Is knowing $E[X]$ and $V[X]$ enough to find $E\left[e^X\right]$?

Question

I have a larger problem that sort of boils down to this question. Without making any assumption on the distribution of the r.v. $X$, can I calculate $E\left[e^X\right]$ knowing only $E[X]$ and $V[X]$?

My intuition tells me the answer is no, but I'm not entirely sure.

Answer 1

Note that by Taylor expansion

$$e^X=\sum_{n=0}^{\infty} \frac{X^n}{n!}$$

So loosely,

$$\Bbb{E}(e^X)= \sum_{n=0}^{\infty} \frac{\Bbb{E}(X^n)}{n!}$$

Thus you need to know EVERY moment to find this expectation.

For a counterexample, let $X$ be a random variable with pdf $f(x)=\frac{3}{x^4} \chi_{[1,\infty)}$

It is easy to compute the first and second moments, however:

$$\int_1^\infty \frac{3e^x}{x^4} dx$$

doesn't even exist.

As a side note,

I have a larger problem that sort of boils down to this question.

is what's called the XY problem.