Let $Y$ be a uniformly distributed random variable on $[0,1]$. I am doing a problem that requires me to simplify $\frac{E[e^{\theta Y}]^n}{e^{c\theta n}}$ for some constant $c$ to get the expression $e^{-nc}$. I am trying to find the numerator which is the expectation $E[e^{\theta Y}]^n$. Along the way, I got stuck an unable to simplify it. Could anybody please give some help?
$E[e^{\theta Y}]^n=(\int_0^1 e^{\theta y}\times1 dy)^n$ since pdf of $Y$ is $\rho(y)=1$.
Then
$$(\int_0^1 e^{\theta y}\times1 dy)^n=\Big(\Big[\frac{1}{\theta}e^{\theta y}\Big]_0^1\Big)^n=\Big(\frac{e^{\theta}}{\theta}-\frac{1}{\theta}\Big)^n$$
Then I was stuck here. How can we perform the multinomial expansion?
Thanks.