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Bernoulli random variables, also called indicator random variables.
Take on only two values, namely, 0 or 1. We say X is Bernoulli(p) random variable if
X = 1 with probability p and X = 0 with probability q = 1 − p.
Expected value
E(X) = 1P (X = 1) + 0P (X = 0) = P (X = 1) = p
Variance
Var(X) = E(X 2 ) − (E(X))2 = p − p2 = p(1 − p) = pq.
Note that X 2 = X always (when X is Bernoulli). Why? If X = 1 then X 2 = 1 too. If
X = 0 then X 2 = 0 too. So E(X 2 ) = E(X) = p.
In general, if X is Bernoulli, then X j = X for any positive integer j. Why? If X = 1
then X j = 1 too. If X = 0 then X j = 0 too.
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