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Xx 0 1/6 1/6 1/6 1/6 1/6 0 0 Find Var(X), Var(Y), and the covariance and correlation of (X,Y). Determine Var(X + Y). Are X and Y independent? Comment on the relation between Var(X) + Var(Y) and Var(X + Y). 1 1/6 2

Xx 0 1/6 1/6 1/6 1/6 1/6 0 0 Find Var(X), Var(Y), and the covariance and correlation of (X,Y). Determine Var(X + Y). Are X and Y independent? Comment on the relation between Var(X) + Var(Y) and Var(X + Y). 1 1/6 2

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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Question

The joint p.f. of (X,Y) is given by: 

### Joint Probability Distribution of X and Y: Below is the joint probability distribution table for two random variables X and Y. | X \ Y | 0 | 1 | 2 | |-------|-----|-----|-----| | 0 | 1/6 | 1/6 | 1/6 | | 1 | 1/6 | 1/6 | 0 | | 2 | 1/6 | 0 | 0 | ### Tasks: a. **Find Var(X), Var(Y), and the covariance and correlation of (X, Y).** b. **Determine Var(X + Y).** c. **Are X and Y independent?** d. **Comment on the relation between Var(X) + Var(Y) and Var(X + Y).** ### Explanation: - **a**) To find the variance and covariance, use the formulas for variance and covariance of discrete random variables. - Var(X) (Variance of X): \( \mathrm{Var}(X) = E(X^2) - [E(X)]^2 \) - Var(Y) (Variance of Y): \( \mathrm{Var}(Y) = E(Y^2) - [E(Y)]^2 \) - Covariance of X and Y: \( \mathrm{Cov}(X, Y) = E(XY) - E(X)E(Y) \) - Correlation of X and Y: \( \mathrm{Corr}(X, Y) = \frac{\mathrm{Cov}(X, Y)}{\sqrt{\mathrm{Var}(X) \mathrm{Var}(Y)}} \) - **b**) To determine \( \mathrm{Var}(X + Y) \): - Use: \( \mathrm{Var}(X + Y) = \mathrm{Var}(X) + \mathrm{Var}(Y) + 2\mathrm{Cov}(X, Y) \) - **c**) To check if X and Y are independent: - If \( P(X = x, Y = y) = P(X = x)P(Y = y) \) for all x, y, then X and Y are independent. - **d**) Comment on the relation between \( \mathrm{Var}(X) + \mathrm{Var}(Y) \) and \( \mathrm{Var}(X + Y
Transcribed Image Text:### Joint Probability Distribution of X and Y: Below is the joint probability distribution table for two random variables X and Y. | X \ Y | 0 | 1 | 2 | |-------|-----|-----|-----| | 0 | 1/6 | 1/6 | 1/6 | | 1 | 1/6 | 1/6 | 0 | | 2 | 1/6 | 0 | 0 | ### Tasks: a. **Find Var(X), Var(Y), and the covariance and correlation of (X, Y).** b. **Determine Var(X + Y).** c. **Are X and Y independent?** d. **Comment on the relation between Var(X) + Var(Y) and Var(X + Y).** ### Explanation: - **a**) To find the variance and covariance, use the formulas for variance and covariance of discrete random variables. - Var(X) (Variance of X): \( \mathrm{Var}(X) = E(X^2) - [E(X)]^2 \) - Var(Y) (Variance of Y): \( \mathrm{Var}(Y) = E(Y^2) - [E(Y)]^2 \) - Covariance of X and Y: \( \mathrm{Cov}(X, Y) = E(XY) - E(X)E(Y) \) - Correlation of X and Y: \( \mathrm{Corr}(X, Y) = \frac{\mathrm{Cov}(X, Y)}{\sqrt{\mathrm{Var}(X) \mathrm{Var}(Y)}} \) - **b**) To determine \( \mathrm{Var}(X + Y) \): - Use: \( \mathrm{Var}(X + Y) = \mathrm{Var}(X) + \mathrm{Var}(Y) + 2\mathrm{Cov}(X, Y) \) - **c**) To check if X and Y are independent: - If \( P(X = x, Y = y) = P(X = x)P(Y = y) \) for all x, y, then X and Y are independent. - **d**) Comment on the relation between \( \mathrm{Var}(X) + \mathrm{Var}(Y) \) and \( \mathrm{Var}(X + Y
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