Expected Value

probability-theory

Definition

Expected Value

The expected value is a weighted mean (average) of the possible outcomes of a random variable $X$.

Properties

Linearity

The expectation is linear, i.e. for all random variables $X,Y$ and scalars $a,b\in \mathbb{R}$ holds:

1. $\mathbb{E}[a\cdot X+b]=a\cdot \mathbb{E}[X]+b$
2. $\mathbb{E}[a\cdot X+b\cdot Y]=a\cdot \mathbb{E}[X]+b\cdot \mathbb{E}[Y]$

Independence

If the two random variables $X$ and $Y$ are independent, then:

$$\mathbb{E}[X\cdot Y]=\mathbb{E}[X]\cdot \mathbb{E}[Y]$$

Law of the Unconscious Statistician

For a measurable function $g$ (assuming absolute convergence) holds:

1. Discrete: $\mathbb{E}[g(X)]={\sum }_{i=1}^{\infty }g(x_i)\cdot f(x_i)$
2. Continuous: $\mathbb{E}[g(X)]={\int }_{-\infty }^{\infty }g(x)\cdot f(x)dx$

Discrete vs Continuous

Discrete

The expected value $\mathbb{E}[X]$ of a discrete random variable $X$ with values $x_1,x_2,\dots$ and probability mass function $f$ is given by:

$$\mathbb{E}[X]=\sum _{i=1}^{\infty }{x}_i\cdot f(x_i)=x_1\cdot f(x_1)+x_2\cdot f(x_2)+\dots$$

Continuous

The expected value $\mathbb{E}[X]$ of a continuous random variable $X$ with density $f$ is given by:

$$\mathbb{E}[X]={\int }_{-\infty }^{\infty }x\cdot f(x)dx$$

Note that absolutely convergence is assumed. Else, the expectation may be $\pm \infty$ or does not exist.