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If X is N(?, ? 2 ), show that the distribution of Y = aX +
Chapter 3, Problem 10E(choose chapter or problem)
QUESTION:
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.
Questions & Answers
QUESTION:
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.
ANSWER:
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