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) = (e^(-λ) * λ^k) / 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.

Knowing about the key properties will help us utilize Poisson distribution better. Here are a few important properties:

• Poisson distribution is discrete, which means we will be dealing with non-continuous, countable values. This is because, the distribution is used to count the number of times an event happens, and an event cannot be halved.

• The events are independent of each other. So the occurrence of one event does not affect the probability of another event happening.

• A Poisson distribution is very interesting because when the mean (λ) increases, the distribution of values becomes more symmetric and, thus, more bell-shaped. When the mean is large enough, we can apply the Central Limit Theorem to approximate the Poisson distribution by a normal distribution N(λ,λ). Here, the approximation improves as the mean increases. 

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.