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Suppose that X and Y are independent random variables. Define Z - max(X,Y). (a) Show, justifying your working, that the cdf of Z, Fz, can be written in terms of the cdf of X, Fx, and the cdf of Y, Fy, as F2(z) - Fx(z) Fy(z). Hint: Recall that Fz(z) - P(Z 0, and that the random variable y follows the uniform distribution with cdf Fy (y)-y on 0

Suppose that X and Y are independent random variables. Define Z - max(X,Y). (a) Show, justifying your working, that the cdf of Z, Fz, can be written in terms of the cdf of X, Fx, and the cdf of Y, Fy, as F2(z) - Fx(z) Fy(z). Hint: Recall that Fz(z) - P(Z 0, and that the random variable y follows the uniform distribution with cdf Fy (y)-y on 0

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.8: Probability
Problem 31E
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Suppose that X and Y are independent random variables. Define Z – max(X,Y). (a) Show, justifying your working, that the cdf of Z, Fz, can be written in terms of the cdf of X, Fx, and the cdf of Y, Fy, as Fz(z) – Fx(z) Fy(2). Hint: Recall that Fz(z) – P(Z < z). (b) Suppose now that the random variable X follows the power distribution with cdf Fx(x) – xº on 0 < x < 1, where 3 > 0, and that the random variable Y follows the uniform distribution with cdf Fy(y) – y on 0< y< 1. What is the support of Z? What is the cdf of Z on its support? To which known family of distributions does this cdf belong?
Transcribed Image Text:Suppose that X and Y are independent random variables. Define Z – max(X,Y). (a) Show, justifying your working, that the cdf of Z, Fz, can be written in terms of the cdf of X, Fx, and the cdf of Y, Fy, as Fz(z) – Fx(z) Fy(2). Hint: Recall that Fz(z) – P(Z < z). (b) Suppose now that the random variable X follows the power distribution with cdf Fx(x) – xº on 0 < x < 1, where 3 > 0, and that the random variable Y follows the uniform distribution with cdf Fy(y) – y on 0< y< 1. What is the support of Z? What is the cdf of Z on its support? To which known family of distributions does this cdf belong?
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