Deriving the Probability Mass Function of the Negative Binomial Distribution from Sum of Bernoulli Trials

The negative binomial distribution (NBrp) is a discrete probability distribution that models the number of successes observed in a series of Bernoulli trials before a fixed number of failures occurs. This article will guide you through the derivation of the probability mass function (PMF) of the negative binomial distribution by summing up Bernoulli trials.

Understanding the Negative Binomial Distribution

The negative binomial distribution, denoted as XBrp, represents the number of successes (denoted as X) observed in a series of independent Bernoulli trials, each with a success probability of p. The trials continue until r failures occur. Mathematically, we denote this as:

X ~ NBrp

The event {X k} signifies that there are exactly k successes before the rth failure occurs.

Relationship with Bernoulli Trials

To derive the PMF, consider a sequence of Bernoulli trials with parameters Yi ~ Bernoulli(p). The PMF of X can be represented as:

P{X k} P{Y1, …, Ykr k}

This equation implies that for X to equal k, exactly k of the Yi outcomes must be successes, and the remaining r outcomes must be failures.

Calculating the Probability of a Specific Sequence

The probability of any specific sequence with k successes and r failures is given by:

p^k(1 - p)^r

This is because each success has a probability of p, and each failure has a probability of (1 - p).

Combinatorial Analysis

The next challenge is to determine the number of ways to arrange k successes and r failures in a sequence of k r trials. This can be done using the combination formula:

C(k r - 1, k)

which represents the number of ways to choose k successes from k r - 1 trials. However, we must take into account that the last trial is determined by the outcome of the previous k r - 1 trials. For X to equal k, those k successes must be among the first k r - 1 trials, and the last trial will be a failure.

Final Probability Mass Function

Combining the above steps, the PMF of the negative binomial distribution is given by:

P{X k} C(k r - 1, k) p^k (1 - p)^r

where C(k r - 1, k) is the binomial coefficient, representing the number of combinations of k successes from k r - 1 trials.

Conclusion

Understanding the negative binomial distribution and its derivation from a series of Bernoulli trials requires a careful combinatorial approach. By leveraging the properties of binomial coefficients and probabilities, we can derive a comprehensive PMF that accurately models the distribution of successes before a fixed number of failures in a series of independent trials.

For more in-depth understanding and application of the negative binomial distribution, refer to the following resources: