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7. Assume that X is a nonnegative, integer-valued random variable with p.g.f. G(z). Let G(¹)(z) denote the nth derivative of G. Let pk = P{X = k} for k = 0, 1, 2,.... Express each of the following as a function of G. (a) E[X]. (b) E[X(X - 1)]. (c) E[X(X - 1)(x - 2)]. (d) Var(X). (e) po. (f) P1. (g) Pn for n = 0, 1, 2,....

7. Assume that X is a nonnegative, integer-valued random variable with p.g.f. G(z). Let G(¹)(z) denote the nth derivative of G. Let pk = P{X = k} for k = 0, 1, 2,.... Express each of the following as a function of G. (a) E[X]. (b) E[X(X - 1)]. (c) E[X(X - 1)(x - 2)]. (d) Var(X). (e) po. (f) P1. (g) Pn for n = 0, 1, 2,....

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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7. Assume that X is a nonnegative, integer-valued random variable with p.g.f. G(z). Let G(¹) (z) denote the nth derivative of G. Let pk = P{X = k} for k = 0, 1, 2,.... Express each of the following as a function of G. (a) E[X]. (b) E[X(X - 1)]. (c) E[X(X - 1)(x - 2)]. (d) Var(X). (e) po. (f) P1. (g) Pn for n = 0, 1, 2,.... (h) Suppose an archer takes X shots at a target. Assume the shots are inde- pendent, and the archer hits the bullseye on any attempt with probability p. What is the probability that the archer is hits the bullseye on all X attempts? (i) Suppose the effective interest rate per period is i%. What is the present value of one dollar received at time X?
Transcribed Image Text:7. Assume that X is a nonnegative, integer-valued random variable with p.g.f. G(z). Let G(¹) (z) denote the nth derivative of G. Let pk = P{X = k} for k = 0, 1, 2,.... Express each of the following as a function of G. (a) E[X]. (b) E[X(X - 1)]. (c) E[X(X - 1)(x - 2)]. (d) Var(X). (e) po. (f) P1. (g) Pn for n = 0, 1, 2,.... (h) Suppose an archer takes X shots at a target. Assume the shots are inde- pendent, and the archer hits the bullseye on any attempt with probability p. What is the probability that the archer is hits the bullseye on all X attempts? (i) Suppose the effective interest rate per period is i%. What is the present value of one dollar received at time X?
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