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2. Let U₁, U₂,... be IID Uniform(0, 1) random variables. Let M₁ = I₁U; be the product of the first n of them. (a) Show that ; = -log U; is distributed as an Exponential random variable with a certain rate. Hint: If U is Uniform(0, 1), then so is 1 - U. (b) Find the PDF of Sn = 15. (c) Finally, find the PDF of Mn. Hint: Mn = exp(-Sn)

2. Let U₁, U₂,... be IID Uniform(0, 1) random variables. Let M₁ = I₁U; be the product of the first n of them. (a) Show that ; = -log U; is distributed as an Exponential random variable with a certain rate. Hint: If U is Uniform(0, 1), then so is 1 - U. (b) Find the PDF of Sn = 15. (c) Finally, find the PDF of Mn. Hint: Mn = exp(-Sn)

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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2. Let U₁, U₂,... be IID Uniform(0, 1) random variables. Let M₁ = II/₁U; be the product of the first n of them. (a) Show that i = -log U; is distributed as an Exponential random variable with a certain rate. Hint: If U is Uniform(0, 1), then so is 1 - U. (b) Find the PDF of Sn = 11 Ši. (c) Finally, find the PDF of Mn. Hint: Mn = exp(-Sn)
Transcribed Image Text:2. Let U₁, U₂,... be IID Uniform(0, 1) random variables. Let M₁ = II/₁U; be the product of the first n of them. (a) Show that i = -log U; is distributed as an Exponential random variable with a certain rate. Hint: If U is Uniform(0, 1), then so is 1 - U. (b) Find the PDF of Sn = 11 Ši. (c) Finally, find the PDF of Mn. Hint: Mn = exp(-Sn)
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