Discrete Random Variables

These are variables that can take on a countable set of discrete values. The sum of the probabilities of each possible outcome is always $1$.

Expected value

The expected value (the mean) of a discrete random variable is given by the formula $$ \begin{equation}\begin{aligned} E(X)&=\sum{xP(X=x)}\\ \end{aligned}\end{equation} $$

where $X$ is the variable e.g. the face obtained from rolling a fair die and $x$ is a particular outcome e.g. $1$, $2$, etc.

Variance

The variance of the discrete random variable is $$ \begin{equation}\begin{aligned} Var(X)&=\sum{x^2P(X=x)}-[E(X)]^2\\ \end{aligned}\end{equation} $$

Transformations

$$ \begin{equation}\begin{aligned} E(2X)&=2E(X)\\ E(2X+1)&=2E(X)+1\\ E(aX)&=aE(X)\\ E(aX+b)&=aE(X)+b\\ \end{aligned}\end{equation} $$ $$ \begin{equation}\begin{aligned} Var(2X)&=2^2Var(X)\\ Var(2X+1)&=2^2Var(X)\\ Var(aX)&=a^2Var(X)\\ Var(aX+b)&=a^2Var(X)\\ \end{aligned}\end{equation} $$