I am looking for $E[X]$ when $X$ has a density function $$f(x) = \frac{5}{x^2}$$
when $x > 5$ and $0$ elsewhere.
But $\int_5^\infty x \frac{5}{x^2}dx = 5\int_5^\infty x^{-1}dx = 5ln(x)\mid_5^\infty$
$=5[ln(\infty) - ln(5)]$
This seems to say that the expected value is $\infty$, which I figure means it's undefined. Have I made a mistake?
The density $f(x)=\dfrac{\alpha x_m^\alpha}{x^{\alpha+1}}$ is of a Pareto distribution. In this case $\alpha =1$ and $x_m=5$.
A Pareto distribution has an infinite mean when $0 \lt \alpha \le 1$ as in this case, and an infinite variance when $1 \lt \alpha \le 2$.