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}



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