Poisson Distribution

A Poisson distribution is used to predict or estimate the number of times an event might occur within a given period of time. This type of distribution method is specifically used when the variables are discrete count variables. We usually use Poisson distribution when dealing with variables, such as economic and financial data.

The formula used to calculate the probability of an event occurring discreetly over a given period of time is:

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

Where:

e is approximately 2.718 (Euler’s number),

λ is the average number of events in the interval

k! is the factorial of k

k is the actual number of occurrences.

In a Poisson distribution, both the mean and variance are equal to λ. Here, λ is greater than 0.

Example:

Imagine you are waiting for text messages from your friends. Based on your history, you know that on average, you receive three text messages per hour.
Even though the average is 3, you know that in any specific hour, the actual number could vary. You might get 0, 3, or 5 texts. The text messages arrive independently of one another (receiving one doesn't change the chance of receiving another).

This is a classic Poisson Distribution problem because:

• We know the average rate (\lambda = 3).
   
• We are looking at a fixed time interval (1 hour).
   
• Events are independent.

How the Probability Looks:

Using the Poisson distribution, we can calculate the likelihood of receiving a specific number of texts in the next hour.

• 0 Texts: There is about a 5% chance you receive absolute silence.
   
• 1 Text: There is about a 15% chance.
   
• 2 Texts (The Average): There is about a 22% chance.
   
• 3 Texts: There is about a 22% chance.
   
• 5 Texts: There is only about a 10% chance.

Calculation:

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

Where:

• \(P(X=k)\) is the probability of getting exactly k texts.
   
• \(\lambda\) (Lambda) is the average rate (3 texts).
   
• e is Euler's number (\approx 2.718).
   
• k! is the factorial of k.

Example Calculation for three texts:

\(P(X=3) = \frac{3^3 \cdot e^{-3}}{3!} = \frac{27 \cdot 0.0498}{6} \approx 0.22\)

Use the Poisson distribution calculator to check.
 

For a scenario to be modeled as a Poisson distribution, it must satisfy these specific properties (often called the “Assumptions of the Poisson Process”):
 

1. Independence: The occurrence of one event does not affect the probability of another event occurring.
   
   Example: Receiving a text message now does not change the probability of receiving another one 5 minutes from now.
    
2. Individuality (Non-Simultaneity): Events occur one at a time. The probability of two or more events occurring at the exact same instant is essentially zero.
   
   Example: Two cars cannot crash into the exact same spot on a bumper at the exact same microsecond; they count as distinct events in a sequence.
    
3. Homogeneity (Constant Rate):The average rate at which events occur (\(\lambda\)) must remain constant throughout the entire time interval or space being observed.
   
   Example: You cannot use a single Poisson distribution to model traffic for a whole day, because the “average rate” changes drastically between Rush Hour and 2:00 AM.
    
4. Proportionality: The probability of an event occurring in a small interval is proportional to the length of that interval.
   
   Example: You are twice as likely to see a shooting star in a 2-hour window as you are in a 1-hour window.
    
5. The “Rare Event” Property (Limit of Binomial): The Poisson distribution is a limiting case of the Binomial Distribution. It applies specifically when:
    
   • The number of trials (n) is very large (indefinitely large).
      
   • The probability of success (p) is very small (rare).
      
   • The product \(np = \lambda\) remains finite.
      
6. The Unique Mathematical Property: Equidispersion
   
   There is one specific mathematical property that is so unique to Poisson that it serves as a test to see if your data fits this model:
   
   \(Mean = Variance\), In a Poisson distribution, the spread of the data (Variance) is exactly equal to the average (Mean).
   
   \(E(X) = Var(X) = \lambda\)
   
   If your data has a variance that is much higher than the mean (overdispersion), a Poisson distribution is not the right tool to use.

Here are the key characteristics of the Poisson Distribution:
 

1. Single Parameter Dependence
   
   Unlike the Normal distribution (which needs a Mean and a Standard Deviation) or the Binomial (which needs n and p), the Poisson distribution is defined entirely by a single parameter: \(\lambda\) (Lambda).
    
   • If you know the average rate (\(\lambda\)), you know everything there is to know about the distribution.
      
2. Equality of Mean and Variance (Equidispersion)
   
   This is the most distinct characteristic of Poisson. The spread of the data is tied directly to the average.
    
   • Mean (Expectation): \(E(X) = \lambda\)
      
   • Variance: \(Var(X) = \lambda\)
      
   • What this means practically: As the average number of events increases, the variability (spread) of those events also increases linearly.
      
3. Standard Deviation
   
   Since the Standard Deviation is the square root of the variance, for a Poisson distribution:
    
   • Standard Deviation: \(\sigma = \sqrt{\lambda}\)
      
4. Skewness (Shape of the Graph)
   
   The shape of a Poisson distribution changes depending on the value of \lambda.
    
   • Small \(\lambda\) (e.g., 1 or 3): The graph is positively skewed (skewed to the right). The “hump” is on the left, and there is a long tail stretching toward higher numbers.
      
   • Large \(\lambda\) (e.g., 20+): As \lambda increases, the graph becomes more symmetrical. Eventually, it looks almost identical to a Normal Distribution (Bell Curve).
      
5. The Additive Property
   
   If you have two independent Poisson variables, their sum is also a Poisson variable.
    
   • If Source A produces events at rate \(\lambda_1\) and Source B produces events at rate \(\lambda_2\), the total count follows a Poisson distribution with rate \(\lambda_{total} = \lambda_1 + \lambda_2.\)
      
   • Example: If you get 3 emails/hour and 2 texts/hour, your total “notifications” follow a Poisson distribution of \(\lambda = 5.\)
      
6. Probability Mass Function (PMF)
   
   The PMF is the formula used to find the probability of exactly k successes.
   
   \(P(X=k) = \frac{e^{-\lambda} \cdot \lambda^k}{k!}\)
    
   • Characteristic Limit: The probabilities sum to 1: \(\sum_{k=0}^{\infty}P(X=k) = 1\)

In a Poisson distribution, the average and the spread are identical. They are both defined by the single parameter \lambda (Lambda).
 

• The mean of the Poisson distribution represents the expected average number of events in the given time interval. It is exactly equal to the rate parameter.
  
  \(E(X) = \lambda\)
   
• The variance of the Poisson distribution represents the spread or variability of the data. Uniquely, it is equal to the mean.
  
  \(Var(X) = \lambda\)
   
• Since standard deviation is always the square root of the variance, the formula is:
  
  \(\sigma = \sqrt{\lambda}\)
   

Example:

If your average rate of texts is \(\lambda = 4:\)
 

• Mean: 4
   
• Variance: 4
   
• Standard Deviation: \(\sqrt{4} = 2\)

A Poisson Distribution Table is a reference chart used to find probabilities without doing complex calculations manually. Think of it as a “Cheat Sheet” for the Poisson formula.

Instead of typing \(P(X=k) = \frac{\lambda^k e^{-\lambda}}{k!}\) into a calculator every time, you simply look up the value where a specific Row and Column meet.

Types of Poisson Distribution Tables

There are two distinct types of tables. It is critical to know which one you are using, as they give very different answers.
 

Poisson PMF Table (Probability Mass Function):

This table gives the probability of an event occurring an exact number of times.
 

• Notation: \(P(X = k)\)
   
• Use this when: The question asks for "exactly 3 emails" or "exactly 0 errors".
   
• Example: If \(\lambda = 2,\) the table value for k=3 tells you the chance of getting exactly 3 events.
   

Poisson CDF Table (Cumulative Distribution Function):

This table gives the probability of an event occurring up to and including a certain number of times. It is the running total of the probabilities.
 

• Notation: \(P(X \le k)\)
   
• Use this when: The question asks for "at most 3 emails," "less than 4 errors," or "3 or fewer".
   
• Example: If \(\lambda = 2\), the table value for k=3 tells you the chance of getting 0, 1, 2, or 3 events combined.

A Poisson distribution graph visualizes the probability of a certain number of events occurring within a fixed interval (like time or space).
 

Here are the key features of this graph:

Discrete Data (Bars or Dots): Unlike a Bell Curve (Normal distribution) which is a smooth line, a Poisson graph is "discrete." This means it uses bars or dots because you count whole events (e.g., 0 calls, 1 call, 2 calls); you cannot have 2.5 calls.

The Axes:

• X-axis (k): Represents the number of events (0, 1, 2, 3...).
   
• Y-axis (P(k)): Represents the probability of that specific number of events happening.
   

The Shape (Skewness):
 

• Right Skewed: When the average number of events (represented by the symbol \(\lambda\) or Lambda) is low (e.g., \(\lambda\) = 1 or 3), the graph leans heavily to the left, with a long tail tapering off to the right.
   
• Symmetric: As \(\lambda\) increases (e.g., \(\lambda\) = 10 or more), the graph starts to look more symmetric and bell-shaped, resembling a Normal distribution.

Poisson distribution is important because of its wide application. Here are few other reasons why it is such a prominent tool in mathematics:

• Poisson distribution is ideal for modeling situations where events occur randomly and independently. One such example is the number of emails you receive per hour.

• It's very useful in predicting the probability of rare events in a particular time period. Such as the number of earthquakes happening in a year.

• Insurance companies use Poisson Distribution to assess risks or accidents happening in a specific time frame. 

It can be quite confusing when learning about Poisson distribution. Here are a few tips and tricks students can use to master Poisson distribution.

• Remember the formula: Learning the formula is very important when learning Poisson distribution. P(x) =(X = k) = e-λ λkk! 

• The value of mean and variance will be the same. This is a very significant and unique property to remember. So if the λ = 7, then the mean and variance = 7.

• Know when to use the Poisson Distribution. We use it when we want to find out how many times an event occurs. So if you want to figure out how many calls you receive per hour, we use Poisson Distribution.
   
• Events Must Be Independent. Always remember that Poisson distribution applies only when events occur independently. For example, the number of cars passing a checkpoint in an hour does not depend on the previous hour.
   
• Check for Rare Events. Poisson distribution is best used for modeling rare events. For example, the number of printing errors per page in a book or the number of accidents at a junction in a day.

Poisson distribution is widely used in the real world. Here are a few examples where we use Poisson Distribution.

• Call Centers: To figure out the number of customer calls received at a call center per minute, we use a Poisson distribution. This helps managers allocate the resources efficiently.

• Natural Disasters: Natural disasters like earthquakes and their frequency of occurrence in a particular place can be analyzed using Poisson distribution.

• Detecting Email Spams: By modeling the frequency of incoming emails per day, a Poisson distribution can be used to enhance filtering algorithms.This will allow users to analyze and predict email traffic patterns.
   
• Traffic Flow: Poisson distribution is used to estimate the number of cars passing through a traffic signal or toll booth in a given period, helping in traffic management and road planning.
   
• Hospital Emergencies: Hospitals use Poisson distribution to predict the number of patients arriving at the emergency room within an hour, which helps in resource allocation.