Last modified: September 21, 2024

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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.

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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:

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

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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:

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

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:

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.

Table of Contents

    Negative Binomial Distribution (Discrete)
    1. Probability Mass Function (PMF)
    2. Cumulative Distribution Function (CDF)
    3. Expected Value and Variance
    4. Moment Generating Function (MGF)
    5. Example: Quality Control Testing
    6. Applications