Linear Transformation of a Normal

A linear transformation of a normal random variable remains normal.

Compute the new mean and variance directly from the transformation.

1. 1. Transform the mean

   Expectation is linear: $E[aX+b]=aE[X]+b=a\mu +b$.

2. 2. Transform the variance

   Adding a constant does not change variance, and scaling by $a$ scales variance by $a^2$.

   $$aX+b\sim N(a\mu +b,a^2{\sigma }^2)$$

Answer: N(a*mu+b, a^2*sigma^2)