The mean of a negative binomial distribution is n(1-p)/p and the variance is n(1-p)/p².
Understanding the Negative Binomial Distribution
The negative binomial distribution is a discrete probability distribution that models the number of failures x in a sequence of independent and identically distributed Bernoulli trials before a specified number of successes, denoted by n, occurs. The probability of success on each trial is p.
Think of it like flipping a coin until you get heads a certain number of times. The negative binomial distribution tells you the probability of needing a specific number of tails (failures) before you reach your target number of heads (successes).
Mean and Variance Formulas
The mean (average) and variance (measure of spread) are key statistics that describe the center and dispersion of the distribution. According to the provided reference, for a negative binomial distribution with parameters n (number of successes) and p (probability of success):
- Mean: The expected number of failures before
nsuccesses. - Variance: A measure of how spread out the number of failures is from the mean.
Here are the formulas for the mean and variance:
| Statistic | Formula |
|---|---|
| Mean | n(1-p)/p |
| Variance | n(1-p)/p² |
These formulas indicate that both the mean and variance depend directly on the required number of successes (n) and the probability of achieving a success on any given trial (p).
n: The number of successes required.p: The probability of success on a single trial.(1-p): The probability of failure on a single trial.
Maximum Likelihood Estimate of p
The provided reference also mentions the Maximum Likelihood Estimate (MLE) for the probability parameter p when you have a sample from a negative binomial distribution. If you have a sample of observed numbers of failures (x₁, x₂, ..., x<subscript>m</subscript>) before n successes in each trial, and x̄ is the sample mean of these observed failures, the MLE of p is given by:
Maximum Likelihood Estimate of p: n / (n + x̄)
This estimate allows you to infer the underlying probability of success p from observed data.
Practical Context
The negative binomial distribution is useful in various fields for modeling events where you wait for a fixed number of occurrences. Examples include:
- Reliability Engineering: Modeling the number of tests until a certain number of component failures occur.
- Quality Control: Determining the number of items inspected until a specific number of defects are found.
- Ecology: Modeling the distribution of species in an area.
Understanding the mean and variance helps in predicting the expected number of failures and quantifying the variability around that expectation in such scenarios.
