$Y$ is a random postivie integer valued random variable.
$U_1,U_2,...\in U(0,1)$ are I.I.D:s and independent from $Y$.
$ M = max(U_1,U_2,...,U_Y) $
I want to prove that $P(M \leq t) = g_Y(t)$ for $t\in [0,1]$ where $g_Y(t)$ is the p.g.f of $Y$.
$ P(M \leq t) = P(max(U_1,U_2,...,U_Y) \leq t) = \{ independent \} = P(U_1 \leq t)P(U_2 \leq t)\cdot\cdot\cdot P(U_Y \leq t)=\{P(U_i \leq t) = t\} = t^Y$
But $g_Y(t) = E[t^Y]$, how do I get an expected value in there? What am I doing wrong?
Your LHS is a real number: $P(M\leq t)$, but your RHS is a random variable: $t^Y$.
This is the way to do it: $$P\left(M\leq t\right)=\sum_{n=1}^{\infty}P\left(M\leq t\mid Y=n\right)P\left(Y=n\right)=\sum_{n=1}^{\infty}t^{n}P\left(Y=n\right)=g_{Y}\left(t\right)$$