The discrete random variable $X$ has p.d.f. given by $P(X=x) = qp^{x}$, where \(x = 0, 1, 2

Question

The discrete random variable $X$ has p.d.f. given by $P(X=x) = qp^{x}$, where $x = 0, 1, 2, \dots$ Express $q$ in terms of $p$. Find $\mathrm{E}(X)$. Find $\mathrm{E}(X(X-1))$ and determine $\mathrm{E}(X^2)$, and hence $\mathrm{Var}(X)$.

Accepted Answer

For $0\leq p<1$: **(i)** $q=1-p$. **(ii)** $\mathrm{E}(X)=\frac{p}{1-p}$. **(iii)** $\mathrm{E}(X(X-1))=\frac{2p^2}{(1-p)^2}$, $\mathrm{E}(X^2)=\frac{p(1+p)}{(1-p)^2}$, $\mathrm{Var}(X)=\frac{p}{(1-p)^2}$.

Computation of variance using transformed variable

Computation of variance using transformed variable

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