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

To compute the expectation of a nonnegative random variable $X$ when one is given its cumulative distribution function $F_X$, one does not need to deduce the probability density function $f_X$ from $F_X$ and to plug $f_X$ into the well known integral formula. Rather, one can rely on the more direct
$$
\mathbb E(X)=\int_0^{+\infty}(1-F_X(x)) \, \mathrm dx.
$$
In the present case, $F_X(x)=0$ if $x\leqslant1$ and $F_X(x)=1-x^{-a}$ if $x\geqslant1$ hence
$$
\mathbb E(X)=1+\int_1^{+\infty}x^{-a} \, \mathrm dx=\ldots
$$
Similar direct formulas exist when $X$ is real-valued, not necessarily almost surely nonnegative, and still others are available when one needs to compute higher moments $E(X^\alpha)$ for $\alpha>1$, all using only the CDF $F_X$ and not the (possibly nonexistent) PDF of $X$.
