Q2(iv.) If X1, X2, · are independent and identically distributed exponential random variables with parameter A and N is a geometric random variable with parameter p independent of the sequence X1, X2, ... (ie P(N =n) = p(1 – p)"-1), then show that N Y =X; i=1 s exponentially distributed with parameter Xp using the moment generating function from part (i). Hint: The m.g.f. of an exponential random variable with parameter A is A/(A – t).
Q2(iv.) If X1, X2, · are independent and identically distributed exponential random variables with parameter A and N is a geometric random variable with parameter p independent of the sequence X1, X2, ... (ie P(N =n) = p(1 – p)"-1), then show that N Y =X; i=1 s exponentially distributed with parameter Xp using the moment generating function from part (i). Hint: The m.g.f. of an exponential random variable with parameter A is A/(A – t).
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.8: Probability
Problem 19E
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