X1, X2, , nare n independent zero mean Gaussian random variables M] / nis with equal variance o². Show that z = = [₁₁ + random variable with zero mean and o² = 0² /n. (Hint: use characteristic functions) + X Inis a Gaussian
X1, X2, , nare n independent zero mean Gaussian random variables M] / nis with equal variance o². Show that z = = [₁₁ + random variable with zero mean and o² = 0² /n. (Hint: use characteristic functions) + X Inis a Gaussian
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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![**Problem Statement:**
Let \( x_1, x_2, \cdots, x_n \) be \( n \) independent zero mean Gaussian random variables with equal variance \( \sigma^2 \).
**Objective:**
Show that
\[
z = \frac{x_1 + \cdots + x_n}{n}
\]
is a Gaussian random variable with zero mean and
\[
\sigma_z^2 = \frac{\sigma^2}{n}
\]
**Hint:** Use characteristic functions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9f5798cd-af25-40fe-9fff-d36841e46773%2F8d39ec36-d3d8-426e-85c6-a09c2e62cd7a%2Fk9b55c_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Let \( x_1, x_2, \cdots, x_n \) be \( n \) independent zero mean Gaussian random variables with equal variance \( \sigma^2 \).
**Objective:**
Show that
\[
z = \frac{x_1 + \cdots + x_n}{n}
\]
is a Gaussian random variable with zero mean and
\[
\sigma_z^2 = \frac{\sigma^2}{n}
\]
**Hint:** Use characteristic functions.
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