Evaluate $\mathbb{E}[e^{e^x}] =\int_\mathbb{R}e^{e^x}f(x)dx$, where $f(x)$ is Gaussian kernel.

Question

Trying to find $\mathbb{E}[e^{e^x}] =\int_\mathbb{R}e^{e^x}f(x)dx$, where $f(x)$ is the Gaussian density, ($\mu,\sigma^2$).

Can't seem to figure out where to start with this one. I've tried a few things, but nothing really comes close.

This post tells the pdf of $e^x$, which means the expectation above is just the MGF evaluated at 1. Still having trouble computing that integral, though.

Any help?

score 2 · Answer 1 · edited Apr 13 '17 at 12:20

2

As you said, this is the moment generating function for $Y=e^{X}$ evaluated at $1$. $Y$ has a log-normal distribution. Unfortunately the MGF, $M_{Y}(t)$, for the log-normal distribution is not defined for any positive $t$.

See, for example, this proof.

edited Apr 13 '17 at 12:20

Community

1

answered Oct 04 '15 at 00:33

Rubarb

555

Evaluate $\mathbb{E}[e^{e^x}] =\int_\mathbb{R}e^{e^x}f(x)dx$, where $f(x)$ is Gaussian kernel.

1 Answers1