I'm going to start out by saying this is a homework problem straight out of the book. I have spent a couple hours looking up how to find expected values, and have determined I understand nothing.


  Let $X$ have the CDF $F(x) = 1 - x^{-\alpha}, x\ge1$.
  Find $E(X)$ for those values of $\alpha$ for which $E(X)$ exists.


I have no idea how to even start this. How can I determine which values of $\alpha$ exist? I also don't know what to do with the CDF (I'm assuming this means Cumulative Distribution Function). There are formulas for finding the expected value when you have a frequency function or density function. Wikipedia says the CDF of $X$ can be defined in terms of the probability density function $f$ as follows:

$F(x) = \int_{-\infty}^x f(t)\,dt$

This is as far as I got. Where do I go from here?

EDIT: I meant to put $x\ge1$.

Usage of the density function is not necessary

Integrate 1 minus the CDF

When you have a random variable $X$ that has a support that is non-negative (that is, the variable has nonzero density/probability for only positive values), you can use the following property:

$$
E(X) = \int_0^\infty \left( 1 - F_X(x) \right) \,\mathrm{d}x
$$

A similar property applies in the case of a discrete random variable.

Proof

Since $1 - F_X(x) = P(X\geq x) = \int_x^\infty f_X(t) \,\mathrm{d}t$,

$$
\int_0^\infty \left( 1 - F_X(x) \right) \,\mathrm{d}x = \int_0^\infty P(X\geq x) \,\mathrm{d}x = \int_0^\infty \int_x^\infty f_X(t) \,\mathrm{d}t \mathrm{d}x
$$

Then change the order of integration:

$$
= \int_0^\infty \int_0^t f_X(t) \,\mathrm{d}x \mathrm{d}t = \int_0^\infty \left[xf_X(t)\right]_0^t \,\mathrm{d}t = \int_0^\infty t f_X(t) \,\mathrm{d}t
$$

Recognizing that $t$ is a dummy variable, or taking the simple substitution $t=x$ and $\mathrm{d}t = \mathrm{d}x$,

$$
= \int_0^\infty x f_X(x) \,\mathrm{d}x = \mathrm{E}(X)
$$

Attribution

I used the Formulas for special cases section of the Expected value article on Wikipedia to refresh my memory on the proof. That section also contains proofs for the discrete random variable case and also for the case that no density function exists.
