---
title: "Normal definition; approximation conditions"
subject: "Probability and Statistics"
chapter: 3
key: "semester_5/probability_and_statistics:normal-definition-approximation-conditions"
source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/normal-definition-approximation-conditions
site: BCT Engineering
---

# Normal definition; approximation conditions

Probability and Statistics · Chapter 3: Continuous Probability Distributions

- Asked in: 2070 Chaitra, 6 marks
- Video: [Normal Approximation To The Binomial Distribution - Approximating A Binomial Probability](https://www.youtube.com/watch?v=A2sd09qCvcg) — Whats Up Dude
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/normal-definition-approximation-conditions
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-3

## Question

Define normal distribution. Give the condition for normal approximation of Binomial distribution and Poisson distribution.

## Short answer

Normal distribution: continuous with pdf f(x)=(1/(σ√(2π)))e^(−(x−μ)²/(2σ²)). Binomial approximation requires large n, np≥5, nq≥5, and continuity correction. Poisson approximation needs λ≥10, also with continuity correction.

## Answer

**Normal distribution:** a continuous random variable *X* is normally distributed with mean μ and standard deviation σ if its pdf is

f(x) = (1 / (σ√(2π))) · e^(−(x − μ)² / (2σ²)), −∞ < x < ∞

written X ~ N(μ, σ²). Its curve is a symmetric bell shape, centred at μ, with mean = median = mode = μ, total area 1, and points of inflection at μ ± σ.

#### Normal approximation to the Binomial
If X ~ B(n, p), then for large n it is approximated by N(μ = np, σ² = npq) with q = 1 − p.

Conditions:
- n is large (n ≥ 30 is commonly used), and
- p is not too close to 0 or 1, so that np ≥ 5 and nq ≥ 5 (some books use 10).

Use the continuity correction: P(X = a) ≈ P(a − 0.5 < X < a + 0.5), and z = (x ± 0.5 − np)/√(npq).

#### Normal approximation to the Poisson
If X ~ Poisson(λ), then for large λ it is approximated by N(μ = λ, σ² = λ).

Condition:
- λ is large, commonly λ ≥ 10 (some books accept λ > 5), and the continuity correction is applied: z = (x ± 0.5 − λ)/√λ.

## Check yourself

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

1. Which describes the normal curve?
   A. Symmetric bell with mean = median = mode = μ
   B. Skewed right with mean > median
   C. Flat with equal density
   D. Mean = variance
   Answer: A. The normal curve is a symmetric bell centred at μ where mean, median and mode coincide. (Key point: Normal curve properties)

2. A B(n, p) variable is approximated by a normal with:
   A. mean np and variance npq
   B. mean p and variance pq
   C. mean np and variance np
   D. mean n and variance npq
   Answer: A. The binomial mean is np and variance npq with q = 1 − p. (Key point: Binomial approximation parameters)

3. Which condition suits the normal approximation to the binomial?
   A. n small and p near 0
   B. n large and np, nq both at least 5
   C. p exactly 0.5 only
   D. np less than 5
   Answer: B. n should be large and p not too close to 0 or 1, so that np ≥ 5 and nq ≥ 5. (Key point: Conditions for binomial)

4. The Poisson(λ) is approximated by the normal when λ is large with:
   A. mean λ and variance λ
   B. mean λ and variance λ^2
   C. mean sqrt(λ) and variance λ
   D. mean 0 and variance 1
   Answer: A. A Poisson has mean and variance both λ, so X ≈ N(λ, λ) for λ ≥ 10 or so. (Key point: Poisson approximation)

## Close variants set in other exams

- [Standard normal; normal approx of Poisson](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Astandard-normal-poisson-approximation) — Probability and Statistics, chapter 3, asked 3×

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