I'm not a mathematician, so my question may be not so clear, sorry.
Let's say we toss up an ideal coin and win 1 dollar on heads and lose 1 dollar on tails. So, expected value is M = 1×0.5 − 1×0.5 = 0. It may be interpreted as: we will gain/lose nothing playing a game for a long time. It is pretty clear.
Now, let's consider bankroll size. It will be changed on every coin flip. For Heads Tails Tails Tails Heads sequence, bankroll size will be 1, 0, -1, -2, -1 dollar. As I understand, bankroll size is a random variable. Is it even possible to evaluate this random variable somehow? How to calculate expected value and variance? May I evaluate minimum and maximum of bankroll size?