14.2 Probability Density Function (PDF) of the Negative Binomial Distribution

The Negative Binomial Distribution is a discrete probability distribution that models the number of trials required to achieve a fixed number of successes in a series of independent Bernoulli trials, where each trial has the same probability of success.

Probability Density Function (PDF) Formula

The probability of observing exactly $x$ trials to achieve the $k$-th success is given by the following formula:

$$P(X = x) = \binom{x-1}{k-1} \theta^k (1-\theta)^{x-k}$$

Where:

• $P(X = x)$: The probability of achieving the $k$-th success on the $x$-th trial.
• $\binom{x-1}{k-1}$: The binomial coefficient, representing the number of ways to choose the positions of the first $k-1$ successes among the first $x-1$ trials. It is calculated as: $$\binom{x-1}{k-1} = \frac{(x-1)!}{(k-1)!(x-k)!}$$
• $\theta$: The probability of success in a single Bernoulli trial.
• $1-\theta$: The probability of failure in a single Bernoulli trial.
• $x$: The total number of trials (where $x \geq k$).
• $k$: The fixed number of successes required.

Valid Range:

• $x = k, k+1, k+2, \dots$
• $0 < \theta < 1$

Interpretation of the Formula

The PDF formula quantifies the likelihood that the process of independent Bernoulli trials will continue until the $k$-th success occurs, and this $k$-th success happens specifically on the $x$-th trial. This implies that out of the first $x-1$ trials, there must have been exactly $k-1$ successes, and the $x$-th trial must be a success.

The formula is derived from the understanding that:

1. Binomial Coefficient ($\binom{x-1}{k-1}$): This term accounts for all possible sequences of outcomes in the first $x-1$ trials that contain exactly $k-1$ successes.
2. Success Probability ($\theta^k$): This represents the probability of achieving $k$ successes in total, with the last one occurring on the $x$-th trial.
3. Failure Probability ($(1-\theta)^{x-k}$): This represents the probability of experiencing $x-k$ failures in the first $x-1$ trials.

Key Assumptions:

• Independence: Each trial is independent of the others.
• Constant Probability of Success: The probability of success ($\theta$) remains the same for every trial.
• Stopping Rule: The experiment stops exactly when the $k$-th success is observed.

Example

Suppose a basketball player has a free throw success probability ($\theta$) of 0.8. What is the probability that their 3rd successful free throw ($k=3$) occurs on their 5th attempt ($x=5$)?

Using the formula: $P(X=5) = \binom{5-1}{3-1} (0.8)^3 (1-0.8)^{5-3}$ $P(X=5) = \binom{4}{2} (0.8)^3 (0.2)^2$ $P(X=5) = 6 \times 0.512 \times 0.04$ $P(X=5) = 0.12288$

So, there is approximately a 12.3% chance that the player makes their 3rd successful free throw on their 5th attempt.

Use Cases in Real Life

The Negative Binomial Distribution is valuable in various scenarios:

• Sales and Marketing: Predicting the number of sales calls required to secure a specific number of deals.
• Manufacturing and Quality Control: Estimating the number of items to inspect before finding a predetermined quantity of defective products.
• Clinical Trials: Modeling the number of patients treated until a certain number of positive treatment outcomes are achieved.
• Reliability Engineering: Determining the number of operational periods until a system fails a specific number of times.
• Gambling: Calculating the probability of winning a specific number of hands in a card game before losing a certain number of times.

Interview Questions

Here are some common interview questions related to the Negative Binomial Distribution:

• What is the probability density function (PDF) formula for the Negative Binomial Distribution?
• How do you interpret the probability of the $k$-th success occurring on the $x$-th trial?
• What does the binomial coefficient $\binom{x-1}{k-1}$ represent in the Negative Binomial Distribution formula?
• What are the underlying assumptions for a process to be modeled by the Negative Binomial Distribution?
• How does the Negative Binomial Distribution differ from the Binomial Distribution? (Hint: Binomial counts successes in a fixed number of trials; Negative Binomial counts trials for a fixed number of successes).
• Can you provide an example of how the Negative Binomial Distribution is applied in sales or marketing?
• Describe a real-life situation where the Negative Binomial Distribution could be useful in quality control.
• Why must the trials in a Negative Binomial Distribution be independent?
• How do the parameters $k$ (number of successes) and $\theta$ (probability of success) influence the shape of the Negative Binomial Distribution?
• What are the valid ranges for the number of trials ($x$) and the probability of success ($\theta$) in the Negative Binomial Distribution?