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Page 197, 3.4.30.* Compute P([X₁ + X2 − 2X3]² > 40) if X₁, X2, X3 are i. i. d. with com distribution N(1, 1).

Page 197, 3.4.30.* Compute P([X₁ + X2 − 2X3]² > 40) if X₁, X2, X3 are i. i. d. with com distribution N(1, 1).

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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please answer page 197 3.4.30

**Problem Statement:** Compute the probability \( P([X_1 + X_2 - 2X_3]^2 > 40) \) given that \( X_1, X_2, X_3 \) are independent and identically distributed (i.i.d.) random variables with a normal distribution \( N(1, 1) \). **Explanation:** This problem involves calculating the probability that the square of the linear combination \( X_1 + X_2 - 2X_3 \) exceeds 40. Variables \( X_1, X_2, \) and \( X_3 \) are normally distributed with mean 1 and variance 1.
Transcribed Image Text:**Problem Statement:** Compute the probability \( P([X_1 + X_2 - 2X_3]^2 > 40) \) given that \( X_1, X_2, X_3 \) are independent and identically distributed (i.i.d.) random variables with a normal distribution \( N(1, 1) \). **Explanation:** This problem involves calculating the probability that the square of the linear combination \( X_1 + X_2 - 2X_3 \) exceeds 40. Variables \( X_1, X_2, \) and \( X_3 \) are normally distributed with mean 1 and variance 1.
Expert Solution
Step 1

Given that X2, X2 and X3 are iid normal distribution N(1,1).

Let Y=X1+X2-2X3

Consider,

EY=EX1+X2-2X3=EX1+EX2-2EX3                              =1+1-2                              =0

VY=VX1+X2-2X3=VX1+VX2+-22VX3                              =1+1+4                              =6

 

 

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