Question # 3:(a) Consider the binomial probability mass function px (k)= ()pk (1-p)n-k. Show that the mean isE[X] = np. (b) Consider the geometric probability mass function px (k) = p(1 - p) for k = 0,1,.... Show that themean is E[X] = (1-P)/p. \text { (c) Consider the Poisson probability mass function } p_{X}(k)=\frac{\lambda^{k}}{k !} e^{-\lambda} \text {. Show that the variance is } \operatorname{Var}[X]=\lambda \text {. } \text { (d) Consider the uniform probability mass function } p X(k)=\frac{1}{L} \text { for } k=1, \ldots, L \text {. Show that the variance } \text { is } \operatorname{Var}[X]=\frac{L^{2}-1}{12} \text {. Hint: } 1+2+\ldots+n=\frac{n(n+1)}{2} \text { and } 1^{2}+2^{2}+\ldots+n^{2}=\frac{n^{3}}{3}+\frac{n^{2}}{2}+\frac{n}{6} \text {. }

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Question # 3:(a) Consider the binomial probability mass function px (k)= ()pk (1-p)n-k. Show that the mean isE[X] = np. (b) Consider the geometric probability mass function px (k) = p(1 - p) for k = 0,1,.... Show that themean is E[X] = (1-P)/p. \text { (c) Consider the Poisson probability mass function } p_{X}(k)=\fra