Random variable that satisfies $\widetilde{x} > 0$ yet $\mathbb{E}\left[\ln(\widetilde{x})\right] = -\infty$

Question

I'm trying to come up with a random variable $\widetilde{x} > 0$ where $\mathbb{E}\left[\ln(\widetilde{x})\right] = -\infty$

As far as I can tell, I'm looking for a distribution that gets $\epsilon$ close to 0 and as long as I can guarantee that, it should work. Would the open uniform distribution from 0,1 be sufficient to make this claim?

Answer 1

Choose a random variable $X$ that takes the value $2^{-2^n}$ with probability $\frac{1}{n(n+1)}$, for each integer $n \geq 1$.

Then $\ln{X}$ takes the value $-2^n$ with probability $\frac{1}{n(n+1)}$ so its expected value is clearly $-\infty$.