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!
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
