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If X is N(?, ? 2 ), show that the distribution of Y = aX +

Probability and Statistical Inference | 9th Edition | ISBN: 9780321923271 | Authors: Robert V. Hogg, Elliot Tanis, Dale Zimmerman ISBN: 9780321923271 41

Solution for problem 10E Chapter 3.3

Probability and Statistical Inference | 9th Edition

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Probability and Statistical Inference | 9th Edition | ISBN: 9780321923271 | Authors: Robert V. Hogg, Elliot Tanis, Dale Zimmerman

Probability and Statistical Inference | 9th Edition

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Problem 10E

If X is N(μ, σ 2 ), show that the distribution of Y = aX + b is N(aμ + b, a2σ 2 ), a ≠ 0. Hint: Find the cdf P(Y ≤ y) of Y, and in the resulting integral, let w = ax+b or, equivalently, x = (w − b)/a.

Step-by-Step Solution:

Answer :

Step 1 of 2 :

A random variable X is  normally distributed with mean  and standard deviation  .

The probability density function is

f(x) =

The claim is to show that the distribution of Y = ax+b is N(), a

Where, P(Xx) =  dt

Step 2 of 2

Chapter 3.3, Problem 10E is Solved
Textbook: Probability and Statistical Inference
Edition: 9
Author: Robert V. Hogg, Elliot Tanis, Dale Zimmerman
ISBN: 9780321923271

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If X is N(?, ? 2 ), show that the distribution of Y = aX +

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