Comprehending the role of the Poisson distribution in linear combinations is vital. This topic illuminates the intriguing property of independent Poisson-distributed random variables summing up to follow a Poisson distribution, a cornerstone concept in statistical analysis.

## Understanding the Poisson Distribution

The Poisson distribution is used to model the probability of a number of events happening in a fixed interval of time or space, given a known constant mean rate of occurrence and the independence of events.

**Key Traits of Poisson Distribution**:- Independence of events.
- Constant average rate (λ, lambda).
- Suitable for low-frequency events over large intervals.

## Linear Combinations with Poisson Distribution

A linear combination in statistics involves adding or subtracting variables, often multiplied by constants. For Poisson distributions, this translates to combining different Poisson processes.

## Examples

### Example 1: Combining Two Processes

**Problem: **Two independent Poisson processes have average rates of 3 and 5 events per hour. What is the distribution for the total events in an hour?

**Solution**:

**Combine the Rates**:- Process $A (λ_A)$ = 3, Process $B (λ_B)$ = 5.
- Total rate $λ_Total = λ_A + λ_B = 3 + 5 = 8.$

**Poisson Distribution for Total Events**:- The combined process follows a Poisson distribution with λ = 8.

### Example 2: Real-World Call Center

**Problem:** A call centre gets Type A calls at a rate of 2 per hour and Type B calls at 4 per hour. Calculate the probability of exactly 5 calls in an hour.

**Solution:**

**Calculate Combined Rate**:

**Probability for 5 Calls**:- Use Poisson formula: $P(X = 5) = (e^(-6) * 6^5) / 5!.$
- Perform the calculation to find P(X = 5).

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.