Let $X$ be the number of people waiting for a bus. Assuming $X$ takes on a geometric distribution that starts at $0$ rather than $1$, we have $p_{X}(k) = (1 - p)^{k}p$ for $k \geq 0$. Suppose that the bus has a capacity of $M$ passengers.
(a) Find the PMF for $Y = \text{max}\{X - M, 0\}$, which represents the number of passengers left behind.
(b) Determine $\mathbb{E}[Y]$
I am stuck on part $(a)$. I'm trying to work backwards:
$$P(Y = y) = P(X - M = 0) = P(X = M) = (1 - p)^{M}p.$$
I don't know if this is right, though. Also it doesn't account for the maximum function.
Then, I think that $E(Y) = 1/p$. But I also think that this is wrong, because this would be the same for $X$.