Last modified: June 25, 2020
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Negative Binomial Distribution (Discrete)
A discrete random variable X follows a negative binomial distribution if it represents the number of trials required to achieve a specified number of successes in a sequence of independent Bernoulli trials. The negative binomial distribution is often denoted as X∼NegBinomial(r,p), where r is the number of successes required and p is the probability of success on each trial.
Probability Mass Function (PMF)
The PMF of a negative binomial distribution is given by:
P(X=k)=(k−1r−1)pr(1−p)k−r
where k∈r,r+1,r+2,… and (k−1r−1) is the binomial coefficient.
Cumulative Distribution Function (CDF)
The CDF of a negative binomial distribution is not as straightforward to express in a closed form but is the sum of its PMF values:
F(k)=P(X≤k)=k∑i=r(i−1r−1)pr(1−p)i−r
Expected Value and Variance
The expected value (mean) of a negative binomial distribution is:
E[X]=rp
The variance of a negative binomial distribution is:
Var(X)=r(1−p)p2
Moment Generating Function (MGF)
The moment generating function (MGF) of a negative binomial distribution is:
MX(t)=(pet1−(1−p)et)r
Example: Quality Control Testing
In quality control testing, an inspector examines products until finding a certain number of defective items. Suppose the probability of a product being defective is 0.1, and the inspector continues testing until 5 defective products are found.
Given:
- Probability of defective product
p = 0.1 - Number of defects required
r = 5
I. What is the probability that exactly 20 products need to be tested?
For exactly 20 products tested:
P(X=20)=(194)(0.1)5(0.9)15≈0.031
II. What is the expected number of products to be tested?
Expected number of products tested:
E[X]=50.1=50
III. What is the variance in the number of products tested?
Variance in the number of products tested:
Var(X)=5×(1−0.1)0.12=450
Applications
Negative binomial distributions are commonly used in scenarios where the number of trials until a specified number of failures occurs is of interest, such as in reliability testing, epidemiology, and ecological studies.