Answered: 4), X2 ∼ N(μ = 1, σ^2 = 16), and X3…

4), X2 ∼ N(μ = 1, σ^2 = 16), and X3 ∼N(μ = −4, σ^2 = 9). Let X1, X2, and X3 be independent and Y = (X1 + 2X2 + X3)^2. What is the mean and variance of Y?

Calculus For The Life Sciences

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter1: Functions

Section1.2: The Least Square Line

Problem 1E

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Question

Let X1 ∼ N(μ = 2, σ^2 = 4), X2 ∼ N(μ = 1, σ^2 = 16), and X3 ∼N(μ = −4, σ^2 = 9). Let X1, X2, and X3 be independent and Y = (X1 + 2X2 + X3)^2. What is the mean and variance of Y?

Expert Solution

Step 1

A random variable follows a Normal distribution with mean $μ$ and variance $σ^{2}$

Random Variable	Mean $μ$	Variance $σ^{2}$
X1	2	4
X2	1	16
X3	-4	9

The random variables X1, X2 and X3 are independent

Given Y is another random variable

$Y = {(X_{1} + 2 X_{2} + X_{3})}^{2}$

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