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X₁ and X₂ are independent random variables having the standard normal distribution. Suppose Y₁ = X₁ and Y₂ = X₁ + X₂. Find the joint p.d.f. of Y₁ and Y₂. Find the mean and the variance of Y₂. If n independent random variables have the same gamma distribution with the pa- rameters a and ß, find the moment-generating function of their sum Y and find the p.d.f. of Y.

X₁ and X₂ are independent random variables having the standard normal distribution. Suppose Y₁ = X₁ and Y₂ = X₁ + X₂. Find the joint p.d.f. of Y₁ and Y₂. Find the mean and the variance of Y₂. If n independent random variables have the same gamma distribution with the pa- rameters a and ß, find the moment-generating function of their sum Y and find the p.d.f. of 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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### Educational Content: Understanding Joint Probability and Gamma Distribution #### Problem 1 **Independent Random Variables and Joint Probability** Consider two independent random variables \( X_1 \) and \( X_2 \), each having a standard normal distribution. Define the following: - \( Y_1 = X_1 \) - \( Y_2 = X_1 + X_2 \) **Tasks:** 1. Find the joint probability density function (p.d.f.) of \( Y_1 \) and \( Y_2 \). 2. Calculate the mean and the variance of \( Y_2 \). #### Problem 2 **Gamma Distribution and Moment-Generating Functions** Suppose you have \( n \) independent random variables, each following a gamma distribution with parameters \( \alpha \) and \( \beta \). **Tasks:** 1. Determine the moment-generating function of their sum \( Y \). 2. Find the probability density function (p.d.f.) of \( Y \).
Transcribed Image Text:### Educational Content: Understanding Joint Probability and Gamma Distribution #### Problem 1 **Independent Random Variables and Joint Probability** Consider two independent random variables \( X_1 \) and \( X_2 \), each having a standard normal distribution. Define the following: - \( Y_1 = X_1 \) - \( Y_2 = X_1 + X_2 \) **Tasks:** 1. Find the joint probability density function (p.d.f.) of \( Y_1 \) and \( Y_2 \). 2. Calculate the mean and the variance of \( Y_2 \). #### Problem 2 **Gamma Distribution and Moment-Generating Functions** Suppose you have \( n \) independent random variables, each following a gamma distribution with parameters \( \alpha \) and \( \beta \). **Tasks:** 1. Determine the moment-generating function of their sum \( Y \). 2. Find the probability density function (p.d.f.) of \( Y \).
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