The Negative Binomial Distribution
Section 3.5
An experiment is called a negative binomial experiment if
it satisfies the following conditions:
1. The experiment of interest consists of a sequence of
sub-experiments (can be infinite) called trials.
2. Each trial can result in one of two outcomes usually
denoted by success (S) or failure (F).
3. These trials are independent.
4. The probability of success, p, is constant from trial to
trial
5. We stop this experiment when a fixed number, r, of
successes occur.
The Negative Binomial Distribution
Section 3.5
What the above is saying:
The experiment consists of a group of independent
Bernoulli sub-experiments, where r (not n), the number of
successes we are looking to observe, is fixed in advance of
the experiment and the probability of a success is p.
What we are interested in studying is the number of
failures that precede the rth success.
Called negative binomial because instead of fixing the
number of trials n we fix the number of successes r.
The Negative Binomial Distribution
Section 3.5
Example: Observe the light-bulb production line testing
each bulb, and stopping the production process for
maintenance after the 3rd observed defective (S). From
experience, the probability that a light bulb is defective
is 0.1.
The Negative Binomial Distribution
Section 3.5
1) Identify the experiment of interest and understand it
well (including the associated population)
The experiment consists of a sequence of
independent Bernoulli sub-experiments, where r = 3,
the number of successes we are looking to observe, is
fixed in advance of the experiment and the probability
of a success is p = 0.1.
So the experiment qualifies as a negative binomial
experiment.
The Negative Binomial Distribution
Section 3.5
2) Identify the sample space (all possible outcomes)
S = {SSS, FSSS, SFSS, SSFS, FFSSS, …, FFFSSS …}
How many?
Still discrete?
The Negative Binomial Distribution
Section 3.5
3) Identify an appropriate random variable that reflects
what you are studying (and simple events based on
this random variable)
X = # of failures until the 3rd success
Snew = {0, 1, 2, …}
The Negative Binomial Distribution
Section 3.5
4) Construct the probability distribution associated
with the simple events based on the random variable
X = 0, how many simple events in association?
Probability is?
X = 1, how many associated simple events (outcomes)?
The probability is?
X = 2, how many associated simple events (outcomes)?
The probability is?
The general equation describing this distribution is?
The Negative Binomial Distribution
Section 3.5
The resulting distribution in table format:
x
1
2
3
4
5
6
0.0010 0.0027 0.0049 0.0073 0.0098 0.0124 0.0149
…
Sum
…
1
0.4
P<=x)
0.015
0.2
0.010
0.005
0.0
0.000
P(X=x)
0.6
0.020
0.8
0.025
1.0
P(X = x)
0
0
20
40
60
x
80
100
0
20
40
60
x
80
100
The Negative Binomial Distribution
Section 3.5
The resulting distribution in table format:
1
2
3
4
5
6
0.005
0.010
0.015
0.020
0.025
0.0010 0.0027 0.0049 0.0073 0.0098 0.0124 0.0149
0.000
P(X = x)
0
P(X=x)
x
0
20
40
60
x
80
100
…
Sum
…
1
The Negative Binomial Distribution
Section 3.5
Notation in association with the negative binomial
experiment:
The negative binomial random variable X = the number
of failures (F’s) until the rth success.
We say X is distributed negative Binomial with parameters
r and p,
The Negative Binomial Distribution
pmf is:
CDF is
Section 3.5
The Negative Binomial Distribution
Mean
Variance
Standard deviation
Section 3.5
The Negative Binomial Distribution
Mean
Variance
Standard deviation
Section 3.5
The Negative Binomial Distribution
Section 3.5
What is the chance that you will observe 3 defective light
bulbs after observing 27 +/- 2*16.43 (-5.86, 59.86) failures?
Approximately using Chebyshev’s rule:
Exactly from using R:
The Negative Binomial Distribution
Section 3.5
A special case of the negative binomial is when r = 1, then
we call the distribution geometric.
Notation in association with the geometric experiment:
The geometric random variable X = the number of
failures (F’s) until the 1st success.
We say X is distributed geometric with parameter p,
The Negative Binomial Distribution
pmf is:
CDF is
Section 3.5
The Negative Binomial Distribution
Mean
Variance
Standard deviation
Section 3.5
The Negative Binomial Distribution
Section 3.5
Example: Observe the light-bulb production line testing
each bulb, and stopping the production process for
maintenance after the 1st observed defective (S). From
experience, the probability that a light bulb is defective
is 0.1.
Now we can start to jump in and solve problems directly
as we have the base! (I hope).
This experiment looks like a geometric one.
So the distribution that we can use to find probabilities
is:
The Negative Binomial Distribution
Section 3.5
P(X > 3) = ?
Mean is?
What does it mean?
Variance and standard deviation are?
Chance of being within two standard deviations from
the mean? Exact and Chebyshev!
The Poisson distribution
Moving away from the solid Bernoulli based trials to
approximate distributions.
Section 3.6