For a given positive integer $N$ define a random variable:
$X_{n}=\frac{1}{n}\sum_{i=1}^{n}e^{-\gamma _{i}^{2}}$
For a given positive integer n and where the $\gamma _{i}$ are iid samples of $Z ∼ N (0, 1)$. Hence we can write:
$\mathbb{E}[X_{n}]=\frac{1}{n}\sum_{i=1}^{n}\mathbb{E}[e^{-\gamma _{i}^{2}}]$
But since the $\gamma _{i}$ are iid samples of $Z ∼ N (0, 1)$ can we say that:
$\mathbb{E}[X_{n}]=\int_{-\infty }^{\infty }e^{-x^2}f(x)dx$
where $f$ is the density of a standard normal random variable and where I have used that:
$\gamma _{i}\sim N(0,1)$