MA/STAT416: Probability
Lecture Notes
Spring 2011
1
Chapter 6: Standard Discrete Distributions
March 23, 2011
6
Poisson and Negative Binomial Distributions
6.4
The Negative Binomial Distribution
Definition.
Let
X
be the number of the first toss at which the
r
th
success is obtained.
Then
X
has a negative binomial distribution;
X
has
NB
(
r, p
), with the pmf
P
(
X
=
x
) =
x

1
r

1
·
p
r
·
(1

p
)
x

r
,
x
=
r, r
+ 1
,
· · ·
.
Theorem.
If
X
has
NB
(
r, p
),
r
≥
1. Then,
E
(
X
) =
r
p
,
and
V ar
(
X
) =
r
(1

p
)
p
2
.
Example
1
.
A couple want to have children till they have two girls. Find the distribution,
expected value and variance of the number of children they will have.
Note.
This is the negative binomial distribution with parameters
r
= 2
, p
=
1
2
.
Example
2
.
Exploratory oil drilling in a certain region will strike oil with a 20% proba
bility. Find the probability that the third strike of oil comes at the sixth drill; that the
third strike comes on or before the sixth drill.
Example
3
.
Onethird of blood donors at a clinic have type O+ blood. Find the prob
ability that the second O+ donor is the fourth donor of the day.
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 Spring '08
 Staff
 Binomial, Normal Distribution, Poisson Distribution, Probability, Probability theory, Binomial distribution, Discrete probability distribution, Negative binomial distribution

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