Application of power series/ binomial theorem in inverse sampling

Question

I have posted this already in other forums. Apologies for cross posting.

In order to establish some properties of inverse sampling, Haldane (1945) uses power series and the binomial theorem I assume. According to the inverse-sampling method (Haldane 1945) you continue sampling until m of the rare items have been found. Let p be the frequency of the rare item and q = 1-p. m is the number of rare items observed. n is the number of observations

you continue sampling until m of the rare items have been found. Let p be the frequency of the rare item and q = 1-p. m is the number of rare items observed. n is the number of observations

What is now the probability that exactly n observations have been made before m rare items are observed? According to Haldane (1945) this probability is

$w_{n} = \binom{n-1}{m-1}p^{m}q^{n-m}$

Haldane then concludes that "this is the coefficient of $t^n$ in $(\dfrac{qt}{1-qt})^m$ " (Haldane 1945: 222)

He does not go more into detail here and just continues with his proof on inverse sampling. However, I just do not see how this coefficient of $t^n$ is related to the probability $w_{n}$.

Assuming a geometric series, I could write

$\dfrac{qt^m}{(1-qt)} = \sum_{i=0} qt^mqt^i$

Furthermore, through differentiation of the power series, we get:

$\dfrac{qt^m}{(1-qt)^m} = \sum_{i=0} \binom{m+i+1}{i}qt^mqt^i$

Here, I get stuck. What is implied? Does the coefficient of $t^n$ simply refer to $\binom{m+i+1}{i}qt^mqt^i$? If so, however, how does it link up with $w_{n} = \binom{n-1}{m-1}p^{m}q^{n-m}$? Thanks for any hints/ helps!

Answer 1

It's convenient to use the coefficient of operator $[t^n]$ to denote the coefficient of $t^n$ in a series. We can write this way \begin{align*} [t^n](1+t)^m=\binom{m}{n} \end{align*}

I think there's a small typo and we have instead \begin{align*} w_n=\binom{n-1}{m-1}p^mq^{n-m}=[t^n]\left(\frac{\color{blue}{p}t}{1-qt}\right) \end{align*}

We obtain \begin{align*} [t^n]\left(\frac{pt}{1-qt}\right)^m&=[t^n]p^mt^m\frac{1}{(1-qt)^m}\\ &=p^m[t^{n-m}]\sum_{k=0}^{\infty}\binom{-m}{k}(-q)^kt^k\tag{1}\\ &=p^m[t^{n-m}]\sum_{k=0}^{\infty}\binom{m+k-1}{m-1}q^kt^k\tag{2}\\ &=\binom{n-1}{m-1}p^mq^{n-m}\tag{3}\\ \end{align*} and the claim follows.

Comment:

• In (1) we use the linearity of the coefficient of operator and the rule $$[t^{n-m}]A(t)=[t^n]t^mA(t)$$

  We obtain for $0\leq m\leq n$ \begin{align*} [t^{n-m}]A(t)&=[t^{n-m}]\sum_{k=0}^{\infty}a_kt^k=a_{n-m}\\ [t^n]t^mA(t)&=[t^n]t^m\sum_{k=0}^{\infty}a_kt^k =[t^n]\sum_{k=0}^{\infty}a_kt^{k+m} =a_{n-m} \end{align*} We also apply the binomial series expansion.

• In (2) we use the binomial identity \begin{align*} \binom{-m}{k}=\binom{m+k-1}{m-1}(-1)^k \end{align*}

• In (3) we select the coefficient of $t^{n-m}$ and take the summand with $k=n-m$.

Answer 2

The expression $w_n$ that is described is the probability mass function of a particular parametrization of the negative binomial distribution. If $W$ is the random number of observations until we obtain the $m^{\rm th}$ rare event, where the probability of observing a rare event is $p$, then $$\Pr[W = n] = w_n = \binom{n-1}{m-1} p^m (1-p)^{n-m}, \quad n = m, m+1, \ldots.$$ This is because there are $\binom{n-1}{m-1}$ ways to arrange $m$ rare events among $n$ total observations such that the final observation is a rare event, and the joint probability of observing any particular arrangement of $m$ rare events among $n-m$ non-rare events is $p^m (1-p)^{n-m}$.

Thus, $$1 = \sum_{n=m}^\infty \Pr[W = n] = \sum_{n=m}^\infty \binom{n-1}{m-1} p^m (1-p)^{n-m}.$$ Now consider the function $$\begin{align*} P_W(t) = \operatorname{E}[t^W] &= \sum_{n=m}^\infty t^n \Pr[W = n] \\ &= (tp)^m \sum_{n=m}^\infty \binom{n-1}{m-1} (t(1-p))^{n-m} \\ &= \frac{(tp)^m}{(1-t(1-p))^m} \sum_{n=m}^\infty \binom{n-1}{m-1} (1-t(1-p))^m (t(1-p))^{n-m} \\ &= \left(\frac{pt}{1-t(1-p)}\right)^m \sum_{n=m}^\infty \binom{n-1}{m-1} (p^*)^m (1-p^*)^{n-m} \\ &= \left(\frac{pt}{1-t(1-p)}\right)^m, \quad 0 < t < \frac{1}{1-p}, \end{align*}$$ where $p^* = 1 - t(1-p)$, and the last summation is equal to $1$ because it is the sum over all outcomes of a modified negative binomial random variable $W^*$ with modified probability of observing a rare event $p^*$. We conclude that $w_n$ is the coefficient of the $t^n$ term in the series expansion of the function $$P_W(t) = \left(\frac{pt}{1-t(1-p)}\right)^m.$$ Note that the numerator should be $(pt)^m$, not $(qt)^m$.