given a density function of $f_x = x^{-n}$ how to compute $\operatorname {var}(X)$?

Question

Say that you are given a density function of $f_x = \frac{1}{5}x^{-n}$ for some $n$ with $x \in [5,\infty]$, how to compute $\operatorname {var}(X)$? Is the gamma function of factorial used in this question?

Answer 1

Remember that $Var(x) = E[E[X] - X] = E[X^2] - E[X]^2$.

$E[X] = \int_5^{\inf}xf_x dx$

$E[X^2] = \int_5^{\inf}x^2f_x dx$

Perform these integrations and combine them as shown above to get the variance.

Answer 2

Note that $n\in(1,2)$, so the random variables $X$ and $X^2$ are not integrable and hence $\operatorname{\mathrm{Var}}X$ is not defined.