# If $X \sim {\rm Bernoulli}(p)$, what is the distribution of $Y=aX-b$?

Let $X \sim {\rm Bernoulli}(p)$ and $Y=aX-b$. I want to find the distribution of $Y$.

I am assigned a homework problem with specific $a$ and $b$. My textbook covers methods for solving this, given that x is continuously distributed. However, $X$ is discretely distributed.

I'm not looking to be shown exactly how to do this. I am just unsure of where to start, as this is not covered in our text.

(i) What are the possible values that $y$ can take? if $x\in\{0,1\}$ then $y=ax+b\in\{...\}$. (ii) You know that $P(x=1)=p$ and $P(x=0)=1-p$ you can then prove that $x=(y+b)/a$, therefore $p=P(x=1)=P[(y+b)/a=1]$, similarly $1-p=P(x=0)=P[(y+b)/a=0]$ ... I hope these hints help. –  user10525 Jun 8 '12 at 22:37