M1 Study Notes: The Poisson Distribution
Hey everyone! Welcome to your study notes for one of the most interesting topics in M1 Statistics: The Poisson Distribution. Don't worry if the name sounds a bit strange. It's a super useful tool for understanding events that happen randomly over time or space.
In this chapter, we'll learn how to predict the probability of things like:
- The number of emails you receive in an hour.
- The number of typos on a page in a book.
- The number of customers arriving at a checkout counter in 10 minutes.
By the end of these notes, you'll understand what the Poisson distribution is, when to use it, and how to calculate probabilities with it. Let's get started!
1. What Exactly is a Poisson Distribution?
Imagine you're trying to count how many times something happens, but the events are random and independent. The Binomial distribution works when you have a fixed number of trials (like flipping a coin 10 times). But what if there's no fixed number of "trials"? What if you're just waiting for events to occur in a fixed interval?
That's where the Poisson distribution comes in! It's a discrete probability distribution that helps us find the probability of a given number of events happening in a fixed interval of time, area, volume, or distance.
The Secret Ingredient: Lambda (λ)
The entire Poisson distribution is built around one single, super important parameter: lambda, written as λ.
λ (Lambda) = The average number of events that occur in a given interval.
Think of it as the "average rate". For example:
- If a call centre receives an average of 10 calls per hour, then λ = 10 (for a one-hour interval).
- If a biologist finds an average of 2 rare flowers per square metre in a field, then λ = 2 (for a 1 m² interval).
When Can We Use the Poisson Distribution?
You can't use it for everything! A situation can be modelled by a Poisson distribution only if it meets these conditions:
- Events occur at a constant average rate (the value of λ doesn't change).
- Events occur independently of each other (one event happening doesn't make another one more or less likely).
- Events occur randomly.
- Two events cannot occur at the exact same instant.
Key Takeaway: Section 1
The Poisson distribution is used for counting the number of random, independent events in a fixed interval (like time or space). Its one and only parameter is λ (lambda), which is the average rate of events in that interval.
2. The Poisson Probability Formula
Okay, now for the main event! If a random variable `X` follows a Poisson distribution with an average rate of λ, we write it as:
$$ X \sim Po(\lambda) $$The probability of observing exactly k events in that interval is given by this formula:
$$ P(X=k) = \frac{e^{-\lambda} \lambda^k}{k!} \quad \text{for } k = 0, 1, 2, ... $$Don't be scared by the formula! Let's break it down:
- P(X=k) is what we want to find: the probability that the number of events is exactly `k`.
- k is the specific number of events you are interested in (e.g., probability of exactly 3 calls).
- λ is the average number of events in the interval.
- e is Euler's number (your calculator knows it, it's approximately 2.718).
- k! is "k factorial", which means k × (k-1) × ... × 2 × 1 (e.g., 3! = 3 × 2 × 1 = 6). Remember that 0! = 1.
Step-by-Step Example
A customer service hotline receives an average of 5 calls per hour. What is the probability that they receive exactly 2 calls in a given hour?
Step 1: Identify the distribution and parameters.
- The events (calls) happen randomly in a fixed interval (one hour). This sounds like Poisson!
- The average rate is 5 calls per hour, so λ = 5.
- We want to find the probability of exactly 2 calls, so k = 2.
- So, we have $$X \sim Po(5)$$ and we need to find $$P(X=2)$$.
Step 2: Plug the values into the formula.
$$ P(X=2) = \frac{e^{-5} \cdot 5^2}{2!} $$Step 3: Calculate the result.
$$ P(X=2) = \frac{e^{-5} \cdot 25}{2 \times 1} $$ $$ P(X=2) = \frac{(0.006738) \cdot 25}{2} $$ $$ P(X=2) \approx 0.0842 $$So, there's about an 8.42% chance of receiving exactly 2 calls in that hour.
Common Mistakes to Avoid
- Mixing up λ and k: Remember, `λ` is the average for the interval, `k` is the specific number you're testing.
- Forgetting 0! = 1: The probability of zero events is $$P(X=0) = \frac{e^{-\lambda} \lambda^0}{0!} = e^{-\lambda}$$, since $$\lambda^0=1$$ and $$0!=1$$.
- Calculator Errors: Be careful when typing factorials and powers of `e`. Use the `e^x` button on your calculator.
3. Properties of the Poisson Distribution
This part is nice and simple, but very important. The syllabus requires you to know the mean and variance of a Poisson distribution. The proofs are not needed!
Mean and Variance
For a random variable $$X \sim Po(\lambda)$$:
- The Mean (or Expected Value) is: $$ E(X) = \lambda $$
- The Variance is: $$ Var(X) = \lambda $$
Memory Aid
Here's an easy way to remember this: "The Poisson is easy and plain, the mean and the variance are the same!"
This is a unique property! If a question tells you that a discrete distribution has a mean that is equal to its variance, it's a huge hint that you might be dealing with a Poisson distribution.
Did you know?
The standard deviation is the square root of the variance. So for a Poisson distribution, the standard deviation is $$\sqrt{\lambda}$$.
Quick Review Box
If $$X \sim Po(3)$$, then:
- The average number of events is 3.
- The mean, `E(X)`, is 3.
- The variance, `Var(X)`, is 3.
See? It's that easy!
4. Adjusting the Rate (λ)
This is a very common trick in exam questions, so pay close attention! The value of λ must match the interval in the question.
If the question gives you an average rate for one interval but asks for a probability in a *different* interval, you must adjust λ first.
Step-by-Step Example
A website receives an average of 180 hits per hour. What is the probability of getting exactly 4 hits in a 1-minute period?
Step 1: Find the original rate.
- The rate is 180 hits per 60 minutes.
Step 2: Adjust the rate (λ) to the new interval (1 minute).
- Average rate per minute = $$\frac{180 \text{ hits}}{60 \text{ minutes}} = 3$$ hits per minute.
- Our new, adjusted λ = 3.
Step 3: Use the new λ in the Poisson formula.
- We want the probability of 4 hits (k=4) in this 1-minute interval.
- So, we need to calculate $$P(X=4)$$ for $$X \sim Po(3)$$.
It's crucial to adjust λ before you start calculating!
5. Using Poisson to Approximate the Binomial Distribution
Sometimes, calculating probabilities with the Binomial distribution $$X \sim B(n, p)$$ can be really difficult, especially when `n` is very large. Imagine calculating $$C(500, 2)$$ by hand!
Thankfully, when certain conditions are met, we can use the much simpler Poisson distribution as an excellent approximation.
The Conditions for Approximation
You can approximate a Binomial distribution with a Poisson distribution if:
- n is large (usually `n > 50` is a good guide).
- p is small (usually `p < 0.1` is a good guide).
Basically, we are counting the number of "successes" for a rare event over a large number of trials.
How to do the Approximation
If $$X \sim B(n, p)$$ and the conditions above are met, you can approximate it with:
$$ Y \sim Po(\lambda) \quad \text{where} \quad \lambda = np $$Why `λ = np`? Because the mean of a Binomial distribution is `np`. We are setting the mean of the Poisson distribution equal to the mean of the Binomial distribution we are approximating. It makes perfect sense!
Step-by-Step Example
A factory produces a very large number of computer chips. The probability that a chip is faulty is 0.005. The chips are packed in boxes of 400. Find the approximate probability that a box contains exactly 3 faulty chips.
Step 1: Identify the original distribution.
- This is a Binomial situation. We have a fixed number of trials (`n=400`) and a constant probability of success (a chip being faulty, `p=0.005`).
- So, $$X \sim B(400, 0.005)$$.
Step 2: Check the conditions for Poisson approximation.
- `n = 400` (which is large).
- `p = 0.005` (which is small).
- The conditions are met! Calculating this with the Binomial formula would be horrible. Let's use Poisson.
Step 3: Calculate λ for the Poisson model.
$$ \lambda = np = 400 \times 0.005 = 2 $$Step 4: Use the Poisson formula with the new λ.
- We can now model this using $$Y \sim Po(2)$$.
- We want the probability of exactly 3 faulty chips, so k=3.
The approximate probability is about 18.0%.
Key Takeaway: Section 5
When you see a Binomial problem with a large `n` and small `p`, think "Poisson approximation!". Simply calculate λ = np and use the Poisson formula. It's much easier!