# An example of continuous random variable X > 0 with finite second moment but Infinite third moment [duplicate]

Can someone construct an example of this?

i.e., $$E[X^2] < \infty$$ but $$E[X^3] = \infty$$. Results could be in terms of pdf, or cdf, or survival function. Justification would be appreciated

So, you want $$\int_0^\infty x^2f(x),dx$$ to exist, but $$\int_0^\infty x^3f(x),dx$$ to be infinite.
We know that the integral $$\int_1^\infty x^{-n}\,dx$$ is finite if $$n>1$$ and infinite if $$n\leq 1$$. So one possibility is $$f(x) \propto 1_{x\geq 1} x^{-4}$$. Or, it would look a bit tidier to use $$f(x) \propto (1+x)^{-4}$$ on $$x>0$$.
Now we need the constant of proportionality. $$\int_0^\infty (1+x)^{-4}\,dx =1/3$$, so $$f(x) = 3(1+x)^{-4}$$
Another approach is to note that a $$t$$ distribution with $$\nu$$ degrees of freedom has $$\nu-1$$ finite moments (https://en.wikipedia.org/wiki/Student%27s_t-distribution). Thus, a $$t_3$$ distribution has finite second moment but not third moment. It doesn't satisfy the non-negativity condition, but its absolute value does.