Exercise 6. Suppose X1 and X2 are independent standard normal randomvariables. For a constant a letWi = cos(a)X1 – sin(a)X2W2 = sin(a)X1 + cos(a)X2|3DShow that W1 and W2 are independent standard normal random variables.Remark Geometrically the given transformation is just a rotation of the planeby an angle a. So the conclusion is simply a reflection of the fact that the jointdistribution has circular symmetry; i.e. is invariant under rotations.

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Answered: Exercise 6. Suppose X1 and X2 are…

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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Exercise 6. Suppose X1 and X2 are independent standard normal random variables. For a constant a let Wi = cos(a)X1 – sin(a)X2 W2 = sin(@)X1 + cos(@)X2 - Show that W and W2 are independent standard normal random variables.