Laplace transform / gaussian random variable

Question

We have a gaussian random variable $X \sim N(0, \sigma^{2})$, with $\sigma^{2}$ unknown.

the Laplace transform given by:

$\phi(t) := \mathbb{E} [e^{tX}]$ = $e^{{\sigma}^{2} t^2/2}$

I need to make i.i.d $N$ copies of X and compute the empirical mean

$\phi_{N} (t) := \frac{1}{N} \sum_{i=1}^{N} e^{t X_i}$ and then calculate the confidence interval, but I am clueless where to start. Which properties of the expected value, gaussian rvs can I apply here?

Answer 1

Mean of average $=\phi(t)$. To get variance you need second moment $\frac{1}{N^2}(\sum\limits_{i=1}^N (E(e^{2tX_i})+2\sum\limits_{j=1}^{i-1} E(e^{tX_(i+j)}))=\frac{\phi(2t)}{N}+\frac{N-1}{N}\phi(t)^2$

Variance then is $\frac{1}{N}(\phi(2t)-\phi(t)^2)$.