Cumulative distribution function of Y

Question

How do you get the cumulative distribution function of $Y$ in terms of $X$?

Let's say that $E[X] = μ, Var[X] = σ^2,$ and $Y =a + bX$.

What would be the process to get the cumulative distribution function?

Answer 1

Let's say that $E[X] = μ, Var[X] = σ^2,$ and $Y =a + bX$.

Wouldn't $E[Y]$ = $a + Bμ$ and $Var[Y] = B^2 σ^2$?

It actually would be that.

What would be the process to get the cumulative distribution function?

You start with the cummulative distribution function for $X$.

If $F_{\small X}(x)$ is the CDF for $X$, and $b>0$, then $F_{\small Y}(y)$, the cdf for $Y$, is:$$F_{\small Y}(y) ~=~ F_{\small X}\left(\dfrac{y-a}b\right)$$

Since $\mathsf P(Y\leqslant y) ~{= \mathsf P(a+bX\leqslant y)\\=\mathsf P\left(X\leqslant (y-a)/b\right)}$