Chapter 4, Problem 4.29TE

Let X be a negative binomial random variable with parameters n and p, and let Y be a binomial random variable with parameters n and p. Show that $P\left\{X>n\right\}=P\left\{Y<r\right\}$

Hint: Either one could attempt an analytical proof of the preceding equation, which is equivalent to proving the identity ${\displaystyle \underset{i=n+1}{\overset{\infty }{\sum }}\left(\begin{array}{c}i-1\\ r-1\end{array}\right)p^r{\left(1-p\right)}^{i-r}=}{\displaystyle \underset{i=0}{\overset{r-1}{\sum }}\left(\begin{array}{c}n\\ i\end{array}\right)\times p^i{\left(1-p\right)}^{n-i}}$ or one could attempt a proof that uses the probabilistic interpretation of these random variables. That is, in the latter case, start by considering a sequence of independent trials having a common probability p of success. Then try to express the events $\left\{X>n\right\}$ and $\left\{Y<r\right\}$ in terms of the outcomes of this sequence.