Understanding the Mean and Variance of the Poisson Distribution is a vital aspect. This detailed exploration focuses on the practical applications and implications of these concepts.

## Overview of the Poisson Distribution

- What is it? A
**probability distribution**used to model the number of events in a fixed time or space interval. - Key Feature: Mean $λ$ = Variance $λ$.
**Mean**$μ$**:**The expected number of events, equal to λ.**Variance**$σ²$**:**The spread of the distribution, also equal to λ.

## Applications

**1. Predicting Event Counts:**

**High λ:**More events, greater spread.**Low λ:**Fewer events, lesser spread.

**Examples**

### Example 1: Library Book Returns

Calculate and visualize the mean and variance of a library that averages 4 book returns per hour (λ = 4).

#### Solution:

**1. Mean (μ) of the Distribution:**

- For a Poisson distribution, the mean (μ) is equal to the rate (λ) of the event.
- In this scenario, λ = 4 (4 book returns per hour).
- Therefore, μ = 4.

**2. Variance (σ²) of the Distribution:**

- The variance (σ²) in a Poisson distribution is also equal to λ.
- Hence, σ² = λ = 4.
- This means the average squared deviation from the mean number of book returns is also 4.

**Answers:**

**Mean and Variance:**Both are 4.**Graph:**Shows the probability of different return counts in an hour.

### Example 2: Email Receipts Probability

A website averages 2 emails per hour (λ = 2). Calculate the probability of receiving exactly 3 emails in an hour and visualize it.

**Solution:**

**Formula:**$P(X = k) = \frac{e^{-λ} λ^k}{k!}$ for $k = 3$.**Calculating for ( k = 3 ):**- Find $P(X = 3)$ given $λ = 2.$
- Plugging in the values:
- $P(X = 3) = \frac{e^{-2} 2^3}{3!}$
- $P(X = 3) = \frac{e^{-2} \times 8}{6}$ (since $3!$ equals 6)
- $P(X=3) = 0.1804$

Therefore, the probability $P(X = 3)$ is about 0.1804, or 18.04%. This means there's an 18.04% chance of receiving exactly 3 emails in any given hour.

**Graph:**Visualizes the probability of receiving different email counts in an hour.

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.