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Expectation of a Negative Binomial RV
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[QUOTE="richievuong, post: 2552540, member: 48333"] [h2]Homework Statement [/h2] 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? [h2]Homework Equations[/h2] 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 [h2]The Attempt at a Solution[/h2] 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... [/QUOTE]
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Expectation of a Negative Binomial RV
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