Last modified: August 12, 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 $X \sim \text{NegBinomial}(r, p)$, 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:
$$P(X=k) = \binom{k-1}{r-1} p^r (1-p)^{k-r}$$
where $k \in {r, r+1, r+2, \dots}$ and $\binom{k-1}{r-1}$ 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:
$$F(k) = P(X \le k) = \sum_{i=r}^{k} \binom{i-1}{r-1} p^r (1-p)^{i-r}$$
Expected Value and Variance
The expected value (mean) of a negative binomial distribution is:
$$E[X] = \frac{r}{p}$$
The variance of a negative binomial distribution is:
$$\text{Var}(X) = \frac{r(1-p)}{p^2}$$
Moment Generating Function (MGF)
The moment generating function (MGF) of a negative binomial distribution is:
$$M_X(t) = \left(\frac{pe^t}{1 - (1 - p)e^t}\right)^r$$
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:
$$P(X = 20) = \binom{19}{4} (0.1)^5 (0.9)^{15} ≈ 0.031$$
II. What is the expected number of products to be tested?
Expected number of products tested:
$$E[X] = \frac{5}{0.1} = 50$$
III. What is the variance in the number of products tested?
Variance in the number of products tested:
$$\text{Var}(X) = \frac{5 \times (1-0.1)}{0.1^2} = 450$$
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.