Question

Let X and Y be independent random variables each having a geometric probability

mass function with parameter 1/2 Let Z = Y - X and M = min(X, Y) . Find the joint

p.m.f of M and Z P(M=m , Z=z) , for integers z and m > 0

Answer 1

$P\left( X=k \right) =P\left( Y=k \right) =\frac{1}{2}\cdot \left( \frac{1}{2} \right) ^{k-1}=\frac{1}{2^k}\\P\left( M=m,Z=z \right) =P\left( \min \left( X,Y \right) =m,X-Y=z \right) \\z\geqslant 0:\\P\left( M=m,Z=z \right) =P\left( Y=m,X=m+z \right) =\frac{1}{2^m}\cdot \frac{1}{2^{m+z}}=\frac{1}{2^{2m+z}}\\z<0:\\P\left( M=m,Z=z \right) =P\left( X=m,Y=m-z \right) =\frac{1}{2^m}\cdot \frac{1}{2^{m-z}}=\frac{1}{2^{2m-z}}\\In\,\,general:\\P\left( M=m,Z=z \right) =\frac{1}{2^{2m+\left| z \right|}}$