Expected Value

Definition

Centeroid of $f_X(x)$.

Mathematical expectation of a discrete random variable $X$ is defined as:

$$E[X]=\sum _{i=1}^{\infty }{x}_{i}P(X=x_i)$$

if ${\sum }_{i=1}^{\infty }|x_i|P(X=x_i)<\infty$, in other words if it converges. Otherwise $E[X]$ doesn’t exist. Similarly,

$$E[X]=\int xf(x)dx$$

if $\int |x|f(x)dx<\infty$.

Existence with bounded random variable.

If $X$ is bounded, $P(|X|<M)=1$, then the $E[X]$ exists.

Existence

For any rv $X$, $E[X]<\infty$ if and only if the integrals ${\int }_0^{\infty }P(X>x)dx$ and ${\int }_{-\infty }^{0}P(X\le x)$ both converge. Then

$$E[X]={\int }_0^{\infty }P(X>x)dx-{\int }_{-\infty }^{0}P(X\le x)$$

Definition

Let $Y=\{y_{ij}\}$ be $n\times p$ matrix. Then,

$$E[Y]=\{E[y_{ij}]\}$$

Properties

• $E[X+Y]=E[X]+E[Y]$
• $E[aX+bY]=aE[X]+bE[Y]$
• $E[AXB]=AE[X]B\ne ABE[X]$ (multivariate case)
• $L\le X\le U⟹L\le E[X]\le U$
• $|E[X]|\le E[|X|]$
• $Y=g(X)⟹E[Y]=E[g(X)]=\int g(x)f(x)dx$

Note that if $g(x)<h(x)$ then $E[g(x)]<E[h(x)]$ for all $x$.

Expectation for non-negative rv

Let $X$ be a non-negative rv. Then

$$E[X]={\int }_0^{\infty }[1-F(X)]dx$$