Expected Value

Definition

$E(X)={\int }_{\Omega }XdP={\int }_{\mathbb{R}}xd{\mu }_X={\int }_{\mathbb{R}}xd{F}_X$ The expected value of the Random Variable $X$ on Probability Space $(\Omega ,\mathcal{F},P)$

Continuous

$E(X)={\int }_{-\infty }^{\infty }x{f}_X(x)dx$ where $f_X(x)$ is a PDF of Random Variable $X$

The expected value of the Random Variable $X$ when $F_X$ satisfies absolute continuous over $\lambda$, ${\mu }_X<<\lambda$

Discrete

$E(X)={\sum }_{x}x{p}_X(x)$ where $p_X(x)$ is a PMF of Random Variable $X$

The expected value of the Random Variable $X$ when $F_X$ satisfies absolute continuous over $\#$, ${\mu }_X<<{\#}_X$ In other words, $F_X$ has at most countable jumps.

Expected Value of a Function

$E(g(x))={\int }_{\mathbb{R}}g(x)d{\mu }_X$

Continuous

$E(g(x))={\int }_{\mathbb{R}}g(x)f_X(x)dx$

Discrete

$E(g(x))=\sum _{x\in S_X}g(x)p_X(x)$

Properties

Linearity

Random Variables

If $\exists k_{i}E[X_i]$, then $E\left[{\sum }_{i=1}^n{k}_i{X}_i\right]={\sum }_{i=1}^n{k}_{i}E(X_i)$

Matrix of Random Variables

Let $W_1,W_2$ be a $m\times n$ matrices of random variables, $A_1,A_2$ be $k\times m$ matrices of constants, and $B$ a $n\times l$ matrix of constant. Then, $E[A_1{W}_1+A_2{W}_2]=A_{1}E[W_1]+A_{2}E[W_2]$

$E[A_1{W}_{1}B]=A_{1}E[W_1]B$

Notations

Expression	Discrete	Continuous
Expression for the event $2^{\Omega }$ and the probability $P$	${\sum }_{\omega \in \Omega }X(\omega )\cdot P(\omega )$	${\int }_{\Omega }X(\omega )\cdot dP(\omega )$
Expression for the measurable space $(\mathbb{R},\mathcal{R})$ and the distribution ${\mu }_X$	${\sum }_{x\in \mathbb{R}}x\cdot f(x)$	${\int }_{\mathbb{R}}f(x)\lambda (x)={\int }_{\mathbb{R}}\frac{d{\mu }_X}{d\lambda }\lambda (x)={\int }_{\mathbb{R}}d{\mu }_X={\int }_{\mathbb{R}}d{F}_X$