You can substitute $ for % in my comment if it helps. If you start with $100, your expected wealth after two throws is the average of $225, $90, $90 and $36.
Re:The Infamous Coin Toss
by didgeoridoo on Wednesday August 02, 2023 @10:40PM
(#36980741)
That average is still greater than $100, because you haven’t yet hit the Kelly point beyond which the downside catastrophe dominates. Play it out a few more rounds and see where the average heads to.
[Edit: delete bad math]
The expected value of this distribution goes up with every iteration, there is no such Kelly point. You could try this with
heads: double your money tails: lose all your money
in which case the expected value is always $1, as you have a 1/2^n chance of having $2^n dollars after n rounds, and 0 otherwise.
The point of discussing ergodicity here, however, is whether you can describe the behavior of the iterated distribution deterministically if you exclude a portion of that distribution which has measure zero.
Try 225% * 90% * 90% * 36% to get the expected value.