Assume random variable $X\sim N(\mu,\delta)$, how to evaluate $E[e^{\lambda X}]$ for all $\lambda > 0$

Question

I've tried applying Taylor expansion to $e^{\lambda X}$, but it seems to end up with an infinite sum of non-standard moments. Directly integrate $\int{f_X(x)e^{\lambda X}dx}$ is hard for me.

Answer 1

To do the integration, first complete the square in the exponent, expressing the integrand $f_X(x)e^{\lambda x}$ as ${1\over\sqrt{2\pi\sigma^2}}\exp(\lambda\mu+\lambda^2\sigma^2/2)\exp(-(x-(\mu+\lambda\sigma^2))^2/2\sigma^2)$. Factor out the first exponential, then use the fact that any normal density function integrates to $1$.