Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Suppose we are give $X$ which is normally distributed with mean $\mu$ and variance $\sigma^2$. How do I show that

$$E[e^{e^X}]=\infty$$

Of course I wanted to find a lower bound, which also explodes to conclude. However I did not find the right one.

Thanks for your help

hulik

share|cite|improve this question
up vote 2 down vote accepted

The problem comes from the divergence of $\int_{\Bbb R}e^{e^x-x^2/2}dx$, which can be seen noticing that $e^x-x^2/2\geqslant x\geqslant 0$ for $x\geqslant 0$.

share|cite|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.