The Problem
Suppose there are $H$ balls in a jar, with $X_i(0)$ white balls and $H-X_i(0)$ blacks balls. At each step, we randomly choose a ball from the jar. If a white ball is chosen, we replace it an additional white ball and remove a black ball. If a black ball is chosen, we simply put it back in, with no replacement. Thus, the number of balls is always $H$. We assume that the process never terminates: we keep adding extra white balls even when there are no black balls left. The number of white balls increase exponentially per step when there are black balls (see end of post); once there are only white balls, the growth is linear. This way, we can also consider what happens for arbitrarily large numbers of steps without worrying about stopping at the correct step.
Notation
Denote by $X_i(k)$ the number of white balls at the $k$th step, so initially there are $X_i(0)=i$ balls. That is,
- $k$ is the number of steps,
- $i$ is the number of white balls initially,
- Finally, $X_i(k)$ is the number of white balls after $k$ steps, for $i$ initially white balls.
Goal
I am trying to calculate/identify the following probability distribution (next paragraph), or even $E[X^2]$ at each step, to ultimately determine the variance (I have derived $E[X]$ as shown later).
What I know
A similar question is asked here, but it is only concerned with the number of iterations till there are only white balls. There are a few highly relevant wiki articles, but none have the exact process I am looking for:
- The Polya Distribution Page outlines that
there are variants of the Moran model that insist that the ball removed from the urn be a different ball from one originally sampled in that step
but I cannot find the relevant resource. Even looking through the page for Moran Process, I didn't find a section where the opposite allele is always removed.
An Attempt
Expected value:
At any step, we have
$$E[X_i(k+1)]= (X_i(k)+1)\cdot \frac{X_i(k)}{H} + X_i(k) \cdot \frac{H-X_i(k)}{H}$$ Solving the recursive equation we get $$ E[X_i(k)] = X_i(0)\cdot \left(1+\frac{1}{H} \right)^k$$
until $X_i(k) \geq H$. This way, I have avoided having to find the probability distribution. However, I can't find a way to apply similar logic for $E[X_i(k)^2]$.