2.9 Geometric and Negative Binomial Distributions
125
Since Formula 2.9 exhausts the possibilities, it must be that
∞
P
x=2
P. X = x = 1. One way to verify this is to notice that
∞
X
x=2
x − 1 1
· q
x−2
· p
2
= p
2 ∞
X
x=2
x − 1 1
· q
x−2
= p
2
· .1 − q
−2
= p
2
· p
−2
= 1
by the binomial theorem with a negative exponent. This series will arise again in our work. We have established the probability distribution for the waiting time for the
second head. What is the average waiting time for the second head? We might reason as follows:
We flip the coin until the first head appears; the average number of flips is
1 p
. But then the situation is exactly the same as it was for the first flip of the coin; the fact that we
flipped the coin and waited for the first head has absolutely no influence on subsequent tosses of the coin. We must wait an average of
1 p
flips again until the second head appears. So the average waiting time for the second head to appear is
1 p
+
1 p
=
2 p
. It follows that if we were to wait for the rth head to appear, the average total waiting time
would be
r p
. We will give a more formal derivation of this result later. What is the probability distribution function for the rth head to appear? Let X
denote the number of tosses until the rth head appears. Since, again, the last toss must be a head and the first x − 1 tosses must contain exactly r − 1 heads,
P. X = x = x − 1
r − 1 · p
r−1
· q
x−r
· p; x = r; r + 1; r + 2; : : : : 2.10
Since P. X = x ≥ 0; we must check the sum of the probabilities to see that we have a probability distribution function. But,
∞
X
x=r
x − 1 r − 1
· p
r−1
· q
x−r
· p = p
r ∞
X
x=r
x − 1 r − 1
q
x−r
= p
r
. 1 − q
−r
= 1;
so P. X = x is a probability distribution. If r = 1 in Formula 2.10, we find that P.X = x reduces to the geometric proba-
bility distribution function. The result in Formula 2.10 is called the negative binomial distribution because of the occurrence of the binomial expansion with a negative
exponent. We now calculate the mean and the variance of this negative binomial random
variable. We reasoned that the mean is
r p
and we now give another derivation of this.
126
Chapter 2 Discrete Random Variables and Probability Distributions
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