STAT 100A Lecture Notes - Lecture 15: Tachykinin Receptor 1, Random Variable
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Negative binomial random variable (negative binomial random variable). De nition trials, each having probability of success 0 < p < 1, are performed until a total of r successes are accumulated. If x is the number of trials required, then x is said to be a negative binomial random variable. The last trial must necessarily result in a success, and there must be r 1 more success in the rst n 1 trials. = (cid:2)n 1 r 1(cid:3)pr 1(1 p)n rp r 1(cid:3)pr(1 p)n r. Theorem n(cid:2)n 1 r(cid:3) r 1(cid:3) = r(cid:2)n. E(cid:4)x k(cid:5) = r r p nk(cid:2)n 1 r 1(cid:3)pr(1 p)n r nk 1(cid:2)n r(cid:3)pr+1(1 p)n r (m 1)k 1(cid:2)m 1 (cid:6)n=r (cid:6)n=r (cid:6)m=r+1 p r (cid:3)pr+1(1 p)m (r+1) E(cid:4)x k(cid:5) = r p r r 1(cid:3)pr(1 p)n r nk(cid:2)n 1 r(cid:3)pr+1(1 p)n r nk 1(cid:2)n (m 1)k 1(cid:2)m 1 (cid:6)n=r (cid:6)n=r (cid:6)m=r+1 p r (cid:3)pr+1(1 p)m (r+1) E[y 1] p (cid:2) r + 1 p.
