2. Recall that for rN1:={1,2,3,} and 0<p<1 arbitrary the density of the negative binomial distribution NB(r,p) with parameters r and p is given by NB(r,p)({k}):=(r+k1k)pr(1p)k,kN. Recall that for r=1 this distribution is also known as geometric distribution with parameter p, Geo(p). Show using characteristic functions that a random variable XNB(r,p) has the same distribution as the sum of r independent geometric random variables with appropriate parameters. Give also an intuitive reason for this result. Hint: You can use the formula (x+y)n=k=0(1)k(nn+k1)xky(nk) without proof, valid for x<y. It actually suffices to use this generalised binomial formula for y=1..

1. 2. Recall that for rN1:={1,2,3,} and 0<p<1 arbitrary the density of the negative binomial distribution
NB(r,p) with parameters r and p is given by NB(r,p)({k}):=(r+k1k)pr(1p)k,kN. Recall that for r=1 this
distribution is also known as geometric distribution with parameter p, Geo(p). Show using
characteristic functions that a random variable XNB(r,p) has the same distribution as the sum of r
independent geometric random variables with appropriate parameters. Give also an intuitive
reason for this result. Hint: You can use the formula (x+y)n=k=0(1)k(nn+k1)xky(nk) without proof,
valid for x<y. It actually suffices to use this generalised binomial formula for y=1.