3. x and y are independent zero mean Gaussian random variables with variance of and o. Let z = (x + y) / 2, w = (x - y) / 2. (a) Find the joint pdf f(z,w), (b) find the marginal pdf fz(=), (c) Are z and w independent?
3. x and y are independent zero mean Gaussian random variables with variance of and o. Let z = (x + y) / 2, w = (x - y) / 2. (a) Find the joint pdf f(z,w), (b) find the marginal pdf fz(=), (c) Are z and w independent?
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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parts A and B in detail. Thanks
Transcribed Image Text:3. \( x \) and \( y \) are independent zero mean Gaussian random variables with variance \( \sigma_x^2 \) and \( \sigma_y^2 \). Let \( z = (x + y) / 2 \), \( w = (x - y) / 2 \).
(a) Find the joint pdf \( f(z, w) \),
(b) Find the marginal pdf \( f_z(z) \),
(c) Are \( z \) and \( w \) independent?
Expert Solution
Step 1
Here we have x and y, two independent zero mean Gaussian random variables with and .
So the pdf of x and y are given by,
Let, and
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