I have a function $Y=e^{-e^{a+bZ}}$, where $a$ and $b$ are constants and $Z$ is a standard normal random variable. I need to find $\mathbb{E}[Y]$. Is there even a closed-form solution to $\mathbb{E}[Y]$?
I tried log transformation. $-\ln Y=e^{a+bZ}$. Then $\mathbb{E}[-\ln Y]=e^{a+0.5b^2}$. But then there doesn't seem to be any easy way to find $\mathbb{E}[Y]$ using $\mathbb{E}[-\ln Y]$.
Any experts out there? Thanks.