Expected value

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.[5] This problem had been debated for centuries, and 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 couldn't 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 a now 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.[6]

Three years later, in 1657, a Dutch mathematician Christiaan Huygens, who had just visited Paris, published a treatise (see Huygens (1657)) "De ratiociniis in ludo aleæ" on probability theory. In this book, he considered the problem of points, and presented a solution based on the same principle as the solutions of Pascal and Fermat. Huygens also 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). In this sense, this book can be seen as the first successful attempt at laying down the foundations of the theory of probability.

In the foreword to his book, 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.

— Edwards (2002)

Thus, Huygens learned about de Méré's Problem in 1655 during his visit to France; later on in 1656 from his correspondence with Carcavi, he learned that his method was essentially the same as Pascal's; so that before his book went to press in 1657, he knew about Pascal's priority in this subject.

The use of the letter ${\displaystyle \mathop {\hbox{E}} }$ to denote expected value goes back to W. A. Whitworth in 1901.[9] The symbol has become popular since then for English writers. In German, ${\displaystyle \mathop {\hbox{E}} }$ stands for "Erwartungswert", in Spanish for "Esperanza matemática", and in French for "Espérance mathématique".[10]

Another popular notation is ${\displaystyle \mu _{X}}$, whereas ${\displaystyle \langle X\rangle }$ is commonly used in physics, and ${\displaystyle \mathop {\hbox{M}} (X)}$ in Russian-language literature.

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 ${\displaystyle X\geq 0}$ is true almost surely, when the probability measure attributes zero-mass to the complementary event ${\displaystyle \left\{X<0\right\}}$.

• For a general random variable ${\displaystyle X}$, define as before ${\displaystyle X^{+}(\omega )=\max(X(\omega ),0)}$ and ${\displaystyle X^{-}(\omega )=-\min(X(\omega ),0)}$, and note that ${\displaystyle X=X^{+}-X^{-}}$, with both ${\displaystyle X^{+}}$ and ${\displaystyle X^{-}}$ nonnegative, then:

${\displaystyle \operatorname {E} [X]={\begin{cases}\operatorname {E} [X^{+}]-\operatorname {E} [X^{-}]&{\text{if }}\operatorname {E} [X^{+}]<\infty {\text{ and }}\operatorname {E} [X^{-}]<\infty$ ;\\\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}}}

• Let ${\displaystyle {\mathbf {1} }_{A}}$ denote the indicator function of an event ${\displaystyle A}$, then ${\displaystyle \operatorname {E} [{\mathbf {1} }_{A}]=1\cdot \operatorname {P} (A)+0\cdot \operatorname {P} (\Omega \setminus A)=\operatorname {P} (A).}$
• Formulas in terms of CDF: If ${\displaystyle F(x)}$ is the cumulative distribution function of the probability measure ${\displaystyle \operatorname {P} ,}$ and ${\displaystyle X}$ is a random variable, then

${\displaystyle \operatorname {E} [X]=\int _{\overline {\mathbb {R} }}x\,dF(x),}$

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. Here, ${\displaystyle {\overline {\mathbb {R} }}=[-\infty ,+\infty ]}$ is the extended real line.

Additionally,

${\displaystyle \displaystyle \operatorname {E} [X]=\int \limits _{0}^{\infty }(1-F(x))\,dx-\int \limits _{-\infty }^{0}F(x)\,dx,}$

with the integrals taken in the sense of Lebesgue.

The proof of the second formula follows.

Proof.

For an arbitrary ${\displaystyle \omega \in \Omega ,}$ ${\displaystyle \displaystyle X^{-}(\omega )=\int \limits _{-X^{-}(\omega )}^{0}dx=\int \limits _{-\infty }^{0}{\mathbf {1} }{\{x\mid X^{-}(\omega )\geq -x\}}\,dx=\int \limits _{-\infty }^{0}{\mathbf {1} }{\{(\omega ,x)\mid X(\omega )\leq x\}}\,dx.}$ The last equality holds because the fact that ${\displaystyle X^{-}(\omega )\geq -x,}$ where ${\displaystyle x\leq 0,}$ implies that ${\displaystyle X^{+}(\omega )=0}$ and hence ${\displaystyle X(\omega )\leq x.}$ Conversely, if ${\displaystyle X(\omega )\leq x,}$ where ${\displaystyle x\leq 0,}$ then ${\displaystyle X^{+}(\omega )=0}$ and ${\displaystyle X^{-}(\omega )\geq -x.}$ The integrand in the above expression for ${\displaystyle X^{-}(\omega )}$ is non-negative, so Tonelli's theorem applies, and the order of integration may be switched without altering the result. We have ${\displaystyle {\begin{aligned}\operatorname {E} (X^{-})&=\int \limits _{\Omega }\left(\int \limits _{-\infty }^{0}{\mathbf {1} }{\{(\omega ,x)\mid X(\omega )\leq x\}}\,dx\right)d\operatorname {P} \\&=\int \limits _{-\infty }^{0}\left(\int \limits _{\Omega }{\mathbf {1} }{\{\omega \mid X(\omega )\leq x\}}\,d\operatorname {P} \right)dx\\&=\int \limits _{-\infty }^{0}\operatorname {P} (X\leq x)\,dx=\int \limits _{-\infty }^{0}F(x)\,dx.\end{aligned}}}$ Arguing as above, ${\displaystyle \displaystyle X^{+}(\omega )=\int \limits _{0}^{\infty }{\mathbf {1} }{\{(\omega ,x)\mid X(\omega )>x\}}\,dx,}$ and ${\displaystyle \displaystyle \operatorname {E} (X^{+})=\int \limits _{0}^{\infty }\operatorname {P} (X>x)\,dx=\int \limits _{0}^{\infty }(1-F(x))\,dx.}$ Recalling that ${\displaystyle \operatorname {E} (X)=\operatorname {E} (X^{+})-\operatorname {E} (X^{-})}$ completes the proof.

• Non-negativity: If ${\displaystyle X\geq 0}$ (a.s.), then ${\displaystyle \operatorname {E} [X]\geq 0}$.
• Linearity of expectation:[3] The expected value operator (or expectation operator) ${\displaystyle \operatorname {E} [\cdot ]}$ is linear in the sense that, for any random variables ${\displaystyle X}$ and ${\displaystyle Y}$, and a constant ${\displaystyle a}$,

${\displaystyle {\begin{aligned}\operatorname {E} [X+Y]&=\operatorname {E} [X]+\operatorname {E} [Y],\\\operatorname {E} [aX]&=a\operatorname {E} [X],\end{aligned}}}$

whenever the right-hand side is well-defined. 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.

• Monotonicity: If ${\displaystyle X\leq Y}$ (a.s.), and both ${\displaystyle \operatorname {E} [X]}$ and ${\displaystyle \operatorname {E} [Y]}$ exist, then ${\displaystyle \operatorname {E} [X]\leq \operatorname {E} [Y]}$.

Proof follows from the linearity and the non-negativity property for ${\displaystyle Z=Y-X}$, since ${\displaystyle Z\geq 0}$ (a.s.).

• Non-multiplicativity: In general, the expected value is not multiplicative, i.e. ${\displaystyle \operatorname {E} [XY]}$ is not necessarily equal to ${\displaystyle \operatorname {E} [X]\cdot \operatorname {E} [Y]}$. If ${\displaystyle X}$ and ${\displaystyle Y}$ are independent, then one can show that ${\displaystyle \operatorname {E} [XY]=\operatorname {E} [X]\operatorname {E} [Y]}$. If the random variables are dependent, then generally ${\displaystyle \operatorname {E} [XY]\neq \operatorname {E} [X]\operatorname {E} [Y]}$, although in special cases of dependency the equality may hold.
• Law of the unconscious statistician: The expected value of a measurable function of ${\displaystyle X}$, ${\displaystyle g(X)}$, given that ${\displaystyle X}$ has a probability density function ${\displaystyle f(x)}$, is given by the inner product of ${\displaystyle f}$ and ${\displaystyle g}$:

${\displaystyle \operatorname {E} [g(X)]=\int _{\mathbb {R} }g(x)f(x)\,dx.}$ [3]

This formula also holds in multidimensional case, when ${\displaystyle g}$ is a function of several random variables, and ${\displaystyle f}$ is their joint density.[3][12]

• Non-degeneracy: If ${\displaystyle \operatorname {E} [|X|]=0}$, then ${\displaystyle X=0}$ (a.s.).
• For a random variable ${\displaystyle X}$ with well-defined expectation: ${\displaystyle |\operatorname {E} [X]|\leq \operatorname {E} |X|}$.
• The following statements regarding a random variable ${\displaystyle X}$ are equivalent:
  • ${\displaystyle \operatorname {E} [X]}$ exists and is finite.
  • Both ${\displaystyle \operatorname {E} [X^{+}]}$ and ${\displaystyle \operatorname {E} [X^{-}]}$ are finite.
  • ${\displaystyle \operatorname {E} [|X|]}$ is finite.

For the reasons above, the expressions "${\displaystyle X}$ is integrable" and "the expected value of ${\displaystyle X}$ is finite" are used interchangeably throughout this article.

• If ${\displaystyle \operatorname {E} [X]<+\infty }$ then ${\displaystyle X<+\infty }$ (a.s.). Similarly, if ${\displaystyle \operatorname {E} [X]>-\infty }$ then ${\displaystyle X>-\infty }$ (a.s.).
• If ${\displaystyle \operatorname {E} |X^{\beta }|<\infty }$ and ${\displaystyle 0<\alpha <\beta ,}$ then ${\displaystyle \operatorname {E} |X^{\alpha }|<\infty .}$
• If ${\displaystyle X=Y}$ (a.s.), then ${\displaystyle \operatorname {E} [X]=\operatorname {E} [Y]}$. 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 ${\displaystyle X=c}$ (a.s.) for some constant ${\displaystyle c\in [-\infty ,+\infty ]}$, then ${\displaystyle \operatorname {E} [X]=c}$. In particular, for a random variable ${\displaystyle X}$ with well-defined expectation, ${\displaystyle \operatorname {E} [\operatorname {E} [X]]=\operatorname {E} [X]}$. 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.
• For a non-negative integer-valued random variable ${\displaystyle X:\Omega \to \{0,1,2,3,\ldots ,+\infty \},}$

${\displaystyle \operatorname {E} [X]=\sum _{n=0}^{\infty }\operatorname {P} (X>n).}$

Proof.

If ${\displaystyle \operatorname {P} (X=+\infty )>0,}$ then ${\displaystyle \operatorname {E} [X]=+\infty .}$ On the other hand, ${\displaystyle \operatorname {P} (X>n)\geq \operatorname {P} (X=+\infty )>0,}$ so the series on the right diverges to ${\displaystyle +\infty ,}$ and the equality holds. If ${\displaystyle \operatorname {P} (X=+\infty )=0,}$ then ${\displaystyle \sum _{n=0}^{\infty }\operatorname {P} (X>n)=\sum _{n=0}^{\infty }\sum _{j=n+1}^{\infty }\operatorname {P} (X=j).}$ Define the infinite upper-triangular matrix ${\displaystyle M={\begin{bmatrix}\operatorname {P} (X=1)&\operatorname {P} (X=2)&\operatorname {P} (X=3)&\cdots &\operatorname {P} (X=n)&\cdots \\&\operatorname {P} (X=2)&\operatorname {P} (X=3)&\cdots &\operatorname {P} (X=n)&\cdots \\&&\operatorname {P} (X=3)&\cdots &\operatorname {P} (X=n)&\cdots \\&&&\ddots &\vdots &\\&&&&\operatorname {P} (X=n)&\cdots \\&&&&&\ddots \end{bmatrix}}.}$ The double series ${\displaystyle \textstyle \sum _{i=1}^{\infty }\sum _{j=i}^{\infty }\operatorname {P} (X=j)}$ is the sum of ${\displaystyle M}$'s elements if summation is done row by row. Since every summand is non-negative, the series either converges absolutely or diverges to ${\displaystyle +\infty .}$ In both cases, changing summation order does not affect the sum. Changing summation order, from row-by-row to column-by-column, gives us ${\displaystyle {\begin{aligned}\sum _{n=0}^{\infty }\sum _{j=n+1}^{\infty }\operatorname {P} (X=j)&=\sum _{j=1}^{\infty }\sum _{n=0}^{j-1}\operatorname {P} (X=j)\\&=\sum _{j=1}^{\infty }j\operatorname {P} (X=j)\\&=\sum _{j=0}^{\infty }j\operatorname {P} (X=j)\\&=\operatorname {E} [X].\end{aligned}}}$

The expectation of a random variable plays an important role in a variety of contexts. For 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. For a different example, in statistics, where one seeks estimates for unknown parameters based on available data, the estimate itself is a random variable. In such settings, a desirable criterion for a "good" estimator is that it is unbiased; that is, the expected value of the estimate is equal to the true value of the underlying parameter.

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. ${\displaystyle \operatorname {P} ({X\in {\mathcal {A}}})=\operatorname {E} [{\mathbf {1} }_{\mathcal {A}}]}$, where ${\displaystyle {\mathbf {1} }_{\mathcal {A}}}$ is the indicator function of the set ${\displaystyle {\mathcal {A}}}$.

The mass of probability distribution is balanced at the expected value, here a Beta(α,β) distribution with expected value α/(α+β).

In classical mechanics, the center of mass is an analogous concept to expectation. For example, suppose X is a discrete random variable with values x_i and corresponding probabilities p_i. Now consider a weightless rod on which are placed weights, at locations x_i along the rod and having masses p_i (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

${\displaystyle \operatorname {Var} (X)=\operatorname {E} [X^{2}]-(\operatorname {E} [X])^{2}.}$

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

Expected values of common distributions

Distribution	Notation	Mean E(X)
Bernoulli	${\displaystyle X\sim ~b(1,p)}$	${\displaystyle p}$
Binomial	${\displaystyle X\sim B(n,p)}$	${\displaystyle np}$
Poisson	${\displaystyle X\sim Po(\lambda )}$	${\displaystyle \lambda }$
Geometric	${\displaystyle X\sim Geometric(p)}$	${\displaystyle 1/p}$
Uniform	${\displaystyle X\sim U(a,b)}$	${\displaystyle (a+b)/2}$
Exponential	${\displaystyle X\sim \exp(\lambda )}$	${\displaystyle 1/\lambda }$
Normal	${\displaystyle X\sim N(\mu ,\sigma ^{2})}$	${\displaystyle \mu }$
Standard Normal	${\displaystyle X\sim N(0,1)}$	${\displaystyle 0}$
Pareto	${\displaystyle X\sim Par(\alpha )}$	${\displaystyle \alpha /(\alpha +1)}$ if ${\displaystyle \alpha >1}$
Cauchy	${\displaystyle X\sim Cauchy(x_{0},\gamma )}$	undefined

The probability density function ${\displaystyle f_{X}}$ of a scalar random variable ${\displaystyle X}$ is related to its characteristic function ${\displaystyle \varphi _{X}}$ by the inversion formula:

${\displaystyle f_{X}(x)={\frac {1}{2\pi }}\int _{\mathbb {R} }e^{-itx}\varphi _{X}(t)\,dt.}$

For the expected value of ${\displaystyle g(X)}$ (where ${\displaystyle g:{\mathbb {R} }\to {\mathbb {R} }}$ is a Borel function), we can use this inversion formula to obtain

${\displaystyle \operatorname {E} [g(X)]={\frac {1}{2\pi }}\int _{\mathbb {R} }g(x)\left[\int _{\mathbb {R} }e^{-itx}\varphi _{X}(t)\,dt\right]\,dx.}$

If ${\displaystyle \operatorname {E} [g(X)]}$ is finite, changing the order of integration, we get, in accordance with Fubini–Tonelli theorem,

${\displaystyle \operatorname {E} [g(X)]={\frac {1}{2\pi }}\int _{\mathbb {R} }G(t)\varphi _{X}(t)\,dt,}$

where

${\displaystyle G(t)=\int _{\mathbb {R} }g(x)e^{-itx}\,dx}$

is the Fourier transform of ${\displaystyle g(x).}$ The expression for ${\displaystyle \operatorname {E} [g(X)]}$ also follows directly from Plancherel theorem.

• Center of mass
• Central tendency
• Chebyshev's inequality (an inequality on location and scale parameters)
• Conditional expectation
• Expectation (the general term)
• Expectation value (quantum mechanics)
• Law of total expectation—the expected value of the conditional expected value of X given Y is the same as the expected value of X.
• Moment (mathematics)
• Nonlinear expectation (a generalization of the expected value)
• Wald's equation—an equation for calculating the expected value of a random number of random variables