Negative Binomial Distribution (Discrete)

Last modified: March 22, 2022

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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 $X \sim NegBinomial (r, p)$ , 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.

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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:

$F (k) = P (X \leq k) = \sum_{i = r}^{k} (\binom{i - 1}{r - 1}) p^{r} (1 - p)^{i - r}$

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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:

$Var (X) = \frac{r (1 - p)}{p^{2}}$

Moment Generating Function (MGF)

The moment generating function (MGF) of a negative binomial distribution is:

$M_{X} (t) = {(\frac{p e^{t}}{1 - (1 - p) e^{t}})}^{r}$

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:

$P (X = 20) = (\binom{19}{4}) (0.1)^{5} (0.9)^{15} \approx 0.031$

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:

$Var (X) = \frac{5 \times (1 - 0.1)}{{0.1}^{2}} = 450$

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.