---
title: "Discrete distribution; compare NB and binomial"
subject: "Probability and Statistics"
chapter: 2
key: "semester_5/probability_and_statistics:discrete-distribution-nb-vs-binomial"
source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/discrete-distribution-nb-vs-binomial
site: BCT Engineering
---

# Discrete distribution; compare NB and binomial

Probability and Statistics · Chapter 2: Discrete Probability Distribution

- Asked in: 2074 Chaitra, 3+3 marks
- 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/discrete-distribution-nb-vs-binomial
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-2

## Question

Define discrete probability distribution with suitable example. Compare Negative Binomial and Binomial probability distributions.

## Short answer

A discrete probability distribution assigns probabilities to each value of a discrete random variable, summing to 1, e.g., number of heads in two coin tosses. Binomial: fixed n trials, counts successes, mean np. Negative binomial: fixed k successes, counts trials until k-th success, mean k/p.

## Answer

#### Discrete probability distribution
A **discrete probability distribution** lists every value x that a discrete random variable can take together with its probability P(X = x), called the probability mass function (pmf). It must satisfy:
1. P(x) ≥ 0 for every x
2. Σ P(x) = 1

**Example:** Toss a fair coin twice and let X = number of heads.

| x | 0 | 1 | 2 |
|---|---|---|---|
| P(x) | 1/4 | 1/2 | 1/4 |

All probabilities are non-negative and sum to 1, so this is a discrete distribution (it is binomial with n = 2, p = 0.5). Its mean is Σ x P(x) = 1.

#### Negative binomial vs binomial
- **Binomial:** P(X = x) = C(n, x) pˣ qⁿ⁻ˣ, x = 0, 1, …, n.
- **Negative binomial:** if X is the trial on which the k-th success occurs, P(X = x) = C(x − 1, k − 1) pᵏ qˣ⁻ᵏ, x = k, k + 1, …

| Basis | Binomial | Negative binomial |
|---|---|---|
| Fixed quantity | Number of trials n | Number of successes k |
| Random variable | Number of successes in n trials | Number of trials needed for k-th success |
| Range of X | 0 to n (finite) | k, k+1, … (infinite) |
| Last trial | Any outcome | Must be a success |
| Mean | np | k/p |
| Variance | npq | kq/p² |
| Special case | n = 1 gives Bernoulli | k = 1 gives geometric |

Both need independent trials with constant p and two outcomes.

## Check yourself

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

1. What must a discrete probability distribution satisfy?
   A. P(x) >= 0 and Sum P(x) = 1
   B. Sum P(x) = 0
   C. P(x) equals x
   D. P(x) > 1
   Answer: A. Both conditions are required. (Key point: Conditions)

2. For X = heads in two fair coin tosses, P(1)?
   A. 1
   B. 3/4
   C. 1/2
   D. 1/4
   Answer: C. P = 1/4, 1/2, 1/4. (Key point: Two coins)

3. Which is true?
   A. Neither
   B. Binomial fixes k; NB fixes n
   C. Both fix k
   D. Binomial fixes n; NB fixes k
   Answer: D. NB counts trials until k-th success. (Key point: Comparison)

4. k = 1 in negative binomial gives?
   A. Geometric
   B. Hypergeometric
   C. Poisson
   D. Bernoulli
   Answer: A. Waiting for the first success. (Key point: Special case)

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