Let's say I have 3 random variables, $X$, $Y$, $Z$ with c.d.f.'s $F_X(x)$, $F_Y(y)$, and $F_Z(z)$. All are supported over $(0,1)$
I want to define $Z$ such that $Z = aX + (1-a)Y$ (i.e. a weighted average of $X$ and $Y$ where $a$ is just some constant).
Does is follow then that $F_Z(z) = aF_X(z) +(1-a)F_Y(z)$?
I am pretty sure this is not true, though it would be true that $\mathbb{E}Z = a\mathbb{E}X +(1-a)\mathbb{E}Y$ but I wanted to check. Thanks for your help!