Geometric-binomial posterior is shifted negative binomial ($X-k\mid Y=k\sim \mathrm{NegBin}(k+1,p+q-pq)$)

Let $\mathbb{P}(X=n)=p(1-p{)}^n$ for $n\ge 0$ and $Y\mid X=n\sim \mathrm{Binomial}(n,q)$, with $0<p,q<1$. Write $D=p+q-pq$. Summing the joint probability mass function with the differentiated geometric series identity ${\sum }_{n=k}^{\infty }\left(\genfrac{}{}{0ex}{}{n}{k}\right)t^{n-k}=(1-t{)}^{-k-1}$ gives $\mathbb{P}(Y=k)=(p/D)((1-p)q/D{)}^k$. By Bayes' theorem,Thus $X-k$, given $Y=k$, has the failure-count negative binomial distribution for $k+1$ successes and success parameter $D$. Its conditional expectation is $k+(k+1)(1-D)/D$ for $X$ itself. The shift by $k$, the number of retained successes, must be retained when identifying the posterior distribution.