---
title: "Negative binomial: definition and difference"
subject: "Probability and Statistics"
chapter: 2
key: "semester_5/probability_and_statistics:nb-define-differ-binomial"
source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/nb-define-differ-binomial
site: BCT Engineering
---

# Negative binomial: definition and difference

Probability and Statistics · Chapter 2: Discrete Probability Distribution

- Asked in: 2075 Chaitra, 6 marks; 2071 Shrawan, 2+3 marks (asked 2 times)
- Video: [Introduction to the Negative Binomial Distribution](https://www.youtube.com/watch?v=BPlmjp2ymxw) — jbstatistics
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/nb-define-differ-binomial
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-2

## Question

Define Negative Binomial distribution and explain characteristics. How does it differ from binomial distribution?

## Short answer

Negative binomial gives probability that rth success occurs on xth trial. It differs from binomial because number of trials is random, not fixed.

## Answer

#### Definition
Consider repeated independent Bernoulli trials, each with probability of success p and failure q = 1 − p. The **negative binomial distribution** gives the probability that the r-th success occurs on the x-th trial.

```text
P(X = x) = C(x − 1, r − 1) pʳ qˣ⁻ʳ ,   x = r, r + 1, r + 2, ...
```

The last (x-th) trial must be a success, and the previous x − 1 trials contain exactly r − 1 successes. Here r is a fixed positive integer and p is the parameter. It is written NB(r, p).

#### Characteristics
1. The trials are independent and have only two outcomes, with constant p.
2. The number of successes r is fixed; the number of trials X is random.
3. The range of X is r, r + 1, r + 2, … (infinite and discrete).
4. Mean: E(X) = r/p.
5. Variance: Var(X) = rq/p², which is greater than the mean when q > p.
6. For r = 1 it reduces to the **geometric distribution**: P(X = x) = q ˣ⁻¹ p.
7. The sum of r independent geometric variables is negative binomial.
8. If the number of failures Y = X − r is counted, P(Y = y) = C(y + r − 1, y) pʳ qʸ, with mean rq/p and variance rq/p².

**Example:** a player shoots until he scores 3 goals, with p = 0.4 per shot. The probability that the 3rd goal comes on the 5th shot is C(4, 2) (0.4)³ (0.6)² = 6 × 0.064 × 0.36 = 0.138.

#### Difference from the binomial distribution

| Basis | Binomial | Negative binomial |
|---|---|---|
| Fixed | Number of trials n | Number of successes r |
| Random variable | Number of successes | Number of trials needed |
| Possible values | 0, 1, …, n | r, r + 1, … |
| pmf | C(n, x) pˣ qⁿ⁻ˣ | C(x − 1, r − 1) pʳ qˣ⁻ʳ |
| Mean, variance | np, npq | r/p, rq/p² |
| Variance vs mean | Variance < mean | Variance > mean when q > p |

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What does negative binomial give?
   A. Number of successes in n trials
   B. Failures only in a lot
   C. Probability of no success
   D. Probability the r-th success occurs on the x-th trial
   Answer: D. X is the trial number of the r-th success. (Key point: Definition)

2. Why does the pmf contain C(x-1, r-1)?
   A. The last trial is a success and the previous x-1 hold r-1 successes
   B. All trials are failures
   C. Any trial may be the last
   D. The pmf is symmetrical
   Answer: A. The x-th trial must be a success. (Key point: Last trial)

3. What does it reduce to when r = 1?
   A. Hypergeometric
   B. Binomial
   C. Geometric
   D. Poisson
   Answer: C. P(X=x) = q^(x-1) p. (Key point: Special case)

4. What is the mean of NB(r,p)?
   A. rq/p^2
   B. rp
   C. np
   D. r/p
   Answer: D. E(X) = r/p. (Key point: Mean)

## Close variants set in other exams

- [Binomial characteristics vs negative binomial](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Abinomial-vs-nb-characteristics) — Probability and Statistics, chapter 2, asked 4×
- [Binomial characteristics vs hypergeometric](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Abinomial-vs-hypergeometric) — Probability and Statistics, chapter 2, asked 4×

## Related in Probability and Statistics

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