A small discrete distribution

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Problem

Let $X\sim\operatorname{Binomial}(3,1/2)$ . List its pmf for $x=0,1,2,3$ , then compute $P(X\ge2)$ , $E[X]$ , and $\operatorname{Var}(X)$ .

Reveal answer or reference solution

P(X=x)=\binom3x2^{-3},

so the probabilities for $x=0,1,2,3$ are $(1,3,3,1)/8$ . Therefore

P(X\ge2)=\frac38+\frac18=\frac12

For a binomial variable, $E[X]=np=3/2$ and $\operatorname{Var}(X)=np(1-p)=3/4$ .

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A small discrete distribution

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