Solution
= Solution
The <expected value> is the <improper integral> $\alpha\int_1^\infty x^{-\alpha}\,dx$. It converges exactly when $\alpha>1$, yielding
$$
\boxed{E[X]=\frac{\alpha}{\alpha-1},\qquad\alpha>1}.
$$
At $\alpha=1$ this <improper integral> diverges logarithmically, and for $0<\alpha<1$ it diverges as a <power> of its upper endpoint. In both cases the nonnegative variable has infinite <mean>. This is the first <Pareto moment threshold>.