(a) Let X be a random variable that follows a Bernoulli distribution withparameter p, 0 < p < 1. Derive the moment generating function(MGF) of X in terms of p and q = (1 − p). (b) Let X₁, X2,..., Xn be independent random variables and supposethat the MGF of X, is M;(t), defined for t € Aj. Let Y = X₁ + X2+j·+ Xn. Show that the MGF of Y is My(t) = [];=1 Mj(t), definedfor t e n=1 Aj.(c) Use the results in (a) and (b) to find the MGF of a binomial randomvariable Y with parameters n and p.(d) Use the result in (d) to find the variance of a binomial random vari-able Y with parameters n and p.

Answered: Image /qna-images/answer/5d62ee82-7047-48df-ad7b-756a65283c07.jpg

(b) Let X₁, X2,..., Xn be independent random variables and suppose that the MGF of X; is Mj(t), defined for t € Aj. Let Y = X₁ + X2+ + Xn. Show that the MGF of Y is My(t) = [];=1 Mj(t), defined for t E-1Aj. =1 (c) Use the results in (a) and (b) to find the MGF of a binomial random variable Y with parameters n and p. (d) Use the result in (d) to find the variance of a binomial random vari- able Y with parameters n and p.

(b) Let X₁, X2,..., Xn be independent random variables and suppose that the MGF of X; is Mj(t), defined for t € Aj. Let Y = X₁ + X2+ + Xn. Show that the MGF of Y is My(t) = [];=1 Mj(t), defined for t E-1Aj. =1 (c) Use the results in (a) and (b) to find the MGF of a binomial random variable Y with parameters n and p. (d) Use the result in (d) to find the variance of a binomial random vari- able Y with parameters n and p.

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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Question

(a) Let X be a random variable that follows a Bernoulli distribution with
parameter p, 0 < p < 1. Derive the moment generating function
(MGF) of X in terms of p and q = (1 − p).

(b) Let X₁, X2,..., Xn be independent random variables and suppose
that the MGF of X, is M;(t), defined for t € Aj. Let Y = X₁ + X2+
j
·
+ Xn. Show that the MGF of Y is My(t) = [];=1 Mj(t), defined
for t e n=1 Aj.
(c) Use the results in (a) and (b) to find the MGF of a binomial random
variable Y with parameters n and p.
(d) Use the result in (d) to find the variance of a binomial random vari-
able Y with parameters n and p.

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