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.


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