Understanding Poisson Distribution

What is Poisson distribution?

The Poisson Distribution is a type of probability distribution that is used to model the number of events occurring in a fixed interval of time or space.

What describes a Poisson distribution?

Discrete events: Poisson distribution applies to situations where evets are countable and they occur independently of each other, meaning that occurrence of second event is not dependent on the first event.

Fixed Interval: The distribution focusses on the number of events occurring within a specific time frame or a designated area. for example, how many customers arrived in a store per hour, or how many defects were found in a car.

What are key characteristics of Poisson distribution?

Single parameter (λ): Poisson distribution is defined by a single parameter lambda (λ), which represents the average (or mean) at which events occur within a fixed interval.

Shape: When visualized Poisson distribution takes the form of a skewed bell curve, with the most likely number of events occurring around the mean (λ) and probability of observing very high or very low number of events decreasing as we move further away from the mean.

What are the use and application of Poisson distribution?

The Poisson distribution is applied in various fields to model and analyze events like:

• Customer arrival: Predicting the customer arrivals at a store or place within a specific time frame.
• Accidents: Estimates of likelihood of a certain number of accidents happening on a road segment in a given period.
• Manufacturing defects: Analyzing the probability of finding a specific number of defects in products being manufactured.
• Radioactive decay: Modeling the number of radioactive decays occurring within a specific time.

What are differences between Poisson, Binomial and Normal Distribution?

Here is a table summarizing the key differences between Poisson, Normal, and Binomial distribution:

What are the parameters of Poisson distribution?

Mean (λ): This represents the average number of occurring within a fixed interval of time or space.

Example: Imagine a Poisson distribution modeling the number of emails received in your work inbox every hour. Let’s say you typically receive an average of 96 emails per day. We can then calculate the average number of emails received per hour by dividing the daily average by the number of hours in a day:

Average emails per hour (λ) = Total daily emails / Number of hours in a day

λ = 96 emails / 24 hours

λ = 4 emails per hour

Therefore, the mean of the Poisson distribution in this scenario is 4. This signifies that, on average, you receive 4 emails per hour.

Standard Deviation (σ): This represents the data spread around the mean, Interestingly, in Poisson distribution, the variance (σ²) is equal to the mean (λ). Therefore, the standard deviation (σ) is simply the square root of the mean.

σ = √λ

Example: Imagine a Poisson distribution modeling the number of customer arrivals at a store in a given hour (fixed interval). If the average arrival rate (mean) is 10 customers per hour which is represented as λ = 10, then the variance (σ²) would also be 10, and the standard deviation (σ) would be:

σ = √λ = √10 ≈ 3.16 (rounded to two decimal places)

This explains that in this scenario, the number of customer arrivals is likely to be within 3.16 customers (one standard deviation) of the average arrival rate of 10 customers per hour.

How to calculate Poisson distribution?

The Poisson probability can be calculated using the below formula, it allows you to calculate the probability (P(X = x)) of getting a specific number of events (x) within a given interval, considering the average number of events (λ) expected during that interval.

The formula is:

P(X = x) = (e^-λ * λ^x) / x!

where:

• e = Euler’s number (approximately 2.71828)
• λ = The average number of events expected in the interval
• x = The specific number of events you’re interested in (0, 1, 2, 3, etc.)
• x! = The factorial of x (x multiplied by all the positive integers less than x)

Here’s how to use the formula:

1. Identify the average number of events (λ): This is the crucial input for the formula. It represents the expected average of events you’re interested in.
2. Define the specific number of events (x): This is the number of events you want to calculate the probability for (e.g., how likely is it to get 2 occurrences?).
3. Plug the values into the formula: Substitute the values of λ and x into the formula.
4. Calculate the result: Using a calculator, compute the expression.

• Remember to use the appropriate function on your calculator for exponentiation (e^λ) and factorial (x!).

Example: Imagine you’re analyzing customer complaints at a call center. The average number of complaints per hour is 3 (λ = 3). You want to know the probability of receiving exactly 2 complaints in an hour (x = 2).

P(X = 2) = (e^-3 * 3^2) / 2!

e^-3 = 0.049 (approximately)
3^2 = 9
2! = 2

P(X = 2) = (0.049 * 9) / 2

P(X = 2) ≈ 0.221

Therefore, the probability of receiving exactly 2 complaints in an hour is approximately 0.221 or 22.1%.

We can also use tools like Python or Excel to calculate the same.

Cumulative v/s Probability

Cumulative Poisson Probability (P(X <= x):

• This option calculates the probability that the number of events (X) will be less than or equal to a specific value (x).
• In simpler terms, it tells you the likelihood of having up to x events happening within the given interval.

Poisson Probability Mass Function (P(X = x):

• This option calculates the probability of getting exactly a specific number of events (x) within the given interval.
• It focuses on the likelihood of having precisely x events occurring.

We hope you found the information helpful! If you learned something valuable, consider sharing it with your friends, family, and social networks.

Reference: Khan Academy, Wikipedia

Also Read:

• Binomial distribution
• Understanding Probability
• Understanding Normal Distribution and Random Variable