---
title: "Standard normal; normal approx of Poisson"
subject: "Probability and Statistics"
chapter: 3
key: "semester_5/probability_and_statistics:standard-normal-poisson-approximation"
source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/standard-normal-poisson-approximation
site: BCT Engineering
---

# Standard normal; normal approx of Poisson

Probability and Statistics · Chapter 3: Continuous Probability Distributions

- Asked in: 2072 Chaitra, 5 marks; 2071 Shrawan, 2+3 marks (asked 2 times)
- Video: [Poisson to Normal Approximation](https://www.youtube.com/watch?v=-YfOip1YB3E) — Mr Camilleri
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/standard-normal-poisson-approximation
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-3

## Question

Define standard normal distribution. Give the condition for normal approximation of Poisson distribution.

## Short answer

Standard normal is Z=(X-mu)/sigma with mean 0 and variance 1. Poisson can be approximated by normal when lambda is large, usually lambda >= 10, using continuity correction.

## Answer

#### Standard normal distribution
If X is normal with mean μ and standard deviation σ, then Z = (X − μ)/σ has the **standard normal distribution** with mean 0 and variance 1, written Z ~ N(0, 1). Its pdf is

f(z) = (1/√(2π)) e^(−z²/2), −∞ < z < ∞.

The curve is bell shaped, symmetric about 0, with total area 1. Areas under it are tabulated, so probabilities for any normal variable are found by standardising.

#### Normal approximation to the Poisson distribution
If X ~ Poisson(λ), then mean = variance = λ. When λ is large, the Poisson distribution becomes nearly symmetric and bell shaped, and

Z = (X − λ)/√λ ≈ N(0, 1), i.e. X ≈ N(μ = λ, σ² = λ).

**Condition:** λ must be large, usually λ ≥ 10 (some books use λ > 15 or 20). The larger the λ, the better the approximation.

Because Poisson is discrete and the normal is continuous, a continuity correction is applied, e.g. P(X ≤ a) ≈ P(Z ≤ (a + 0.5 − λ)/√λ).

## Check yourself

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

1. If X is normal, Z = (X - mu)/sigma has:
   A. Mean 1 and variance 0
   B. Mean mu and variance 1
   C. Mean 0 and variance sigma
   D. Mean 0 and variance 1
   Answer: D. This is N(0,1). (Key point: Standard normal)

2. The normal approximates Poisson when:
   A. lambda is large (about 10 or more)
   B. lambda is very small
   C. lambda equals 1
   D. n is small
   Answer: A. Large lambda makes it symmetric and bell shaped. (Key point: Condition)

3. The approximating normal has:
   A. mean lambda and variance lambda^2
   B. mean 0 and variance 1 for X
   C. mean lambda and variance lambda
   D. mean sqrt(lambda)
   Answer: C. Poisson mean = variance = lambda. (Key point: Parameters)

4. Why is a continuity correction used?
   A. Poisson is continuous
   B. lambda is small
   C. Poisson is discrete and normal is continuous
   D. To make variance 1
   Answer: C. E.g. P(X <= a) uses a + 0.5. (Key point: Continuity correction)

## Close variants set in other exams

- [Normal definition; approximation conditions](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Anormal-definition-approximation-conditions) — Probability and Statistics, chapter 3, asked 3×

## Related in Probability and Statistics

- [Normal distribution; approximation of Poisson](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Anormal-approx-poisson-define) — Probability and Statistics, chapter 3
- [Poisson definition; approximation to binomial](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Apoisson-definition-binomial-approx) — Probability and Statistics, chapter 2, asked 2×
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- [Normal characteristics; relation to standard normal](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Anormal-characteristics-standard-normal-relation) — Probability and Statistics, chapter 3
- [Standard normal definition and importance](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Astandard-normal-importance-engineering) — Probability and Statistics, chapter 3
- [Standard normal with area property](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Astandard-normal-area-property) — Probability and Statistics, chapter 3

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