Problem 2 Let X be Uniform(0, 1) and Y be Exponential(1). Assume that X and Y are independent. i. Find the PDF of Z = X + Y using convolution. ii. Find the moment generating function, Mz(s), of Z by evaluating E[e®Z]. Assume that s < 0. iii. Check that the moment generating function of Z is the product of the moment gen- erating functions of X and Y.

A First Course in Probability (10th Edition)
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ISBN:9780134753119
Author:Sheldon Ross
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Chapter1: Combinatorial Analysis
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Problem 2 Let X be Uniform(0, 1) and Y be Exponential(1). Assume that X and Y are
independent.
i. Find the PDF of Z = X +Y using convolution.
ii. Find the moment generating function, Mz(s), of Z by evaluating E[esZ]. Assume
that s < 0.
iii. Check that the moment generating function of Z is the product of the moment gen-
erating functions of X and Y.
Transcribed Image Text:Problem 2 Let X be Uniform(0, 1) and Y be Exponential(1). Assume that X and Y are independent. i. Find the PDF of Z = X +Y using convolution. ii. Find the moment generating function, Mz(s), of Z by evaluating E[esZ]. Assume that s < 0. iii. Check that the moment generating function of Z is the product of the moment gen- erating functions of X and Y.
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