Expected value

Let ${\displaystyle X}$ represent the outcome of a roll of a fair six-sided . More specifically, ${\displaystyle X}$ will be the number of pips showing on the top face of the die after the toss. The possible values for ${\displaystyle X}$ are 1, 2, 3, 4, 5, and 6, all of which are equally likely with a probability of 1/6. The expectation of ${\displaystyle X}$ is $${\displaystyle \operatorname {E} [X]=1\cdot {\frac {1}{6}}+2\cdot {\frac {1}{6}}+3\cdot {\frac {1}{6}}+4\cdot {\frac {1}{6}}+5\cdot {\frac {1}{6}}+6\cdot {\frac {1}{6}}=3.5.}$$ If one rolls the die ${\displaystyle n}$ times and computes the average (arithmetic mean) of the results, then as ${\displaystyle n}$ grows, the average will almost surely converge to the expected value, a fact known as the strong law of large numbers.

die

The 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 ${\displaystyle X}$ represents the (monetary) outcome of a $1 bet on a single number ("straight up" bet). If the bet wins (which happens with probability 1/38 in American roulette), the payoff is $35; otherwise the player loses the bet. The expected profit from such a bet will be $${\displaystyle \operatorname {E} [\,{\text{gain from }}\$1{\text{ bet}}\,]=-\$1\cdot {\frac {37}{38}}+\$35\cdot {\frac {1}{38}}=-\${\frac {1}{19}}.}$$ That is, the expected value to be won from a $1 bet is −$1/19. Thus, in 190 bets, the net loss will probably be about $10.

roulette

Non-negativity: If ${\displaystyle X\geq 0}$ (a.s.), then ${\displaystyle \operatorname {E} [X]\geq 0.}$

Linearity of expectation: 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. 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 ${\displaystyle N}$ random variables ${\displaystyle X_{i}}$ and constants ${\displaystyle a_{i}(1\leq i\leq N),}$ we have ${\textstyle \operatorname {E} \left[\sum _{i=1}^{N}a_{i}X_{i}\right]=\sum _{i=1}^{N}a_{i}\operatorname {E} [X_{i}].}$ 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.

[34]

Monotonicity: If ${\displaystyle X\leq Y}$ , 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.).

(a.s.)

Non-degeneracy: If ${\displaystyle \operatorname {E} [|X|]=0,}$ then ${\displaystyle X=0}$ (a.s.).

If ${\displaystyle X=Y}$ , 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.

(a.s.)

If ${\displaystyle X=c}$ for some real number c, 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.

(a.s.)

As a consequence of the formula |X| = X⁺ + X⁻ as discussed above, together with the , it follows that for any random variable ${\displaystyle X}$ with well-defined expectation, one has ${\displaystyle |\operatorname {E} [X]|\leq \operatorname {E} |X|.}$

triangle inequality

Let 1_A denote the of an event A, then E[1_A] is given by the probability of A. This is nothing but a different way of stating the expectation of a Bernoulli random variable, as calculated in the table above.

indicator function

Formulas in terms of CDF: If ${\displaystyle F(x)}$ is the of a random variable X, then $${\displaystyle \operatorname {E} [X]=\int _{-\infty }^{\infty }x\,dF(x),}$$ 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. As a consequence of integration by parts as applied to this representation of E[X], it can be proved that $${\displaystyle \operatorname {E} [X]=\int _{0}^{\infty }(1-F(x))\,dx-\int _{-\infty }^{0}F(x)\,dx,}$$ with the integrals taken in the sense of Lebesgue.^[35] As a special case, for any random variable X valued in the nonnegative integers {0, 1, 2, 3, ...}, one has $${\displaystyle \operatorname {E} [X]=\sum _{n=0}^{\infty }\Pr(X>n),}$$ where P denotes the underlying probability measure.

cumulative distribution function

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 , 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.

independent

: 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 ${\displaystyle g}$:^[34] $${\displaystyle \operatorname {E} [g(X)]=\int _{\mathbb {R} }g(x)f(x)\,dx.}$$ This formula also holds in multidimensional case, when ${\displaystyle g}$ is a function of several random variables, and ${\displaystyle f}$ is their joint density.^[34][36]

Law of the unconscious statistician

Central tendency

Conditional expectation

Expectation (epistemic)

– related to expectations in a way analogous to that in which quantiles are related to medians

Expectile

– the expected value of the conditional expected value of X given Y is the same as the expected value of X

Law of total expectation

– a generalization of the expected value

Nonlinear expectation

Population mean

Predicted value

– an equation for calculating the expected value of a random number of random variables

Wald's equation