Integrable random variable X such that X^2 is not integrable

Question

can anyone help me? I have to show an example of an integrable random variable X such that X^2 is not integrable (it is not specified discrete or continuos case) I know that a r.v. is integrable if and only if its expected value is well-defined and is said to be square integrable if the expected value of its square exists and it is well-defined. But I have not idea what type of random variable respects as I requested. Thank you in advance

Answer 1

Let $X$ have support $[1,\infty)$ and pdf $f_X(x)=\frac{2}{x^3}.$ $X$ is integrable since $\int_1^\infty x\frac{2}{x^3}dx = 2,$ but $X^2$ is not since $\int_1^\infty x^2\frac{2}{x^3}dx$ diverges.