$exp$ of Half-Normal Distribution

Question

I know that the Half-Normal Distribution has moments of all orders - that is, if $X\sim\mathcal{N}(\mu,\sigma)$, then, $$ E[|X|^p]<\infty $$ However, do we also have $$ E[e^{|X|}]<\infty $$ ? Thank you in advance!

Answer 1

Assume $X \sim N(\mu,\sigma^2)$. The distribution of $|X|$ is called a folded normal distribution. Its moment generating function is (https://en.wikipedia.org/wiki/Folded_normal_distribution)

$$\mathbb{E}[e^{t|X|}]=e^{\frac{\sigma^2t^2}{2}+\mu t}\Phi(\frac{\mu}{\sigma}+\sigma t)+e^{\frac{\sigma^2t^2}{2}-\mu t}\Phi(-\frac{\mu}{\sigma}+\sigma t)$$

Here $\Phi$ is the CDF of the standard normal distribution. Hence

$$\mathbb{E}[e^{|X|}]=e^{\frac{\sigma^2t^2}{2}+\mu}\Phi(\frac{\mu}{\sigma}+\sigma)+e^{\frac{\sigma^2}{2}-\mu}\Phi(-\frac{\mu}{\sigma}+\sigma) < \infty$$