1918)
 * The Efficiency Expert (1921)
 * The Girl from Hollywood (1922)
 * Marcia of the Doorstep (1924)
 * You Lucky Girl! (1927)
 * Pirate Blood (1970; written in 1932)
 * Forgotten Tales of Love and Murder (2001; stories from 1910 to 1944)
 * Brother Men (2005; nonfiction)

title: Edgar Rice Burroughs