Poisson Distribution

The Poisson distribution is a fundamental probability distribution used in statistics to model the number of events occurring within a fixed time or space, given a constant average rate of occurrence and independence between events. It’s widely used in fields such as queueing theory, telecommunications, biology, and business forecasting.

 

Table of Contents

• What is Poisson Distribution?
• Poisson Distribution Formula
• Poisson Distribution Table
• Poisson Distribution Characteristics
• Poisson Distribution Graph
• Poisson Distribution Mean and Variance
• Poisson Distribution Mean
• Poisson Distribution Variance
• Poisson Distribution Examples
• Conclusion
• FAQs on Poisson Distribution

 

What is Poisson Distribution?  

The Poisson distribution is a probability distribution that describes how many events happen in a set amount of time or space. These events must happen at a known constant average rate and must be independent of the time since the last event. This distribution is widely used in areas like statistics, physics, engineering, and business to model the frequency of rare events.  

If you're asking what is Poisson distribution is, it works well for counting occurrences like the number of emails received per hour, the number of decay events in a radioactive substance, or the number of calls at a call center during a specific timeframe.  

 

Poisson Distribution Formula  

The Poisson distribution formula calculates the chance of a certain number of events occurring in a fixed interval.  

P(X = x) = (e^-λ * λ^x) / x!  

Where:  

• P(X = x) is the probability of observing x events  

• λ is the average number of occurrences (rate)  

• e ≈ 2.718 is Euler’s number  

• x! is the factorial of x  

 

You apply the Poisson distribution formula when:  

• The events are independent.  

• The average rate (λ) is constant.  

• Two events cannot occur at the same time.  

You will frequently see poisson distribution formula used in practice problems and real-life situations, especially for modeling rare but countable events.  

 

Poisson Distribution Table  

A Poisson distribution table serves as a quick reference to find probabilities for different values of x given values of λ. It saves you from having to compute the formula each time.  

 

 

The Poisson distribution table helps solve problems quickly, especially during exams.  

 

Poisson Distribution Characteristics  

Some important features of the Poisson distribution are:  

• Discrete Distribution: It deals with whole numbers (0, 1, 2, and so on).  

• Skewed Distribution: It is often right-skewed, particularly when λ values are smaller.  

• Mean = Variance: A unique feature of this distribution is that both the mean and variance equal λ.  

• Non-Negative Values: It cannot take negative values.  

• Rare Events: It is suitable for modeling infrequent events.  

Knowing these characteristics helps in deciding when to use the Poisson distribution in real-life situations.  

 

Poisson Distribution Graph  

The Poisson distribution graph is a bar chart since it is a discrete distribution. The graph's shape changes with the value of λ:  

• For small λ (e.g., λ = 1), the graph skews heavily to the right.  

• As λ increases, the graph becomes more symmetric and bell-shaped, resembling a normal distribution.  

The height of each bar shows the probability of getting that specific value of x.  

 

Poisson Distribution Mean and Variance  

An important property of the Poisson distribution is that both its mean and variance equal λ. This feature makes it easy to work with in practical problems.  

Mean of Poisson Distribution = λ  

Variance of Poisson Distribution = λ  

So, when asked about the mean and variance of the Poisson distribution, just remember they are the same and based on the rate of occurrences (λ).  

 

Mean of Poisson Distribution  

The mean of a Poisson distribution indicates the expected number of occurrences (or events) within a fixed time, space, area, or volume. The mean is represented by the Greek letter λ.  

Formula:  

Mean (μ) = λ  

This means that, on average, λ events are expected to happen within the given interval.  

Example:  

A hospital receives an average of 5 emergency calls per hour. Here, the mean of the Poisson distribution is:  

λ = 5  

This means that over a long period, the hospital will likely get around 5 emergency calls every hour, though in any single hour, that number may vary slightly. The mean helps illustrate the central tendency or average behavior of the distribution. It represents the long-term expected number of events.  

 

Variance of Poisson Distribution  

The variance shows how much the data points (number of events) differ from the mean. In the Poisson distribution, the variance is also equal to λ, just like the mean.  

Formula:  

Variance (σ²) = λ  

This indicates that in the Poisson distribution, the variability in the number of events matches the average number of events.  

Interpretation:  

When λ is small, the number of events usually clusters around 0 or 1 with a tight spread. The data is more concentrated.  

When λ is large, the number of events can vary more, causing a larger spread. The distribution becomes flatter and closer to symmetric.  

Example:  

If a toll booth records an average of 10 cars per minute, then:  

Mean = λ = 10  

Variance = λ = 10  

This means that over time, the average number of cars per minute is 10, and the variability around this mean is also 10.  

 

Poisson Distribution Examples  

Example 1: Emails Received per Hour  

A company receives an average of 3 emails per hour. What is the chance they receive exactly 4 emails in one hour?  

Given:  

λ = 3 (mean number of emails per hour)  

x = 4 (number of emails) 

Use the Poisson distribution formula:  

P(X = x) = (e^–λ × λ^x) / x!  

Solution:  

P(X = 4) = (e^–3 × 3^4) / 4!  

= (0.0498 × 81) / 24  

= 4.034 / 24  

≈ 0.1681  

Answer: The probability of receiving exactly 4 emails in one hour is 0.1681.  

 

Example 2: Calls at a Call Center  

A call center receives an average of 6 calls every 10 minutes. What is the probability that exactly 8 calls come in 10 minutes?  

Given:  

λ = 6  

x = 8  

Solution:  

P(X = 8) = (e^–6 × 6^8) / 8!  

= (0.00248 × 1679616) / 40320  

= 4163.8 / 40320  

≈ 0.1032  

Answer: The probability of receiving exactly 8 calls is 0.1032.  

 

Example 3: Typographical Errors  

A proofreader finds an average of 2 typos per page. What is the probability that a given page has no typos?  

Given:  

λ = 2  

x = 0  

Solution:  

P(X = 0) = (e^–2 × 2^0) / 0!  

= (0.1353 × 1) / 1  

= 0.1353  

Answer: The probability that there are no typos on the page is 0.1353.  

 

Example 4: Buses Arriving at a Stop  

On average, 5 buses arrive at a stop per hour. What is the probability that exactly 2 buses arrive in one hour?  

Given:  

λ = 5  

x = 2  

Solution:  

P(X = 2) = (e^–5 × 5^2) / 2!  

= (0.00674 × 25) / 2  

= 0.1685 / 2  

= 0.0842  

Answer: The probability of exactly 2 buses arriving is 0.0842.  

 

Example 5: Customers at a Store  

A store has an average of 4 customers arriving every hour. What is the chance that no customer shows up in an hour?  

Given:  

λ = 4  

x = 0  

Solution:  

P(X = 0) = (e^–4 × 4^0) / 0!  

= (0.0183 × 1) / 1  

= 0.0183  

Answer: The probability of no customer arriving is 0.0183.  

 

Conclusion  

The Poisson distribution is a valuable tool for modeling discrete events over a fixed interval. With the unique property the mean and variance of poisson distribution 

are equal, it offers a straightforward way to calculate probabilities for rare real-world events. By mastering the Poisson distribution formula, using the Poisson distribution table, and examining poisson distribution examples, you gain a solid understanding of probability modeling. Whether calculating the number of accidents per month or studying the failure rate of machines, the Poisson distribution is an essential idea in statistics.  

 

Related Links

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Frequently Asked Questions on Poisson Distribution

1. What is meant by Poisson distribution?

Ans: Poisson distribution is a probability distribution that measures the likelihood of a given number of events occurring within a fixed interval of time or space, provided the events happen independently and at a constant average rate.

 

2. When should you use Poisson distribution?

Ans: You should use Poisson distribution when:

• Events are independent of each other

• The average rate (λ) of occurrence is constant

• Two events cannot occur at exactly the same time

• The number of events is a count over a given time or space

 

3. Where is Poisson distribution used in real life?

Ans: Poisson distribution is commonly used in real-life scenarios such as:

• Number of calls received at a call center per hour

• Number of emails received in a day

• Number of customer arrivals at a bank

• Number of decay events per second from a radioactive source

• Number of accidents at a traffic signal in a month

 

4. What is the main difference between binomial and Poisson distribution?

Ans:

• Binomial distribution deals with a fixed number of trials (n) and two possible outcomes (success or failure), with constant probability (p).

• Poisson distribution does not have a fixed number of trials but focuses on the number of times an event occurs in a fixed interval, based on a constant rate (λ).

 

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