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 :

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