# Combining geometric distribution with binomial

Let $$N \sim \operatorname{Geo}(p)$$, so $$\mathbb{P}(N=k)=(1-p)^k p$$, $$k=0,1,2,\dots$$, and $$N_R | N \sim \operatorname{Bin}(\alpha,N)$$.

I am trying to find a distribution of $$N_R$$, it is fairly easy to write down $$\mathbb{P}(N_R=k)=\sum_{n=k}^\infty \binom{n}{k} \alpha^k(1-\alpha)^{n-k}(1-p)^n p,$$ but I do not see an easy way of evaluating it.

Another approach is to look at the probability generating functions, to deduce that $$G_{N_R}(z)=G_N(\alpha z+1-\alpha)=\frac{p}{1-(1-p)(\alpha z+1-\alpha)},$$ and I can't recognise this as a PGF of anything familiar.

Is the distribution of $$N_R$$ well-known distribution (or a mixture of two well-known distributions), but I just don't see it?

Thank you.

## 1 Answer

Hint: There exists $q$ depending on $p$ and $\alpha$ such that $G_{N_R}(z)=\dfrac{q}{1-(1-q)z}$, hence $N_R$ is geometric with parameter $q$.

• Is there a way to see that such q exists straight away? May 9, 2012 at 16:13
• Yes, the pgf is a rational function with constant numerator and linear denominator, like that of the geometric rv. This, along with pgf(1) = 1 fixes all the coefficients uniquely. May 9, 2012 at 16:17
• Of course, Maclaurin expansion will make it a geometric RV! May 9, 2012 at 16:20