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Problems 5.45 The random variables X and Y are independent and exponentially distributed with parameter u. Let V = X + Y and W = yy. What is the joint density function of V and W? Prove that V and W are independent. 5.16 Th

Problems 5.45 The random variables X and Y are independent and exponentially distributed with parameter u. Let V = X + Y and W = yy. What is the joint density function of V and W? Prove that V and W are independent. 5.16 Th

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 number 5.45

The density of fw (w) = = = = 100 \v\ e = {(1 + ³²) ²2, fu 2π dv = Using the change of variable u = equal to fe-(1+²) u du = Cauchy density. 2 7/7 5.0° V 2π ve¯ {(1+w²}x² dv. dy 1², we have that f ve¯ {(1+x²)x² dv is 12. This shows that X/Y has the two-sided Problems X X+Y* 5.45 The random variables X and Y are independent and exponentially distributed with parameter u. Let V = X + Y and W = What is the joint density function of V and W? Prove that V and W are independent. 5.46 The random variables X and Y are independent and uniformly distributed on (0, 1). Let V = X+Y and W = X/Y. What is the joint density function of V and W? Are V and W independent? 5.47 The random variables V and W are defined by V = Z² + 2 and W = Z-Z2, where Z₁ and Z₂ are independent random variables each having the standard normal distribution. What is the joint density function of V and W? Are V and W independent? 5.48 The continuous random variables X and Y have the joint density function f(x, y) = xe-x(1+y²) for x, y > 0 and f(x, y) = 0 otherwise. Show that TC the random variables Y. (Y Jy
Transcribed Image Text:The density of fw (w) = = = = 100 \v\ e = {(1 + ³²) ²2, fu 2π dv = Using the change of variable u = equal to fe-(1+²) u du = Cauchy density. 2 7/7 5.0° V 2π ve¯ {(1+w²}x² dv. dy 1², we have that f ve¯ {(1+x²)x² dv is 12. This shows that X/Y has the two-sided Problems X X+Y* 5.45 The random variables X and Y are independent and exponentially distributed with parameter u. Let V = X + Y and W = What is the joint density function of V and W? Prove that V and W are independent. 5.46 The random variables X and Y are independent and uniformly distributed on (0, 1). Let V = X+Y and W = X/Y. What is the joint density function of V and W? Are V and W independent? 5.47 The random variables V and W are defined by V = Z² + 2 and W = Z-Z2, where Z₁ and Z₂ are independent random variables each having the standard normal distribution. What is the joint density function of V and W? Are V and W independent? 5.48 The continuous random variables X and Y have the joint density function f(x, y) = xe-x(1+y²) for x, y > 0 and f(x, y) = 0 otherwise. Show that TC the random variables Y. (Y Jy
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