3.2.5 Negative Binomial Distribution
In a sequence of independent Bernoulli(p) trials, let the random variable X denote the trial at which the rth success occurs, where r is a fixed integer. Then
P (X = x|r, p) =
µx − 1 r − 1
¶
pr(1 − p)x−r, x = r, r + 1, . . . , (1)
and we say that X has a negative binomial(r, p) distribution.
The negative binomial distribution is sometimes defined in terms of the random variable Y =number of failures before rth success. This formulation is statistically equivalent to the one given above in terms of X =trial at which the rth success occurs, since Y = X − r. The alternative form of the negative binomial distribution is
P (Y = y) =
µr + y − 1 y
¶
pr(1 − p)y, y = 0, 1, . . . .
The negative binomial distribution gets its name from the relationship µr + y − 1
y
¶
= (−1)y µ−r
y
¶
= (−1)y(−r)(−r − 1) · · · (−r − y + 1)
(y)(y − 1) · · · (2)(1) , (2)
which is the defining equation for binomial coefficient with negative integers. Along with (2), we have
X
y
P¡
Y = y¢
= 1
from the negative binomial expansition which states that
(1 + t)−r = X
k
µ−r k
¶ tk
= X
k
(−1)k
µr + k − 1 k
¶ tk
1
EY = X∞
y=0
y
µr + y − 1 y
¶
pr(1 − p)y
= X∞
y=1
(r + y − 1)!
(y − 1)!(r − 1)!pr(1 − p)y
= X∞
y=1
r(1 − p) p
µr + y − 1 y − 1
¶
pr+1(1 − p)y−1
= r(1 − p) p
X∞
z=0
µr + 1 + z − 1 z
¶
pr+1(1 − p)z
= r1 − p p . A similar calculation will show
VarY = r(1 − p) p2 .
Example 3.2.6 (Inverse Binomial Sampling
A technique known as an inverse binomial sampling is useful in sampling biological popula- tions. If the proportion of individuals possessing a certain characteristic is p and we sample until we see r such individuals, then the number of individuals sampled is a negative bnomial rndom variable.
0.1 Geometric distribution
The geometric distribution is the simplest of the waiting time distributions and is a special case of the negative binomial distribution. Let r = 1 in (1) we have
P (X = x|p) = p(1 − p)x−1, x = 1, 2, . . . ,
which defines the pmf of a geometric random variable X with success probability p.
X can be interpreted as the trial at which the first success occurs, so we are “waiting for a success”. The mean and variance of X can be calculated by using the negative binomial formulas and by writing X = Y + 1 to obtain
EX = EY + 1 = 1
P and VarX = 1 − p p2 . 2
The geometric distribution has an interesting property, known as the “memoryless” property.
For integers s > t, it is the case that
P (X > s|X > t) = P (X > s − t), (3)
that is, the geometric distribution “forgets” what has occurred. The probability of getting an additional s − t failures, having already observed t failures, is the same as the probability of observing s − t failures at the start of the sequence.
To establish (3), we first note that for any integer n,
P (X > n) = P (no success in n trials) = (1 − p)n,
and hence,
P (X > s|X > t) = P (X > s and X > t)
P (X > t) = P (X > s) P (X > t)
= (1 − p)s−t = P (X > s − t).
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