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

# Poisson definition; limiting case of binomial

Probability and Statistics · Chapter 2: Discrete Probability Distribution

- Asked in: 2068 Chaitra, 2+2 marks
- Video: [Poisson Distribution | Limiting case of Binomial Distribution](https://www.youtube.com/watch?v=DWdzROYQZrs) — Dr. Harish Garg
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/poisson-limiting-case-binomial
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-2

## Question

Define Poisson distribution. Write the limiting case of Poisson distribution as a Binomial distribution.

## Short answer

Poisson distribution has P(X=x)=e^-lambda lambda^x/x!, mean=variance=lambda. It is the limit of B(n,p) as n->infinity, p->0 with np=lambda, giving the same Poisson probability mass function.

## Answer

#### Poisson distribution
A discrete random variable X, the number of occurrences of a rare event in a given interval (of time, length, area), follows a **Poisson distribution** with mean λ if

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

Mean = variance = λ.

#### Poisson as the limiting case of the binomial
Let X ~ B(n, p) and let n → ∞, p → 0 so that np = λ stays constant. Then

P(X = x) = C(n, x) pˣ (1 − p)ⁿ⁻ˣ

Put p = λ/n:

```text
P(x) = [n(n-1)...(n-x+1)/x!] (λ/n)ˣ (1 - λ/n)ⁿ⁻ˣ
     = (λˣ/x!) · [1(1-1/n)...(1-(x-1)/n)] · (1-λ/n)ⁿ · (1-λ/n)⁻ˣ
```

As n → ∞: each factor (1 − k/n) → 1, (1 − λ/n)ⁿ → e⁻ˡ and (1 − λ/n)⁻ˣ → 1. Hence

lim P(X = x) = **e⁻ˡ λˣ / x!**

So for large n and small p, B(n, p) ≈ Poisson(λ = np).

## Check yourself

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

1. Poisson arises from B(n, p) when:
   A. n -> 0 and p -> 1
   B. n and p both fixed
   C. n -> infinity, p -> 0 with np = lambda constant
   D. p -> infinity
   Answer: C. The limit keeps the mean np fixed at lambda. (Key point: Limiting conditions)

2. In the proof, (1 - lambda/n)^n tends to:
   A. e^lambda
   B. 1
   C. e^-lambda
   D. lambda
   Answer: C. This standard limit gives the exponential factor. (Key point: Key limit)

3. The limiting pmf is:
   A. e^-lambda lambda^x / x!
   B. C(n,x) p^x q^(n-x)
   C. lambda^x / x!
   D. e^-lambda lambda^n / n!
   Answer: A. Substituting p = lambda/n and letting n grow gives the Poisson pmf. (Key point: Result)

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