# The Bernoulli Shortcut

If $X$ has a Standard Bernoulli Distribution, then it can only have values 0 or 1 with probabilities $q$ and $1-q$.  Any random variables that can only have 2 values is a scaled and translated version of the standard bernoulli distribution.

Expected Value and Variance:

For a standard bernoulli distribution, $E[X] = q$ and $Var(X) = q(1-q)$.  If $Y$ is a random variable that can only have values $a$ and $b$ with probabilities $q$ and $(1-q)$ respectively, then

$\begin{array}{rl} Y &= (a-b)X +b \\ E[Y] &= (a-b)E[X] +b \\ Var(Y) &= (a-b)^2Var(X) \\ &= (a-b)^2q(1-q) \end{array}$