Random Variables and Expectation 1

Definition 2.1

A random variable $X$ on a sample space $\Omega$ is a real-valued (measurable) function on $\Omega$: that is, $\Omega \rightarrow \mathbb{R}$. A discrete random variable is a random variable that takes on only a finite or countably infinite number of values.

Since the random variables are functions, they are usually denoted by capital letters, while real numbers are usually denoted by lowercase letters.

Defination 2.2

Two random variables $X$ and $Y$ are independent if and only if

$Pr((X=x)\cap(Y=y))=Pr(X=x)\cdot Pr(Y=y)$

for all values x and y. Similarly, random variables $X_1$, $X_2$,.... are mutally independent if and only if, for any subset $I\subseteq[i,k]$ and any values $x_i$, $i\in I$,

$Pr\left(\bigcap_{i\in I}(X_i=x_i)\right)=\prod_{i \in I}Pr(X_i=x_i)$

Definition 2.3

The expectation of a discrete random variable X, denoted by $E[X]$ , is given by

$E[X]=\sum_iiPr(X=i)$