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7.15 Suppose that X and Y are continuous random variables such that the density function of X is fx(x) 2x for 0 < x < 1 and the conditional distribution of Y given that X = x is the uniform distribution on (0,r). Show that the conditional density of X given that Y = y is the uniform density on (y, 1). =

7.15 Suppose that X and Y are continuous random variables such that the density function of X is fx(x) 2x for 0 < x < 1 and the conditional distribution of Y given that X = x is the uniform distribution on (0,r). Show that the conditional density of X given that Y = y is the uniform density on (y, 1). =

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 7.15

7.12 Consider again Problem fx(x y) and fy (y|x)? 7.13 A point X is randomly chosen in (0, 1) and then a point Y is randomly chosen in (1 - X, 1). What are the probabilities P(X + Y > 1.5) and P(Y> 0.5)? 7.14 The candidates A and B participate in an election. The random variables X and Y denote the proportion of votes cast for the candidates A and B. The joint density function of X and Y is f(x, y) = 3(x+y) for x,y > 0 and x+y< 1 and f(x, y) = 0 otherwise. What is the conditional density function fx(x | y)? ctions 7.15 Suppose that X and Y are continuous random variables such that the density function of X is fx(x) = 2x for 0 < x < 1 and the conditional distribution of Y given that X = x is the uniform distribution on (0,x). Show that the conditional density of X given that Y = y is the uniform density on (y, 1). noney.920 7.16 Let X and Y be two continuous random variables with joint probability density f(x, y) = 2y = are the conditional density functions fx (x | y) and fy (y | x)? How do you simulate a random observation from f(x, y)? for 0 < y ≤ x< 1 and f(x, y) x² 7.17 Let the discrete random = 0 otherwise. What variable Y
Transcribed Image Text:7.12 Consider again Problem fx(x y) and fy (y|x)? 7.13 A point X is randomly chosen in (0, 1) and then a point Y is randomly chosen in (1 - X, 1). What are the probabilities P(X + Y > 1.5) and P(Y> 0.5)? 7.14 The candidates A and B participate in an election. The random variables X and Y denote the proportion of votes cast for the candidates A and B. The joint density function of X and Y is f(x, y) = 3(x+y) for x,y > 0 and x+y< 1 and f(x, y) = 0 otherwise. What is the conditional density function fx(x | y)? ctions 7.15 Suppose that X and Y are continuous random variables such that the density function of X is fx(x) = 2x for 0 < x < 1 and the conditional distribution of Y given that X = x is the uniform distribution on (0,x). Show that the conditional density of X given that Y = y is the uniform density on (y, 1). noney.920 7.16 Let X and Y be two continuous random variables with joint probability density f(x, y) = 2y = are the conditional density functions fx (x | y) and fy (y | x)? How do you simulate a random observation from f(x, y)? for 0 < y ≤ x< 1 and f(x, y) x² 7.17 Let the discrete random = 0 otherwise. What variable Y
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