Cumulative distribution transformation

[mrkb80]

Homework Statement

Let $F$ be the cumulative distribution function of a random variable $X$. Find the cumulative distribution function of $Y= {\alpha}X+\beta, where \, \alpha \gt 0$

Homework Equations

The Attempt at a Solution

I think this a fairly easy question, I just want to make sure I understand:
$F(x)=P(X \leq x)=P(\dfrac{Y-\beta}\alpha \leq x)=P(\dfrac{Y-\beta}\alpha \leq y)=F(y)$

 

[mrkb80]

I'm thinking I made one small mistake:
$F(x)=P(X \leq x)=P(\dfrac{Y-\beta}\alpha \leq x)=P(Y\leq y)=F(y)$