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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 (2) 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 (2) 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

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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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(z). 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? (c) Now consider the situation where X and Y are independent and are also identically distributed, each following the uniform distribution U(0, 1). In this case, it can be shown that Z has the cdf Fz(z) – 22 on 0 <z < 1. What is the distribution of Z in this case? (d) Find the expected values of X, Y and Z, and hence comment on their values relative to each other. Hint: You don't need to derive these expected values by integration. Instead, refer to the table of known distributions in the M347 Handbook.
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(z). 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? (c) Now consider the situation where X and Y are independent and are also identically distributed, each following the uniform distribution U(0, 1). In this case, it can be shown that Z has the cdf Fz(z) – 22 on 0 <z < 1. What is the distribution of Z in this case? (d) Find the expected values of X, Y and Z, and hence comment on their values relative to each other. Hint: You don't need to derive these expected values by integration. Instead, refer to the table of known distributions in the M347 Handbook.
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