Assume that X is a nonnegative, integer-valued random variable. Let G(z) = E[z]. To simplify notation,...

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Assume that X is a nonnegative, integer-valued random variable. Let G(z) = E[z]. To simplify notation, let Pk = px (k)= P{X = k} for k = 0, 1, 2,.... (a) Use LOTUS to express G(z) = E[zX] as a power series. (Your answer should look something like 80 Pk = k=0 where I left question marks for something that's missing.) (b) What is G(0)? (c) What is G(1)? (d) What is G'(z)? Give two answers: one is a series, the other is E[of something]. (e) What is G'(0)? What is G'(1)? (g) What is G"(z)? Give two answers: one is a series, the other is E[of something]. (h) What is G" (0)? (i) What is G"(1)? (j) Compute G(z) = E[z*] where ?? Pk e-λrk k! for k= 0, 1, 2,... and λ > 0. (k) What is G'(1)? (1) What is G" (1)? (m) Compute the mean and variance of X from G'(1) and G"(1). (I think this is an easier way of computing the mean and variance of this distribution than the way we did in class.) Assume that X is a nonnegative, integer-valued random variable. Let G(z) = E[z]. To simplify notation, let Pk = px (k)= P{X = k} for k = 0, 1, 2,.... (a) Use LOTUS to express G(z) = E[zX] as a power series. (Your answer should look something like 80 Pk = k=0 where I left question marks for something that's missing.) (b) What is G(0)? (c) What is G(1)? (d) What is G'(z)? Give two answers: one is a series, the other is E[of something]. (e) What is G'(0)? What is G'(1)? (g) What is G"(z)? Give two answers: one is a series, the other is E[of something]. (h) What is G" (0)? (i) What is G"(1)? (j) Compute G(z) = E[z*] where ?? Pk e-λrk k! for k= 0, 1, 2,... and λ > 0. (k) What is G'(1)? (1) What is G" (1)? (m) Compute the mean and variance of X from G'(1) and G"(1). (I think this is an easier way of computing the mean and variance of this distribution than the way we did in class.)