Random Variable

a> [!NOTE] Definition

A random variable on a probability space $\Omega$ is a function

$$X:\Omega \to \mathbb{R}.$$

It assigns a real number $X(\omega )$ to each outcome $\omega \in \Omega$.

So a random variable is not itself an event. It is a numerical quantity defined on the outcomes of a Probability Space.

Events defined by a random variable

From a random variable $X$, we get events such as

$$\{X=a\},\{X\le a\},\{X\in S\}.$$

For a discrete random variable, its distribution is given by

$$\mathrm{Pr}[X=a]=\mathrm{Pr}(\{\omega \in \Omega :X(\omega )=a\}).$$

Special case: indicator variable

An indicator variable of an event $A$ is the random variable

$$1_A(\omega )=\{\begin{array}{ll}1,& \omega \in A,\\ 0,& \omega \notin A.\end{array}$$

Its expected value is just: $\mathbb{E}[1_A]=\mathrm{Pr}[A]$

Density and cumulative distribution function

• probability density function (PDF): a function $f_X$ such that probabilities are obtained by integration
• cumulative distribution function (CDF) : $F_X(x)=\mathrm{Pr}[X\le x].$

If $X$ is continuous and has density $f_X$, then

$$\mathrm{Pr}[a\le X\le b]={\int }_a^b{f}_X(x)dx$$

and

$$F_X(x)={\int }_{-\infty }^x{f}_X(t)dt.$$

When $F_X$ is differentiable, the density is its derivative:

$$f_X(x)=F_X^{\prime }(x).$$

Further properties such as Expected value, conditional expectation, linearity of expectation, and moments are collected in Expected value.