Find a sequence of random variables

Question

I have a problem with finding an example of a sequence of nonnegative random variables $X_1, \ldots$ with values in integers s.t. $\mathbb{E}X_n \rightarrow \infty$ and

• $\limsup_{n\rightarrow\infty}\mathbb{P}(X_n>0)<1$ (first case),
• $\limsup_{n\rightarrow\infty}\mathbb{P}(X_n>0)=1$ (second case).

I think in the second case it can be a sequence $X_n=n$. Is it good reasoning? What with the first case?

Answer 1

Hint: your answer for the second case is good. Can you modify it to obtain an example such that $P(X_n = 0) = P(X_n > 0) = \frac{1}{2}$ for all $n$ and $\Bbb{E}(X_n) = \frac{n}{2}$?