Let $X$be a negative binomial random variable with parameters $r$and $p$, and let $Y$be a binomial random variable with parameters $n$and $p$. Show that

$P\{X>n\}=P\{Y<r\}$

Hint: Either one could attempt an analytical proof of the preceding equation, which is equivalent to proving the identity

$\begin{array}{r}\sum _{i=n+1}^{\mathrm{\infty }} \left(\begin{array}{c}i-1\\ r-1\end{array}\right)p^r(1-p{)}^{i-r}=\sum _{i=0}^{r-1} \left(\begin{array}{c}n\\ i\end{array}\right)\\ \times p^i(1-p{)}^{ni}\end{array}$

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 to express the events $\{X>n\}$and $\{Y<r\}$in terms of the outcomes of this sequence.

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