S2.1: Poisson Distribution - Study Notes

Hello there! Welcome to the Poisson Distribution chapter. This topic is super useful because it helps us model events that happen randomly and independently over a fixed period of time or space—like the number of emails you receive in an hour, or the number of typos on a page.


Don't worry if the formula looks intimidating; the concepts are quite straightforward. Once you understand the core idea and the key assumptions, solving problems becomes much easier!

1. What is the Poisson Distribution?

The Poisson distribution is a discrete probability distribution used to model the number of occurrences of an event in a fixed interval of time or space, given that these events occur with a known constant rate and independently of the time since the last event.

Key Notation and Parameter
  • We denote the random variable \(X\) as having a Poisson distribution using the notation:
    \(X \sim \text{Po}(\lambda)\)
  • The symbol \(\lambda\) (lambda) is the only parameter needed for this distribution.
    • \(\lambda\) represents the mean rate of occurrence (or average number of events) in the specified interval.
    • \(\lambda\) must be a positive constant (\(\lambda > 0\)).

Example: If, on average, 4 cars pass a specific point on the road every minute, then the number of cars passing in a minute, \(X\), is modeled by \(X \sim \text{Po}(4)\), and \(\lambda = 4\).

2. Conditions for Applying the Poisson Distribution

You can only use the Poisson distribution if the events you are counting meet four strict conditions. Think of these as the "rules of the road" for using Poisson:

  1. Events occur singly: Events must happen one at a time. It's impossible for two events to occur at exactly the same instant.

    (E.g., Two telephone calls cannot arrive simultaneously, though in reality, they might seem close.)

  2. Events occur randomly: There is no discernible pattern to when the events happen.

    (E.g., Earthquakes don't happen on a schedule; they are random.)

  3. Events occur independently: The occurrence of one event does not affect the probability of another event happening.

    (E.g., The fact that one typo occurred on line 1 does not make a typo more or less likely on line 2.)

  4. The rate is constant (\(\lambda\) is uniform): The average rate of occurrence (\(\lambda\)) must remain the same throughout the entire interval of time or space being considered.

    (E.g., If you are counting accidents per month, the average rate must be consistent across all months.)

Quick Review: When NOT to use Poisson

If the rate changes significantly (e.g., counting website hits at 3 AM vs. 3 PM), or if the events influence each other (e.g., counting a disease spread through a population), the Poisson distribution is not appropriate.

3. The Poisson Probability Formula

To calculate the probability that exactly \(x\) events occur, we use the formula:

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

Where:

  • \(P(X=x)\) is the probability of exactly \(x\) events happening.
  • \(\lambda\) is the mean rate of occurrence (the parameter).
  • \(x\) is the specific number of events we are interested in (\(x = 0, 1, 2, 3, \ldots\)).
  • \(e\) is Euler's constant (\(e \approx 2.71828\)). You will calculate \(e^{-\lambda}\) using your calculator.
  • \(x!\) is \(x\) factorial (\(x! = x \times (x-1) \times \ldots \times 1\)). Remember \(0! = 1\).
Step-by-Step Calculation Guide

Suppose \(X \sim \text{Po}(3.5)\). Find \(P(X=2)\).

  1. Identify \(\lambda\) and \(x\): \(\lambda = 3.5\), \(x = 2\).
  2. Substitute into the formula:

    $$P(X=2) = \frac{e^{-3.5} (3.5)^2}{2!}$$

  3. Calculate the parts:
    • \(3.5^2 = 12.25\)
    • \(2! = 2\)
    • \(e^{-3.5} \approx 0.030197\)
  4. Compute the final result:

    $$P(X=2) = \frac{0.030197 \times 12.25}{2} \approx 0.185$$

Key Takeaway: The formula allows us to calculate exact probabilities for specific counts (\(x\)).

4. Mean, Variance, and Standard Deviation

One of the most elegant and useful features of the Poisson distribution is the relationship between its mean and variance.

If \(X \sim \text{Po}(\lambda)\), then:

  • Mean: \(E(X) = \lambda\)
  • Variance: \(\text{Var}(X) = \lambda\)
  • Standard Deviation: \(\sigma = \sqrt{\lambda}\)

This means that if you know the average rate (\(\lambda\)), you automatically know the spread (variance) of the data too!

Did you know? This equality of mean and variance is a key diagnostic test in statistics. If a real-world dataset has a mean significantly different from its variance, it probably can't be modeled accurately by a Poisson distribution.

Key Takeaway: For Poisson, Mean = Variance = \(\lambda\).

5. Dealing with Changes in the Interval

The parameter \(\lambda\) is tied to a specific interval. If the interval changes, you must adjust \(\lambda\) proportionally.

Example: The number of potholes on a 1 km stretch of road is \(X \sim \text{Po}(2)\).

What is the distribution for a 3 km stretch?

  • The rate for 1 km is 2.
  • The rate for 3 km must be \(3 \times 2 = 6\).
  • Let \(Y\) be the number of potholes in 3 km. Then \(Y \sim \text{Po}(6)\).

Example: The number of calls per hour is \(X \sim \text{Po}(12)\). What is the distribution for the number of calls in 15 minutes?

  • 1 hour = 60 minutes. 15 minutes is \(\frac{15}{60} = 0.25\) of an hour.
  • The new rate is \(\lambda_{new} = 12 \times 0.25 = 3\).
  • The number of calls in 15 minutes, \(Y\), is \(Y \sim \text{Po}(3)\).

Common Mistake to Avoid: Always make sure the \(\lambda\) you use matches the time or space unit required by the question!

6. Poisson as an Approximation to the Binomial Distribution

The syllabus requires you to understand how the Poisson distribution acts as a limit (or approximation) of the Binomial distribution. This is important for choosing the correct model in exam questions.

When is the Approximation Valid?

If \(X\) follows a Binomial distribution, \(X \sim B(n, p)\), the Poisson distribution provides a good approximation if:

  1. The number of trials, \(n\), is large (\(n > 50\) is a good rule of thumb).
  2. The probability of success, \(p\), is small (\(p < 0.1\) is a good rule of thumb).
The Link Between Parameters

When the approximation is valid, the Poisson parameter \(\lambda\) is calculated directly from the Binomial parameters:

$$\lambda = np$$

(Remember that \(np\) is the mean of the Binomial distribution, and since \(p\) is small, the variance \(np(1-p)\) is very close to \(np\). This confirms why Mean \(\approx\) Variance, allowing the Poisson approximation.)

Example: A production line makes 2000 items daily. The probability that any single item is defective is \(p=0.001\). If \(X\) is the number of defective items, \(X \sim B(2000, 0.001)\).

  • Since \(n=2000\) (large) and \(p=0.001\) (small), we use the Poisson approximation.
  • Calculate \(\lambda = np = 2000 \times 0.001 = 2\).
  • We approximate \(X \sim \text{Po}(2)\).

Key Takeaway: Use Poisson to approximate Binomial when dealing with rare events across a large number of trials.

7. Sum of Independent Poisson Random Variables

This rule simplifies problems where events from multiple independent sources are combined.

If you have two independent random variables, \(X\) and \(Y\), that are both Poisson distributed, then their sum, \(X+Y\), is also Poisson distributed, and their parameters simply add up.

  • If \(X \sim \text{Po}(\lambda_X)\)
  • And \(Y \sim \text{Po}(\lambda_Y)\)
  • And \(X\) and \(Y\) are independent

Then:

$$X + Y \sim \text{Po}(\lambda_X + \lambda_Y)$$

Example: During the lunch hour, a restaurant receives an average of 5 delivery orders (\(D \sim \text{Po}(5)\)) and an average of 3 walk-in orders (\(W \sim \text{Po}(3)\)). If these are independent, the total number of orders, \(T = D + W\), is:

$$T \sim \text{Po}(5 + 3)$$

$$T \sim \text{Po}(8)$$

This rule extends to the sum of any number of independent Poisson variables.

Key Takeaway: The average rates of independent Poisson processes can be added together.

8. Using Cumulative Poisson Tables

While the formula gives \(P(X=x)\), many questions ask for cumulative probabilities (e.g., \(P(X \le x)\) or \(P(X > x)\)). You will typically use statistical tables for these, which usually give cumulative probabilities, \(P(X \le x)\).

Always remember the laws of probability when manipulating inequalities:

  1. Probability of 'Less Than or Equal To' (Direct Table Read):

    $$P(X \le 5) = \text{Read value directly from table for } x=5$$

  2. Probability of 'Less Than' (Must Adjust):

    Since \(X\) is discrete (it can only be integers), \(P(X < 5)\) is the same as \(P(X \le 4)\).

    $$P(X < 5) = P(X \le 4)$$

  3. Probability of 'Greater Than or Equal To' (The Complement Rule):

    The table only shows "less than or equal to," so you must use the complement rule: \(P(A) = 1 - P(A')\).

    $$P(X \ge 3) = 1 - P(X < 3)$$

    Since \(X\) is discrete, \(P(X < 3)\) is \(P(X \le 2)\).

    $$P(X \ge 3) = 1 - P(X \le 2)$$

  4. Probability of 'Greater Than' (The Complement Rule and Adjustment):

    $$P(X > 4) = 1 - P(X \le 4)$$

  5. Probability of a Range:

    $$P(2 \le X \le 5) = P(X \le 5) - P(X \le 1)$$ (This calculation removes the probabilities for \(x=0\) and \(x=1\), leaving \(x=2, 3, 4, 5\).)

Tip for Struggling Students

When unsure how to adjust the inequalities, draw a quick number line! If you want \(X \ge 3\), you want 3, 4, 5, ... The complement (what you don't want) is 0, 1, 2. So, you subtract \(P(X \le 2)\).

Key Takeaway: Be meticulous with inequality signs and remember that tables usually provide \(P(X \le x)\) values.