Expectation of $X$ given a Cumulative Function I've (hard) to find the expectation of $X$:
CDF (Cumulative Function) = $1 - x^{-a}$, $1 \leqslant x < \infty$
$E[X]$ = ??

First you have to establish the proper domain for the random variable $X$. That can be done by using $F(x_{\min}) = 0$. You random variable will be supported on $[x_{\min}, \infty)$. 


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*Find the probability density function, $f_X(x)$ by differentiating the cumulative function, $f_X(x) = F_X^\prime(x)$.

*Apply the law of the unconscious statistician: $$ \mathbb{E}(X) = \int_{x_\min}^\infty x f_X(x) \mathrm{d}x $$ 

*For what values of parameter $a$ does the integral exist?

*Once you found the answer, you have found the mean of the Pareto distribution.
