---
title: "Poisson definition; approximation to binomial"
subject: "Probability and Statistics"
chapter: 2
key: "semester_5/probability_and_statistics:poisson-definition-binomial-approx"
source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/poisson-definition-binomial-approx
site: BCT Engineering
---

# Poisson definition; approximation to binomial

Probability and Statistics · Chapter 2: Discrete Probability Distribution

- Asked in: 2070 Asar, 2+3 marks
- Video: [Poisson approximation to the Binomial Distribution : ExamSolutions](https://www.youtube.com/watch?v=2-30SSlOuk4) — ExamSolutions
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/poisson-definition-binomial-approx
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-2

## Question

Define poisson probability Distribution. Write the conditions for poisson approximation to Binomial Distribution.

## Short answer

A Poisson distribution gives P(X=x)=e^-lambda lambda^x/x! for x=0,1,2,... with mean=variance=lambda. Binomial approximation to Poisson requires n large, p small, and np=lambda moderate.

## Answer

#### Poisson distribution
A discrete random variable X that counts the number of occurrences of a rare event in a fixed interval of time, length, area or volume, has a **Poisson distribution** with parameter λ if

P(X = x) = e⁻ˡ λˣ / x!, x = 0, 1, 2, …

Here λ > 0 is the average number of occurrences, and mean = variance = λ. Examples: calls received per minute, defects per metre of cable.

#### Conditions for Poisson approximation to the binomial
A binomial B(n, p) can be approximated by a Poisson distribution with λ = np when:
1. The number of trials n is very large (n → ∞, usually n ≥ 20 or more).
2. The probability of success p is very small (p → 0, usually p ≤ 0.05).
3. The product λ = np is finite and moderate (commonly less than 5 to 10).

Then C(n, x) pˣ qⁿ⁻ˣ ≈ e⁻ⁿᵖ (np)ˣ / x!.

Example: if 0.2% of bulbs are defective, the number of defectives in a lot of 1000 is approximately Poisson with λ = 1000 × 0.002 = 2.

## Check yourself

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

1. The Poisson distribution describes:
   A. Trials until the first success
   B. Successes in n fixed trials
   C. Continuous waiting time
   D. The count of occurrences of a rare event in a fixed interval
   Answer: D. X counts rare events in an interval of time, length or area. (Key point: Definition)

2. For Poisson, the mean and variance are:
   A. Mean lambda, variance lambda^2
   B. Both equal lambda
   C. Mean np, variance npq
   D. Mean 1/lambda, variance 1/lambda^2
   Answer: B. Mean = variance = lambda. (Key point: Mean and variance)

3. Poisson approximates B(n, p) when:
   A. n is large, p is small and np is moderate
   B. n is small and p is near 0.5
   C. n is large and p is near 1
   D. p is large and n is small
   Answer: A. Then lambda = np. (Key point: Approximation conditions)

## Close variants set in other exams

- [Hypergeometric distribution; binomial approximation](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Ahypergeometric-define-binomial-approx) — Probability and Statistics, chapter 2, asked 2×

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