Negative Binomial Distribution

Each success takes on average $\frac{1}{p}$ trials (this is just the expectation of a geometric distribution). So if you need $n$ successes, and each is independent:

\[ \mathbb{E}[\text{trials}] = \underbrace{\frac{1}{p} + \frac{1}{p} + \cdots + \frac{1}{p}}_{n \text{ times}} = \frac{n}{p} \]

Example

If the probability of success is 0.2 (e.g., hitting a target with 20% accuracy), and you want 5 hits: $$ \mathbb{E}[\text{trials}] = \frac{5}{0.2} = 25 $$ On average, you'd expect to take 25 shots to land 5 hits.

Relationship with [[Geometric Distribution]]

The Negative Binomial Distribution generalizes this by modeling the number of trials needed to achieve r successes. It’s essentially the sum of r independent geometric distributions (each representing the trials until a single success). The Geometric Distribution is a special case of the Negative Binomial Distribution where r=1r = 1r=1. The Negative Binomial extends the Geometric Distribution by counting the trials needed for more than one success, not just the first one.

Markov Processes Optimal Substructure Reasoning

On This Page

Why? (First Principles) Example Relationship with [[Geometric Distribution]]