From Algorithms to Z-Scores - matloff - University of California, Davis

3.12. PARAMETERIC FAMILIES OF PMFS 65
“outside world,” i.e. to the remaining n-k nodes. So, the distribution of T is binomial with
trials and success probability p.
k(n − k) +
k
2
3.12.3 The Negative Binomial Family of Distributions
(3.110)
Recall that a typical example of the geometric distribution family (Section 3.12.1) arises as N, the
number of tosses of a coin needed to get our first head. Now generalize that, with N now being
the number of tosses needed to get our r th head, where r is a fixed value. Let’s find P(N = k), k
= r, r+1, ... For concreteness, look at the case r = 3, k = 5. In other words, we are finding the
probability that it will take us 5 tosses to accumulate 3 heads.
First note the equivalence of two events:
{N = 5} = {2 heads in the first 4 tosses and head on the 5 th toss} (3.111)
That event described before the “and” corresponds to a binomial probability:
P (2 heads in the first 4 tosses) =
4 1
2 2
4
(3.112)
Since the probability of a head on the k th toss is 1/2 and the tosses are independent, we find that
P (N = 5) =
4 1
2 2
5
= 3
16
(3.113)
The negative binomial distribution family, indexed by parameters r and p, corresponds to random
variables that count the number of independent trials with success probability p needed until we
get r successes. The pmf is
We can write
P (N = k) =
k − 1
(1 − p)
r − 1
k−r p r , k = r, r + 1, ... (3.114)
N = G1 + ... + Gr
(3.115)