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You generate three random numbers from (0, 1). Let X be the smallest of these three numbers and Y the largest. What are the conditional expected values E(X | Y = y) and E(Y | X = x)?

You generate three random numbers from (0, 1). Let X be the smallest of these three numbers and Y the largest. What are the conditional expected values E(X | Y = y) and E(Y | X = x)?

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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Question 7.39

7.36 Consider again Example 7.1. What is the expected value of the number of consolation prizes won, given that no main prize has been won? 7.37 Consider again Example 7.3. What is the expected value of the lifetime of the second circuit, given that the first circuit has failed after s time units? 7.38 You choose three different numbers at random from the number 1,2,..., 100. Let X be the smallest of these three numbers and Y the largest. What are the conditional expected values E(X | Y = y) and E(Y|X=x)? 7.39 You generate three random numbers from (0, 1). Let X be the smallest of these three numbers and Y the largest. What are the conditional expected values E(X | Y = y) and E(Y | X = x)? 7.40 Verify that E(X | Y = y) = E(X) for all y if X and Y are independent. Also, verify that cov(X, Y) = 0 if E(X | Y = y) does not depend on y. 7.41 Let (X, Y) have a bivariate normal density with parameters (41, 42,0 o2.p), where o? = 02. What is E(X | X + Y = v)? Hint: Use the result of Problem 6.9. 7.42 Let the joint density function f(x, y) of the random variables X and Y be equal to 1 for -y < x < y₂0 < x < 1 and otherwise. What are t the
Transcribed Image Text:7.36 Consider again Example 7.1. What is the expected value of the number of consolation prizes won, given that no main prize has been won? 7.37 Consider again Example 7.3. What is the expected value of the lifetime of the second circuit, given that the first circuit has failed after s time units? 7.38 You choose three different numbers at random from the number 1,2,..., 100. Let X be the smallest of these three numbers and Y the largest. What are the conditional expected values E(X | Y = y) and E(Y|X=x)? 7.39 You generate three random numbers from (0, 1). Let X be the smallest of these three numbers and Y the largest. What are the conditional expected values E(X | Y = y) and E(Y | X = x)? 7.40 Verify that E(X | Y = y) = E(X) for all y if X and Y are independent. Also, verify that cov(X, Y) = 0 if E(X | Y = y) does not depend on y. 7.41 Let (X, Y) have a bivariate normal density with parameters (41, 42,0 o2.p), where o? = 02. What is E(X | X + Y = v)? Hint: Use the result of Problem 6.9. 7.42 Let the joint density function f(x, y) of the random variables X and Y be equal to 1 for -y < x < y₂0 < x < 1 and otherwise. What are t the
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