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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)?

Holt Mcdougal Larson Pre-algebra: Student Edition 2012
Holt Mcdougal Larson Pre-algebra: Student Edition 2012
1st Edition
ISBN:9780547587776
Author:HOLT MCDOUGAL
Publisher:HOLT MCDOUGAL
Chapter11: Data Analysis And Probability
Section: Chapter Questions
Problem 8CR
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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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