3.4 Practice Problems
Problem 3.4.1. Let \(X\) and \(Y\) be independent, each geometric with \(P(X=x) = pq^{x}\) for \(x=0,1,2,\dots \) where \(q=1-p\). Use convolution to find the distribution of \(Z=X+Y\), and confirm the answer with generating functions.
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Solution.
By convolution
\[P(Z=z) = \sum _{x=0}^{z} pq^{x}\cdot pq^{z-x} = p^{2}q^{z}\sum _{x=0}^{z}1 = (z+1)\,p^{2}q^{z},\qquad z=0,1,2,\dots \] the summand being free of \(x\), so the sum simply counts its \(z+1\) terms. This is the negative binomial with \(k=2\).
By generating functions
\[G_X(t) = \sum _{x=0}^{\infty }t^{x}pq^{x} = \frac {p}{1-qt},\] so \(G_Z(t) = p^{2}/(1-qt)^{2}\), which is the generating function of NBIN\((2,p)\). The two agree, and the second route needed no summation.
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