question number 5.45 The density offw (w) = = = = 100 \v\ e = {(1 + ³²) ²2,fu2πdv =Using the change of variable u =equal to fe-(1+²) u du =Cauchy density.27/7 5.0° V2πve¯ {(1+w²}x²dv.dy1², we have that f ve¯ {(1+x²)x² dv is12. This shows that X/Y has the two-sidedProblemsXX+Y*5.45 The random variables X and Y are independent and exponentiallydistributed with parameter u. Let V = X + Y and W = What is thejoint 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 distributedon (0, 1). Let V = X+Y and W = X/Y. What is the joint density functionof 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 havingthe standard normal distribution. What is the joint density function of Vand W? Are V and W independent?5.48 The continuous random variables X and Y have the joint density functionf(x, y) = xe-x(1+y²) for x, y > 0 and f(x, y) = 0 otherwise. Show thatTCthe random variables Y. (YJy

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)

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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