discrete random variable with expected value of $\pm\infty$?

Question

I am currently studying discrete random variables and I was wondering if there is any probability space with two indipendent discrete random variables $X$ and $Y$ with expected values $E[X]=-\infty,E[Y]=\infty$? Are there any real ones?

Answer 1

Let $Y$ be a random variable with $$\mathbb{P}(Y = 2^n) = 2^{-n}, \qquad n = 1, 2, 3, \ldots$$ Clearly this sums to $1$, so it defines a probability distribution, but the expectation is $$\mathbb{E}\,Y = \sum_{n=0}^\infty 2^n 2^{-n} = \sum_{n=0}^\infty 1 = \infty,$$ and then define $X$ as $$\mathbb P(X = -2^n) = \mathbb P(Y = 2^n) = 2^{-n}, \qquad n = 1, 2, 3, \ldots$$ This particular example is related to the St. Petersburg paradox.

Answer 2

For $Y$, let $P(2)=1/2, P(4)=1/4, P(8)=1/8$, etc. Negate that for $X$. If you're looking for independent random variables on a single sample space, let that sample space simply consist of pairs of power-of-two values for $X$ and $Y$, where the probability of each outcome is $\frac{1}{XY}$.