What is the example of $X$ being in $L_1$ but mgf of $X$ being infinity?

Question

I have spent quite a few hours trying to come with the example that shows the following:

"There exists a non-negative random variable $X$ such that $X$ is in $L_1$ (integrable), but mgf of $X = \infty$ for all $t > 0$."

Can anyone say or hint at an answer for this question?

I know that the mgf for Cauchy is infinity, but its expectation is infinity for all $t > 0$. Nothing else with mgf being infinity comes to mind. Thanks!

Answer 1

What about a power law distribution with a thinner tail than Cauchy, but still power law? Like $$ f_X(x) = \frac{2}{x^3}$$ for $x>1.$ This is integrable, but its MGF doesn't exist for $t>0.$