Poisson Distribution Formula, Definition, Table, Examples

Poisson Distribution Formula is a fundamental concept in statistics and probability theory. It is used to model the number of events that are likely to occur within a fixed interval of time or space when these events occur at a known, constant average rate. Here, I'll provide an overview of the Poisson distribution, its formula, mean, variance, and some examples.

Poisson Distribution Definition

The Poisson distribution is a discrete probability function, meaning it deals with specific and often countable values. It is used to model the number of events that are likely to occur within a given period of time or space. In other words, it calculates the probability of observing a certain number of events in a Poisson experiment.

In a Poisson experiment, events are categorized as either successes or failures. The Poisson random variable "x" represents the number of successes in the experiment. This distribution is especially useful when dealing with events that do not have a fixed number of possible outcomes.

The Poisson distribution is a limiting process of the binomial distribution, which means it arises when specific conditions are met:

The number of trials "n" approaches infinity.

The probability of success "p" approaches zero.

The product of "n" and "p," denoted as "np," remains a finite constant (np = 1).

In simpler terms, as the number of trials becomes very large and the probability of success for each individual trial becomes very small, the Poisson distribution can be used to approximate the probability of a certain number of successes occurring within a specified interval.

This distribution is commonly applied in various fields, including statistics, biology, physics, and engineering, to model rare events or occurrences within a fixed timeframe or space.

Poisson Distribution Formula

The formula for the Poisson distribution function is given by:

f(x) =(e ^{– λ} λ ^x )/x!

Where,

e is the base of the logarithm

x is a Poisson random variable

λ is an average rate of value

Poisson Distribution Formula Table

A Poisson distribution table is a reference table that provides the probabilities associated with the Poisson distribution for different values of the random variable "x" and a given average rate of occurrence, often denoted as "λ" (lambda). These tables are useful for quickly looking up probabilities without the need for complex calculations.

However, it's important to note that with modern technology and statistical software, such as spreadsheet programs or specialized statistical packages, you can easily calculate Poisson probabilities without the need for printed tables.

Poisson Distribution Formula Mean and Variance

In a Poisson experiment, we focus on the average number of successful events occurring within a specified range, and we denote this average as "λ" (lambda). Within the context of the Poisson distribution, this "λ" plays a crucial role as the key parameter.

The probability of observing a specific number of events, denoted as "x," in a Poisson distribution with an average rate of "λ," can be calculated using the Poisson probability formula:

P(x, λ ) =(e ^{– λ} λ ^x )/x!

In Poisson distribution, the mean is represented as E(X) = λ.

For a Poisson Distribution, the mean and the variance are equal. It means that E(X) = V(X)

Where,

V(X) is the variance.

Poisson Distribution Expected Value

A random variable that follows a Poisson distribution with the parameter "λ" is associated with an expected value, which is often denoted as E(X) or the mean of the distribution. This expected value (E(X)) of the Poisson distribution is equivalent to the parameter "λ."

In other words, for a Poisson-distributed random variable, the average number of events, often represented by "λ," is also the expected value. Therefore, you can express it as follows:

E(x) = μ = d(e ^λ(t-1) )/dt, at t=1.

E(x) = λ

Therefore, the expected value (mean) and the variance of the Poisson distribution is equal to λ.

Poisson Distribution Formula Examples

Example:

Imagine you work at a helpdesk, and on average, you receive 5 customer service calls every 15 minutes. You want to calculate the probability of receiving exactly 3 calls within the next 15 minutes.

Using the Poisson Distribution Formula:

In this case, "λ," the average rate of calls per 15 minutes, is 5, and you want to find "P (x = 3)," the probability of receiving 3 calls.

The Poisson distribution formula is as follows:

P (X =x) = (e ^{– λ} λ ^x )/x!

Here, "x" represents the number of events, "λ" is the average rate, and "e" is the mathematical constant (approximately 2.71828).

Now, substitute the values into the formula:

P(3, 5) = (e ^-5 * 5 ³ ) / 3!

Now, let's calculate each part of the formula:

e ^-5 is approximately 0.006738.

5 ³ equals 125.

3! is equal to 6.

Now, plug these values back into the formula:

P (3, 5) = 0.006738 * (125 / 6)

Calculate the final result:

P (3, 5) is approximately 0.08422.

So, the probability of receiving exactly 3 customer service calls within the next 15 minutes is approximately 0.08422, or 8.42%. This is how you can use the Poisson distribution formula to calculate probabilities for different numbers of events when you know the average rate of occurrence.

Poisson Distribution Formula Application

The Poisson distribution formula finds numerous applications in various fields. It's used to model random events or occurrences over time or space when these events happen at a known average rate. Here are some practical applications of the Poisson distribution formula:

• Telecommunications and Network Traffic: In the field of telecommunications, the Poisson distribution is applied to model the number of phone calls, network requests, or data packets arriving at a switch or router within a specific time interval. This is important for network capacity planning and resource allocation.
• Call Centers: Call centers to use the Poisson distribution to predict call arrivals. They can staff their call centers appropriately based on the expected number of incoming calls during different time periods.
• Quality Control: Manufacturing and production processes often employ the Poisson distribution to analyze the occurrence of defects or errors in products. It helps in setting quality control standards and identifying when a process is out of control.
• Web Traffic and Server Load: For websites and online services, the Poisson distribution is used to model website hits, server requests, and traffic patterns. It helps ensure that servers are adequately equipped to handle web traffic.
• Epidemiology: Epidemiologists use the Poisson distribution to study the occurrence of diseases or events in a population over time. It can be used to model disease outbreaks, natural disasters, or other rare events.
• Insurance and Risk Assessment: Insurance companies use the Poisson distribution to estimate the number of insurance claims they are likely to receive within a given time frame. This helps in setting premiums and reserves.
• Finance and Stock Market: In finance, the Poisson distribution is used to model the arrival of stock market trades or price changes. It helps in analyzing market volatility and risk.
• Environmental Science: Environmental scientists use the Poisson distribution to study events like earthquakes, floods, or wildlife population changes. It helps in predicting and managing natural disasters.