The Poisson distribution is a vital concept in probability theory, particularly useful for modeling the frequency of events occurring in a fixed interval of time or space.

## The Basics of Poisson Distribution

The Poisson distribution helps us understand how likely it is for a certain number of events to happen in a fixed interval, given the average number of times these events usually occur. It's useful when we're looking at events that happen independently and at a constant average rate.

### Formula

The probability of observing exactly $k$ events is given by:

$P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!}$

- $e$ is the base of the natural logarithm (about 2.71828).
- $k$ is the number of events we're interested in.
- $\lambda$ is the average number of events.
- $k!$ is $k$ factorial, the product of all positive integers up to $k$.

### Key Points

**Discrete:**It deals with counts of events, so $k$ can be 0, 1, 2, and so on.**Versatile:**Used in various fields for analyzing rare events over time or space.

## Application Steps

1.** Identify **$\lambda$** :** Find the average event rate.

2. **Determine **$k$**:** Decide the specific number of events you're interested in.

3. **Calculate Probability:** Use the formula to calculate the likelihood of (k) 3. events.

## Examples

### Example 1: Bookstore Customers

Suppose a bookstore averages 3 customers per hour. What is the probability exactly 5 customers arrive in an hour?

#### Solution:

- Average Rate $(\lambda)$: 3 customers/hour.
- Interested in $(k)$: 5 customers.
- Probability Calculation: Using the formula with $\lambda = 3$ and $k = 5$, we find the probability is about 10.08%.

### Example 2: Network Faults

In a network operation centre, an average of 2 faults occur per day. What is the probability that there will be no faults in a day?

#### Solution:

- Average Rate $(\lambda)$: 2 faults/day.
- Interested in $(k)$: 0 faults.
- Probability Calculation: Using the formula with $\lambda = 2$ and $k = 0$, we find the probability is about 13.53%.

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.