Derivation of Mean and Variance for Negative Binomial Distribution Using Probability Generating Functions

The negative binomial distribution is a discrete probability distribution which models the number of failures observed before achieving a specified number of successes. This article provides a detailed derivation of the mean and variance for the negative binomial distribution using its probability generating function (PGF).

Probability Generating Function (PGF)

The PGF for a negative binomial distribution that counts the number of failures before achieving a fixed number of successes (r) is given by:

G(s) left(frac{p}{1 - qs}right)^r

where:

p is the probability of success; q 1 - p is the probability of failure; r is the number of successes.

Mean of the Negative Binomial Distribution

To find the mean of the distribution, we use the first derivative of the PGF evaluated at (s 1):

(mu G'(1))

First, we calculate the first derivative of the PGF:

(G'(s) r left(frac{p}{1 - qs}right)^{r-1} cdot frac{pq}{1 - qs^2})

Evaluating this at (s 1), we get:

(G'(1) r left(frac{p}{1 - q}right)^{r-1} cdot frac{pq}{1 - q^2})

Since (p 1 - q), we simplify the expression:

(G'(1) r left(frac{p}{p}right)^{r-1} cdot frac{pq}{p^2} frac{rq}{p})

Therefore, the mean (mu) of the negative binomial distribution is:

(mu frac{rq}{p})

Variance of the Negative Binomial Distribution

To find the variance, we need the second derivative of the PGF:

(sigma^2 G''(1) cdot G'(1) - (G'(1))^2)

First, we derive the second derivative of the PGF:

(G''(s) left[frac{r}{s} cdot frac{rq^2}{1 - qs^2}right] cdot left[frac{ps}{1 - qs}right]^r)

Evaluating at (s 1), we get:

(G''(1) -r frac{rq^2}{1 - q^2} cdot r frac{rq}{1 - q} -r^2 q^3 frac{1}{p^2})

Now, substituting the values in the variance formula:

(sigma^2 -r^2 frac{rq^3}{p^2} cdot frac{rq}{p} - left(frac{rq}{p}right)^2 -r^2 q^2 frac{r}{p})

Thus, the variance (sigma^2) of the negative binomial distribution is:

(sigma^2 frac{rq}{p^2})

Conclusion

The mean and variance of a negative binomial distribution with parameters (r) (number of successes) and (p) (probability of success) can be derived using its probability generating function. The mean is (frac{rq}{p}) and the variance is (frac{rq}{p^2}).

Two alternative forms of the negative binomial distribution are:

First Form

The mean using the first form of the PGF is (frac{rp}{q}) and the variance is (frac{rp}{q^2}).

Second Form

The mean using the second form of the PGF is (frac{rq}{p}) and the variance is (frac{rq}{p^2}).

Understanding the mean and variance of the negative binomial distribution is crucial for applications in various fields, including statistics, probability theory, and data science.