Consider a Negative Binomial random variable Y ~ NB(r, p). Show (from first principles!) that E[Y] is r/p. Why does this imply Y is proper?
I have no idea how to use latex, so this may be messy:
pmf of Y: [ (k+r-1)! / k!(r-1)! ] * (1-p)^r * p^k
The Attempt at a Solution
E[Y] = sigma(r=1 to k); r * (1-p)^r * p^k * [ (k+r-1)! / k!(r-1)! ]
I have no idea of how to manipulate the factorials...I know that sum of geometric random variables is a negative binomial rv. Since the expectation of a geometric random variable is 1 / p - sum them to r and we get r / p (expectation of NB). However I have to do this with first principles and I'm stumped.
I see a lot of possibilities.... p^k / k! is a known series..I even thought about this
for the r * (1-p)^r * p^k part:
= r * q^r * p^k
=r * q^(r-1) * q * p^k
= d/dq(q^r) * q * p^k
Hope this makes some sense....