The Power of Poisson Distribution: Discover Its Vast Applications

Poisson distribution is a fascinating topic that often comes into play in various real-world scenarios. Whether you're analyzing customer arrivals at a store or the number of emails you receive in a day, the Poisson distribution can offer valuable insights.

In this article, we'll delve into what Poisson distribution means, its relevance in statistics, practical examples, and the conditions required for its application.

"Poisson" is French for "fish". Poisson distribution can help calculate the probability of your next catch ;)

Poisson Distribution is a statistical model that helps us understand how many times an event is likely to occur within a specific time frame. It's particularly useful for events that happen relatively rarely or sporadically but at a predictable average rate. The distribution is defined by a single parameter, the average rate of occurrence, and it assumes that each event is independent of the others.

Key Characteristics

• Fixed Time Frame: The events occur within a specific time or space.

• Known Average: There's a known average rate of occurrence.

• Independence: Each event is independent of the others.

Question

Emily owns a small bakery. She notices that she usually sells around 15 cupcakes every hour. She's wondering how likely it is that she'll sell exactly 18 cupcakes in the next hour. She wants to know this so she can prepare the right amount of frosting.

Answer

To solve Emily's problem, we can use Microsoft Excel.

Open Excel: Start by opening a new Excel workbook.

Desired Sales: Emily wants to know the likelihood of selling exactly 18 cupcakes. Type 18 in cell B1.

Average Sales: Emily sells an average of 15 cupcakes per hour. In Excel, type 15 in cell B2 to represent this average.

Poisson Formula: In cell A3, type the following formula to calculate the probability using the Poisson distribution:

18 is the number of cupcakes Emily wants to know the likelihood of selling.

15 is the average number of cupcakes sold per hour.

FALSE indicates that we want the probability of exactly 18 cupcakes being sold.

After entering the formula, cell A3 will display the probability. Let's say it shows 0.0706. This means there's an 7.06% chance that Emily will sell exactly 18 cupcakes in the next hour.

Practical Implication: Knowing there's an 7.06% chance of selling exactly 18 cupcakes, Emily can decide whether to prepare extra frosting for this scenario. Given that the likelihood is relatively low, she might stick to preparing based on her average sales instead. This is a practical way to use Excel and the Poisson distribution to make data-backed decisions for a small business.

When to Use Poisson Distribution:

The Poisson distribution is a powerful tool for modelling the number of events that occur within a fixed period of time or space. However, it's crucial to know when its application is appropriate. Here are some guidelines:

• Fixed Interval: The events should occur within a fixed interval of time or space. This could be an hour, a day, a square meter, etc.

• Known Average Rate: You should know the average rate of occurrence for the events you're studying. This is the λ parameter in the Poisson formula. When events follow a Poisson distribution, λ is the only thing you need to know to calculate the probability of an event occurring a certain number of times.

• Rare or Infrequent Events: Poisson is often used for events that are relatively rare or infrequent, but it can also be applied to more common events as long as the average rate is known.

• Independence: Each event should be independent of the others. The occurrence of one event should not affect the occurrence of another.

• Discrete Events: The Poisson distribution is for modelling discrete events, meaning events that are countable, like the number of emails received, customers arriving, etc.

• No Simultaneous Occurrences: The distribution assumes that more than one event cannot occur at exactly the same instant.

• Constant Rate: The rate of occurrence should be constant throughout the period being studied. Seasonal or trending data may not be suitable.

Does It Depend on the Type of Data?

Yes, the suitability of the Poisson distribution does depend on the type of data you have :

• Numerical Data: Poisson is used for count-based numerical data, not for categorical or continuous data.

• Non-Negative Data: Since you're counting occurrences, the data should be non-negative integers (0, 1, 2, ...).

• No Extreme Values: The distribution is sensitive to extreme values, which could skew the average rate and make the Poisson model less accurate.

If you're a call centre manager and you know that, on average, 10 calls come in every hour, you could use the Poisson distribution to estimate the likelihood of receiving a different number of calls. If you're studying the heights of people in a city, the Poisson distribution would not be appropriate because the data is continuous, not count-based.

By following these guidelines, you can make an informed decision about whether the Poisson distribution is the right tool for your data analysis needs.

Lets look at another example so that you are confident of the concept.

Question

Let's say you manage a contact centre and have observed that the average rate of incoming calls is 30 calls per hour. You want to know the probabilities of receiving different numbers of calls to better manage your staffing levels.

P(x) = (λ^x * e^(-λ)) / x!

λ = Average rate of occurrence

x = Actual number of occurrences

e = Base of the natural logarithm (approx. 2.71828)

x! = Factorial of x

Lets use MS Excel which is readily available for most of us to solve this problem.

By understanding the Poisson distribution, its conditions and applications, you can make more informed decisions in various professional settings. Whether you're a business analyst, a data scientist, or just a curious mind, the Poisson distribution offers a robust method for making sense of the world around us.