Random variables and expectations

Question

I am not even sure on how to get started on this:

Let $k \ge 1$.

(i) find a rv X for which $E|X^k|< \infty$ but $E|X^{k+1}|= \infty$

(ii) Is it possible to find a rv Y for which $E|Y^k|= \infty$ and $E|Y^{k+1}|< \infty$

I feel like this is probably not possible..

Any idea?

Answer 1

Hint: $\int_0^1 x^{-1} dx = \infty$ but $\int_0^1 x^{-p} dx < \infty$ for $p<1$. Can you find a random variable $X$ with these two expectations?

For the second part, here's an idea. Let $Y$ be a random variable with $E[|Y^k|]=\infty$ and define $A=\{ |Y| \leq 1 \}$. Then $E(|Y|^{k+1})=E(|Y|^{k+1}|A)P(A)+E(|Y|^{k+1}|A^c)P(A^c)$ Now show the second term is $\infty$.

Answer 2

Student's t-distribution with 2 degrees of freedom has $E(T) = 0$, but $E(|T|^2) = \infty.$ This is a 'famous' example of lower moments having an expectation and higher moments not. This family of distributions is widely used in statistical inference.