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

# Negative binomial definition and characteristics

Probability and Statistics · Chapter 2: Discrete Probability Distribution

- Asked in: 2073 Chaitra, 5 marks; 2072 Kartik, 5 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-definition-characteristics
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-2

## Question

Define negative binomial distribution with its important characteristics.

## Short answer

Negative binomial: number of trials to get r successes. Characteristics: infinite range, mean r/p, variance rq/p². It reduces to geometric when r=1.

## Answer

#### Definition
In a sequence of independent Bernoulli trials with a constant probability of success p (and q = 1 − p), the random variable X, the number of trials needed to obtain a fixed number r of successes, follows the **negative binomial distribution**:

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

Reasoning: the x-th trial is the r-th success (probability p), and in the first x − 1 trials exactly r − 1 are successes, in C(x − 1, r − 1) ways with probability pʳ⁻¹ qˣ⁻ʳ.

#### Important characteristics
1. **Parameters:** r (number of successes wanted) and p (probability of success).
2. **Conditions:** independent trials, two outcomes, constant p, trials continue until r successes occur.
3. **Range:** X takes the values r, r + 1, r + 2, … (discrete and unbounded).
4. **Mean:** E(X) = r/p.
5. **Variance:** Var(X) = rq/p², so the variance is larger than the mean when q > p. The standard deviation is √(rq)/p.
6. **Special case:** when r = 1 it becomes the geometric distribution, P(X = x) = q ˣ⁻¹ p, with mean 1/p.
7. **Additive property:** the sum of independent NB(r₁, p) and NB(r₂, p) variables is NB(r₁ + r₂, p).
8. **Alternative form:** if Y = X − r (the number of failures before the r-th success), P(Y = y) = C(y + r − 1, y) pʳ qʸ with mean rq/p. The name "negative binomial" comes from the fact that the probabilities are the terms of the expansion of pʳ (1 − q)⁻ʳ.
9. The probabilities add to 1: Σ C(x − 1, r − 1) pʳ qˣ⁻ʳ = 1.

**Example:** the probability that a coin gives the 2nd head on the 4th toss (p = 0.5) is C(3, 1)(0.5)²(0.5)² = 3/16 = 0.1875. The expected number of tosses to get 2 heads is 2/0.5 = 4.

## Check yourself

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

1. What is X in negative binomial?
   A. Trials needed to obtain r successes
   B. Time between events
   C. Successes in n trials
   D. Failures in a trial
   Answer: A. r is fixed and trials are random. (Key point: Random variable)

2. What is the sum of independent NB(r1,p) and NB(r2,p)?
   A. NB(r1+r2, p)
   B. Geometric
   C. NB(r1 r2, p)
   D. Binomial
   Answer: A. Waiting for r1+r2 successes in total. (Key point: Additive property)

3. Probability the 2nd head comes on the 4th toss, p = 0.5?
   A. 3/16
   B. 1/2
   C. 3/8
   D. 1/4
   Answer: A. C(3,1)(0.5)^2(0.5)^2 = 3/16. (Key point: Example)

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