The approximation of the Poisson distribution using a normal distribution is an invaluable tool in statistics, particularly beneficial when dealing with large data sets. This approximation simplifies complex statistical problems, making them more approachable and understandable.

## Introduction

- When the mean $(λ)$ of a Poisson distribution is large (usually λ≥15), using a normal distribution for approximation is effective.
- This method simplifies calculations for large λ values.

## Normal Approximation Criteria

- Applicable when λ is large (≥15).
- As λ increases, the Poisson distribution looks more like a normal distribution.

## Continuity Correction

- Needed because Poisson is discrete and normal distribution is continuous.
- Adjust the range of values slightly for a more accurate approximation.

## Examples

### Example 1: Call Centre Probability

A call centre receives an average of 18 calls per hour. Find the probability of receiving at most 20 calls in an hour.

#### Solution:

**Given:**$\lambda = 18, X \leq 20$.**Applying Continuity Correction**: Adjust X to 20.5.**Converting to Z-Score**:- $Z = \frac{X - λ}{√λ} $</li></ul></li><li><strong>Calculate: </strong>$Z = \frac{20.5 - 18}{\sqrt{18}} \approx 0.59.$</li><li><strong>Result: </strong>Probability for$Z = 0.59$is ~72.24%. So, 72.24% chance of$\leq 20$calls.</li></ul><img src="https://tutorchase-production.s3.eu-west-2.amazonaws.com/8edee42e-1233-401c-bab5-38e832b7c69c-file.jpg" alt="Call Centre Probability Graph" style="width: 500px; cursor: pointer"><h3>Example 2: Bakery Sales Probability</h3><p>A bakery typically sells an average of 16 special cakes per day. Calculate the probability of selling more than 20 cakes on a given day.</p><h4>Solution:</h4><ul><li><strong>Given:</strong>$\lambda = 16, X > 20$. </li><li><strong>Applying Continuity Correction: </strong>Adjust to$X = 20.5$.</li><li><strong>Calculate: </strong>$Z = \frac{20.5 - 16}{\sqrt{16}} = 1.125$.</li><li><strong>Result: </strong>Probability for$Z \geq 1.125$ cakes is ~13.03%. So, 13.03% chance of selling ( > 20 ).

Written by: Dr Rahil Sachak-Patwa

LinkedIn

Oxford University - PhD Mathematics

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.