Welcome to the Poisson Distribution!

In your A Level Statistics journey so far, you’ve likely met the Binomial distribution, which counts "successes" in a fixed number of trials. But what if there is no fixed number of trials? What if we are just counting how many times something happens over a certain amount of time or space?

That is where the Poisson distribution comes in! Whether it's the number of shooting stars in an hour, the number of typos on a page, or how many customers enter a shop at lunch, the Poisson distribution is our go-to tool. Don't worry if it feels a bit abstract at first; we will break it down step-by-step.

1. What Makes a "Poisson" Situation? (Syllabus SB1)

To use the Poisson model, the events we are counting must follow four strict rules. You can remember these using the mnemonic "RISC":

R – Randomly: Events must occur at random.
I – Independently: One event happening doesn't make it more or less likely for another to happen.
S – Singly: Events cannot happen at the exact same time/place.
C – Constant Rate: The average number of events per interval (\(\lambda\)) must stay the same.

Terminology and Notation

If a random variable \(X\) follows a Poisson distribution, we write it as:
\(X \sim \text{Po}(\lambda)\)

The symbol \(\lambda\) (the Greek letter "lambda") represents the mean (the average number of occurrences in a given interval).

Analogy: Imagine raindrops landing on a single pavement slab. If the rain is steady (constant rate), one drop doesn't "tell" another where to land (independent), and they land one by one (singly), the number of drops on that slab follows a Poisson distribution.

Key Takeaway: Before you start calculating, always check if the situation is random, independent, single, and at a constant rate.

2. Calculating Probabilities (Syllabus SB2)

To find the probability of exactly \(x\) events happening, we use this formula:

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

Where:
\(e\) is a constant (approx 2.718, found on your calculator).
\(\lambda\) is the average rate.
\(x\) is the number of successes you are looking for.
\(x!\) is "\(x\) factorial" (e.g., \(4! = 4 \times 3 \times 2 \times 1\)).

Using Your Calculator

In the exam, you will mostly use your calculator's distribution functions:
1. Poisson PD: Use this for "exactly" questions (e.g., \(P(X = 3)\)).
2. Poisson CD: Use this for "cumulative" questions (e.g., \(P(X \leq 3)\)).

Quick Review: Common Calculator Mistakes
- If the question asks for \(P(X > 3)\), your calculator can't do this directly. You must calculate \(1 - P(X \leq 3)\).
- If the question asks for \(P(X \geq 3)\), you must calculate \(1 - P(X \leq 2)\).

Did you know? The Poisson distribution was named after the French mathematician Siméon Denis Poisson, but he actually didn't use it for counting events—he used it to model legal trials!

3. Mean and Variance (Syllabus SB3)

One of the most beautiful (and helpful!) things about the Poisson distribution is how simple its properties are. For \(X \sim \text{Po}(\lambda)\):

Mean: \(E(X) = \lambda\)
Variance: \(\text{Var}(X) = \lambda\)

Yes, they are the same! This is a unique feature of Poisson. If you are given data where the mean is roughly equal to the variance, it's a strong sign that a Poisson model is appropriate.

Standard Deviation: Since variance is \(\lambda\), the standard deviation is \(\sqrt{\lambda}\).

Key Takeaway: If \(X \sim \text{Po}(5)\), then the mean is 5 and the variance is 5. Simple!

4. Adding Independent Poisson Distributions (Syllabus SB4)

Sometimes you have two different Poisson variables and you want to know the total. As long as they are independent, you can just add their means together.

If \(X \sim \text{Po}(\lambda_1)\) and \(Y \sim \text{Po}(\lambda_2)\), then:
\(X + Y \sim \text{Po}(\lambda_1 + \lambda_2)\)

Example: If you get an average of 2 emails per hour on your work account (\(X\)) and 3 emails per hour on your personal account (\(Y\)), the total number of emails you get per hour is \(X + Y \sim \text{Po}(2 + 3)\), which is \(X + Y \sim \text{Po}(5)\).

Don't forget: This only works if the events are independent. If receiving a work email somehow caused you to receive a personal email, you couldn't use this rule!

5. Hypothesis Testing with Poisson (Syllabus SB5)

We use hypothesis testing to see if the average rate (\(\lambda\)) has changed based on a single observation.

Step-by-Step Process:

1. State your Hypotheses:
\(H_0\): \(\lambda = \text{old rate}\)
\(H_1\): \(\lambda > \text{old rate}\) (or \(< \text{old rate}\) or \(\neq \text{old rate}\))

2. Identify the Distribution:
Assume \(H_0\) is true, so \(X \sim \text{Po}(\lambda)\).

3. Calculate the P-value:
Find the probability of getting the observed value or more extreme.
- If testing for an increase, calculate \(P(X \geq \text{observed})\).
- If testing for a decrease, calculate \(P(X \leq \text{observed})\).

4. Compare to Significance Level (\(\alpha\)):
- If P-value \(\leq \alpha\), Reject \(H_0\). There is significant evidence that the rate has changed.
- If P-value \(> \alpha\), Accept \(H_0\) (or "Fail to reject"). There is not enough evidence to say the rate has changed.

Example: A road averages 4 accidents a year. After a new speed camera is installed, there is only 1 accident in the following year. Is the camera working? You would test \(H_0: \lambda = 4\) against \(H_1: \lambda < 4\) by calculating \(P(X \leq 1)\) using your calculator.

Quick Review: Common Mistake
Students often calculate \(P(X = \text{observed})\) instead of the cumulative probability. In hypothesis testing, we always look for the probability of that result or more extreme.

Summary Takeaway Box

Terminology: \(X \sim \text{Po}(\lambda)\)
Conditions: Random, Independent, Singly, Constant Rate.
Property: \(\text{Mean} = \text{Variance} = \lambda\).
Formula: \(P(X=x) = \frac{e^{-\lambda} \lambda^x}{x!}\)
Summing: Add the lambdas for independent variables.
Testing: Use the Poisson distribution to find the probability of a "tail" to test the mean.