# Expected value of X*exp(X) for X normally distributed

## Main Question or Discussion Point

Assume we have $X\sim\exp(\mu,\sigma^2)$.
How does one compute $\mathbb{E}\left(Xe^X\right)$ and/or what is the outcome value?

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CompuChip
Homework Helper
The definition would say that

$$\mathbb{E}[ X e^X ] = \int_{-\infty}^{\infty} x e^{x} e^{-(x - \mu)^2 / \sigma^2} \, dx$$

That's a tricky one, I don't think any of the common integration strategies really work.
WolframAlpha does give me an exact result for the standard distribution ##(\mu, \sigma) = (0, 1)##.

Maybe this will also help.

Sorry for the incomplete answer, hoping that this will get you started.

D H
Staff Emeritus