Lectures on Elementary Probability

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28 CHAPTER 4. THE BERNOULLI PROCESS Theorem 4.8 The expectation of the negative binomial random variable is n=r E[W ] = r 1 p . (4.23) Proof: This is the series ∞∑ E[T r ] = nP [T r = n] = r ∞∑ ( n (1 − p) p r) n−r p r+1 = 1 p . (4.24) n=r This is because P [T r+1 = n + 1] = ( n r) (1 − p) n−r p r+1 is the probability that the r + 1st success occurs on the n + 1st trial. If we grant that there is eventually going to be a second success, then these probabilities should add to one. Alternative proof: Use the fact that the mean of a geometric random variable is 1/p and the fact that the mean of a sum is the sum of the means. Theorem 4.9 The variance of the negative binomial random variable is σT 2 r = r 1 (1 − p). (4.25) p2 Proof: Consider the series ∞∑ E[T r (T r +1)] = n(n+1)P [T r = n] = r(r+1) 1 ∑ ∞ ( ) n + 1 p 2 n (1−p) n−r p r+2 = r + 1 n=r n=r (4.26) This is because P [T n+2 = n + 1] = ( n+1 r+1) (1 − p) n−r p r+2 is the probability that the r+2nd success occurs on the n+2nd trial. If we grant that there is eventually going to be a r + 2nd success, then these probabilities should add to one. Thus σT 2 r = E[Tr 2 ]−E[T r ] 2 = E[Tr 2 ]+E[T r ]−E[T r ]−E[T r ] 2 r(r + 1) = p 2 − r p − r2 p 2 = r p 2 (1−p). (4.27) Alternative proof: Use the fact that the variance of a geometric random variable is 1/p 2 (1 − p) and the fact that the variance of a sum of independent random variables is the sum of the variances. These formula for the mean and expectation of the negative binomial waiting time are quite important. When r = 1 this is the geometric waiting time, so we already know the facts. So consider the case when we are waiting for the rth success and r is large. The formula for the mean is r 1/p, which is a quite intuitive result. The formula for the variance is also important. It is better to look at the standard deviation. This is √ r 1/p √ 1 − p. Note that the standard deviation is considerably smaller than the mean. This means that the distribution is somewhat peaked about the mean, at least in a relative sense. Very small values are unlikely, so are very large values (relative to the mean). The reason for this behavior is that the waiting time for the rth success is the sum of r independent geometric waiting times. These are individually quite variable. But the short geometric waiting times and the long geometric waiting times tend to cancel out each other. The waiting time for the rth success is comparatively stable. r(r + 1) p 2 .
4.7. SUMMARY 29 4.7 Summary Consider a sequence of independent success-failure trials, with probability p of success on each trial. The random variable N n that counts the number of successes in the first n trials is a binomial random variable. (When n = 1 the random variable N 1 is a Bernoulli random variable.) The random variable T r that gives the number of the trial on which the rth success occurs is a negative binomial random variable. (When r = 1 this is a geometric random variable.) The relation between N n and T r is that N n ≥ r is the same event as the event T r ≤ n.
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