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Expectation of a Negative Binomial RV

  1. Jan 28, 2010 #1
    1. The problem statement, all variables and given/known data
    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?

    2. Relevant equations
    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

    3. 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....
  2. jcsd
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