How can a positive random variable $X$ which never takes on the value $+\infty$, have expected value $\mathbb{E}[X] = +\infty$?
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1$\begingroup$ The realization of a random variable has to be a finite number. $\endgroup$– JACKY88Nov 17, 2012 at 15:35
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2$\begingroup$ The term I've come across for such variables is that their distributions have a "too fat tail", signifying that high-value outcomes have such a big probability that even though it all sums up to $1$, once you calculate the expected value, the big outcomes kick in and inflate the result. $\endgroup$– ArthurNov 17, 2012 at 16:05
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1$\begingroup$ The result is quite counter-intuitive. How can summing up products of finite numbers (the values of the random variable) with finite numbers (the probability of the random variable taking on that value) be infinite? $\endgroup$– lodhbNov 25, 2012 at 11:08
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$\begingroup$ See here for a statistical perspective: stats.stackexchange.com/questions/94402/… $\endgroup$– kjetil b halvorsenNov 15, 2015 at 15:25
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3 Answers
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Let $X$ be a random variable that is equal to $2^n$ with probability $2^{-n}$ (for positive integer $n$). Then $${\mathbb E} X = \sum_{n=1}^\infty 2^{-n} \cdot 2^n = \sum_{n=1}^\infty 1 = \infty.$$
Cauchy Distribution is an example of a continuous distribution that doesn't have an expectation.
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$\begingroup$ This example ignores the loading of absolute-summability in the def'n of expected value of a random variable taking countably infinite values. Without such loading, "expected value of a random variable taking countably infinite values" doesn't have plausible meaing due to Riemann Rearrangement Thm, and irresistant to change of the terms in the series itself. With this explanation in mind; since $${\mathbb E} |X| = \sum_{n=1}^\infty |2^{-n} \cdot 2^n| = \sum_{n=1}^\infty 1 = \infty$$; it is more wise to say the expected value in Yury's example does not exist rather than it is infinite. $\endgroup$ Aug 18, 2019 at 11:42
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1$\begingroup$ I believe Riemann Rearrangement theorem applies to conditionally convergent series, and since the terms here are all strictly positive, it is not applicable here(I believe). $\endgroup$ Apr 18, 2020 at 21:28
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$\begingroup$ We have $E(X) := \sup_{Y \leq X} E(Y)$, where $Y$ are simple random variables. That is, we take the supremum over finite sums. But I agree that your example gives $\sup_{Y \leq X} E(Y) = \infty$. $\endgroup$– harisfJun 3, 2022 at 7:20
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Once you consider probabilistic experiments with infinite outcomes, it is easy to find random variables with an infinite expected value. Consider the following example (which is just a game that yields an example similar to the one Yuri provided):
- You throw a coin until it lands tails.
- You then get paid $2^{n}$ dollars, where $n$ is the amount of heads you got.
It is easy to construct the expected value funcion of your payment (let's name it $X$):
$$E[X] = \frac{1}{2} \times 2^0 + \frac{1}{4} \times 2^1 + \dots = \sum_{n=1}^{\infty} 2^{-n}\times 2^{n-1} = \sum_{n=1}^{\infty} \frac{1}{2} = \infty $$
This game is also known as St. Petersburg paradox. Why does this occur and how can we interpret it?
From a construction point of view, it is easier to understand. In this particular case, the probability of each outcome decreases exponentially. Since the number of outcomes is infinite, the payout scheme only has to grow at the same rate as the probability of the outcome decreases in order for the series to diverge.
What this means in practice is that, although the payout is always finite, if you average the payouts from $k$ consecutive games, this average will (with high probability) be higher the greater $k$ is. As $k$ approaches infinity, so does the average of the $k$ payouts. Behind this boundless growth is the fact that everytime an unlikely outcome happens, the payout is so large that, when averaged with the payout of more likely outcomes, the average is skewed up.
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In your comments, you asked how summing up finite things (or products of finite things with other finite things) can give you something infinite. This shouldn't be too counterintuitive; the idea is that you're summing up infinitely many of those finite things. If anything, it's counterintuitive that you can sometimes sum up infinitely many things and get a finite result.
I do agree, however, that the idea of a game like the St. Petersburg Game seems counterintuitive. Part of it might be because of the word "expectation." In common usage, when we expect something to happen, we think it's more likely to happen than not. But in probability, that's clearly not the case, because we're taking a weighted average of possible outcomes, and the weighted average itself might be an unlikely, or even impossible outcome. For example, when you roll a die, you "expect" the value of the number shown to be 3.5, even though you know that will never happen. Similarly, we can "expect" the outcome of an experiment to be infinite, even though we know it will be finite. That explanation might not fully satisfy your intuition, but it's a start at least.
