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In a stochastic processes class, we're studying a theorem which required that a random variable $T$ have finite mean. The notes presented a counterexample where a R.V. $T$ was such that $P(T<\infty) = 1$ but for which the theorem was not valid. It then went on to explain that $E[T] = \infty$.

This could be an elementary question but if the random variable is less than infinity with probability 1, how can the mean be infinite? Both intuitive answers and math are welcome, thank you!

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Think of a variable that can take any positive value but can become arbitrarelly large with a reasonable probability. For example, think about playing a game where we throw two dice with the following rules:

If we get 12 on the first turn, you owe me $1 and we stop playing. Otherwise we continue

If we get 12 on the second turn, you owe me $1,000 and we stop playing. Otherwise we continue

If we get 12 on the third turn, you owe me $1,000,000 and we stop lyaing. Otherwise we continue.

And so on....

It is guaranteed that, given enough turns, a 12 will eventually come and the game will stop, with you paying me a finite amount of money, however, you can easily calculate that my expected winnings are infinite.

Indeed, expectation is $\sum_{k=1}^{infinity} [(\frac{1}{36})^k * 1000^k]$, with this series being divergent

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  • $\begingroup$ Great thank you! As long as your winnings grow by a factor of 36 or more every turn (proportional to the probability) , your expected winnings would be infinite. But then the actual payout is never infinite because this game is guaranteed to end in finite time. Is this right? Is there a way we can tweak this example to put some positive probability on your payout being infinite? $\endgroup$ Mar 8, 2019 at 11:52
  • $\begingroup$ I think you got it right. However, I don't know what an "Inifnite payout" would mean. We could of course flip a coin before the game starts and if If it shows "tails", you owe me infinite money. I am not sure however that this notion makes sense. $\endgroup$
    – David
    Mar 11, 2019 at 12:48

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