What Is the cumulative distribution function Of $Y = aX+b$ if $X\sim N(m,\sigma^2)$?

Question

How do I calculate the cumulative distribution function of the random variable $$ Y = aX+b $$ with $$ X\sim N(m,\sigma^2)$$ and $a,b\in\mathbb R$, $a\ne0$? Thanks for your help!

Answer 1

For $t\in\mathbb R$ we have for $a>0$ \begin{align} \mathbb P(Y\leqslant t) &= \mathbb P(aX+b\leqslant t)\\ &= \mathbb P\left(X\leqslant \frac{t-b}a\right)\\ &= \int_{-\infty}^{(t-b)/a}\frac1{\sqrt{2\pi\sigma^2}} e^{-\frac12\left(\frac{x-m}{\sigma}\right)^2}\ \mathsf dx\\ &= \frac{1}{2} \text{erfc}\left(\frac{ a m+b-t}{\sqrt{2\sigma^2} a}\right), \end{align} and for $a<0$ \begin{align} \mathbb P(Y\leqslant t) &= \mathbb P\left(X\geqslant \frac{t-b}a\right)\\ &=\int_{(t-b)/a}^\infty\frac1{\sqrt{2\pi\sigma^2}} e^{-\frac12\left(\frac{x-m}{\sigma}\right)^2}\ \mathsf dx\\ &= \frac12\left(\text{erfi}\left(\frac{a m+b-t}{\sqrt{2} a \sigma }\right)-1\right). \end{align}