pdf of binomial random varible with binomial random variable number of trials

Question

Given a random sequence $B_k$ where:

$B_{0} = 0$

$B_{k+1} = B_{k} + D_k$

where $D_k \sim Binomial(n - B_k, p) $ and n is positive. Is there a simple non-recursive formula for $B_{k+1}$?

Answer 1

So $B_0=0$, $B_{k+1}=B_k+D_k$, and $D_k\mid B_k\sim \mathcal{Bin}(n-B_k, p)$

$$\begin{align}\mathsf P(B_{k+1}-B_k=d_k\mid B_k=b_k) ~=~& \binom{n-b_k}{d_k}p^{d_k}(1-p)^{n-b_k-d_k}~\mathbf 1_{0\leq b_k\leq b_k+d_k\leq n} \end{align}$$

This looks like: We hold $n$ trials in the first round and keep the $D_0$ successes $(B_1:=B_0+D_0)$.   In the second round we retry the $n-B_1$ failures from the first round and keep the $D_1$ successes ($B_2:=B_1+D_1$). … In round $k$ we retry the failures left from the round $k-1$ and keep the $D_{k-1}$ successes $(B_{k}:=B_{k-1}+D_{k-1} = \sum_{i=0}^{k-1} D_i)$.

Thus $B_k$ is the count of successes from $k$ cumulative retries of a sequence of $n$ trials, where we keep the successes and retry the failures.

Let $X_{i,k}$ be the indicator function that trial $i$ has succeeded at least once in $k$ retries. $(i\in\{1,..,n\})$.   The probability that any particular trial has done so is, $p_k:=\mathsf P(X_{i, k}=1)$, which is identical and independent for each trial.

Thus $B_k$ is the count success of $n$ retrials each with success rate $p_k$.

$$B_k\sim\mathcal {Bin}(n, p_k)$$