Finding the mean of a function of a random variable

Question

Suppose that $X$~$exp(\lambda=1/2)$.

I found the pdf of $U=+\sqrt{X}$ to be$f_U(u)=ue^{-(1/2)u^{2}}$.

Now, how do I find the mean of U?

EDIT: I've just noticed that the pdf of U is similar to the normal distribution pdf, which means I'm on the right track. Not sure how to use that information though to work out the mean of U.

Answer 1

By definition:

$$ \mathbb{E} \left[ g \left( x \right) \right] = \int g \left( x \right) {f}_{X} \left( x \right) dx $$

So you could have chosen:

$$ E \left[ +\sqrt{X} \right] = \int +\sqrt{x} {f}_{X} \left( x \right) dx $$

Or

$$ E \left[ U \right] = \int u {f}_{U} \left( u \right) du $$

Both will yield the same result.