A positive square integrable random variable whit non square integrable inverse

Question

I'm looking for an example of a Square Integrable Random Variable, whose multiplicative inverse is not Square Integrable.

Answer 1

How about an exponential random variable? (Or any other positive square-integrable random variable with a density function that is bounded away from $0$ in a neighborhood of $0$.)

More detail: Let $X$ be a positive random variable with $\Bbb E[X^2]<\infty$ and density $f_X$ such that $f_X(x)\ge\delta>$ for $0<x<x_0$. (An exponential random variable is a specific example.) Then $$ \eqalign{ \Bbb E[X^{-2}] &=\int_0^\infty x^{-2} f_X(x)\,dx =\int_0^\infty f_X(t^{-1})\,dt\cr &\ge\int_{1/x_0}^\infty f_X(t^{-1})\,dt \ge\int_{1/x_0}^\infty \delta\,dt=\infty.\cr } $$