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p. Let Z Let X, Y be independent geometrically distributed random variables with parameter = E[X|X + Y]. Find the expected value and the variance of Z. Hint: E[X + Y|X+Y] = X + Y. =

p. Let Z Let X, Y be independent geometrically distributed random variables with parameter = E[X|X + Y]. Find the expected value and the variance of Z. Hint: E[X + Y|X+Y] = X + Y. =

MATLAB: An Introduction with Applications
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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Let \( X, Y \) be independent geometrically distributed random variables with parameter \( p \). Let \( Z = \mathbb{E}[X \mid X + Y] \). Find the expected value and the variance of \( Z \). Hint: \(\mathbb{E}[X + Y \mid X + Y] = X + Y\).
Transcribed Image Text:Let \( X, Y \) be independent geometrically distributed random variables with parameter \( p \). Let \( Z = \mathbb{E}[X \mid X + Y] \). Find the expected value and the variance of \( Z \). Hint: \(\mathbb{E}[X + Y \mid X + Y] = X + Y\).
Expert Solution
Step 1: Determine the given variable

Given:
X, Y are independent geometrically distributed random variables with parameter p.
The expected value and variance for a geometrically distributed random variable are:

E left parenthesis X right parenthesis equals 1 over p Var left parenthesis X right parenthesis equals fraction numerator 1 minus p over denominator p squared end fraction

Likewise, for Y:

E left parenthesis Y right parenthesis equals 1 over p Var left parenthesis Y right parenthesis equals fraction numerator 1 minus p over denominator p squared end fraction

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