Pdf calculation of two random variables

Question

If $X = aY + b$, both $X$ and $Y$ are random variable. The pdf of $Y$ is given, can anybody please tell how to find pdf of $X$ ?

Answer 1

You calculate questions like these always via the cumulative distribution function, because $P(Y=y) = 0$ for all $y$. Thus if $a>0$: $$ F_Y(y) = P(Y\le y) = P(aX+b\le y) = P(X\le\frac{y-b}{a}) = F_X(\frac{y-b}{a}). $$ Thus $$ f_Y(y) = \frac{d}{dy}F_Y(y) = \frac{d}{dy}F_X(\frac{y-b}{a}) = f_X(\frac{y-b}{a})\frac{1}{a}. $$ The case $a<0$ works the some and the case $a=0$ is quite straight forward.