# Giving Great Deals Using Poisson Distribution I always wanted to start my own business. The big adrenalin rush came when I was struck with the idea of combining my two passions – clothes and data to start a small weekend fashion store out of my garage. I started around three months back. I identified a supplier who shares my taste and I sent out samples to my friends and got their opinion on the collection. Now I’ve opened shop and my business hours are 9 AM to 11 PM every weekend.

I was the sales girl and my friend Maria is the cashier. After my first day, I realized most of my customers ask for discounts. I want to give my customers a good deal but I do not wish to give into intensive bargaining and lose money. It did not go well the first couple of weeks. I figured out I needed a sales strategy that works for both of me and my customers. To give them good deals and to get me the profit, I realized I need to deep dive into what my customers are truly interested in.  So I decided to collect some data.

From the next day onwards, Maria kindly agrees to collect the data while I became the sales girl and the cashier.

I have the data on the number of visitors looking at the clothes I sell (tops, skirts, jackets, dresses, pants, and intimates) every hour for 10 days. The following table shows data for Day 1.

Poisson Distribution Workbook The average number of views each product gets during each working hour. Based on the data I have for 10 days, I want to set up discounts and deals for every hour. For example, if for a particular hour, the probability of visitors looking for tops is high, then I want to give a good deal on tops or throw in a skirt with the top at a discounted price only for that hour.

This is where Poisson distribution comes handy.

### Let’s first understand the Poisson distribution a bit

It was introduced by Siméon Denis Poisson, a French mathematician in 1837. To find the probability using Poisson distribution, our event must satisfy certain conditions.

1. Data must be discrete – Number of visitors is finite or countably infinite
2. The outcome of one event does not depend on another – One customer looking at a product category does not depend on another customer looking at it.
3. The outcomes of the event must be a success or a failure –  A customer can either look at any product category or not.
4. Outcomes of interest are rare events compared to the total number of events that could occur – Compared to the total number of people who would want to shop for tops, the number of people walking into our store is rare.
5. The rate of successes is constant between an interval – The rate of people looking at tops from 9 AM to 10 AM is 3.

### Coming Back to our business…

We want to find the probability based on the number of customers looking at a product category. Let’s say we want to check the probability around 5 customers. There are several ways to look at this.

Refer to sheet ‘PD with Upper Limit’ in the workbook for calculation. 1. The probability of up to 5 customers looking at tops.

Find the Poisson cumulative distribution function of x = 5.

1. The probability of fewer than 5 customers looking at tops.

Find the Poisson cumulative distribution function of x = 4.

1. The probability of more than 5 customers or more looking at tops.

Find Poisson cumulative distribution of x = 5.

1. The probability of exactly 5 customers looking at tops.

Find Poisson probability mass function of x = 5.

1. The probability of more than 5 customers looking at tops.

Find the Poisson cumulative distribution of x = 4.

Now, what if you want to get more detailed on the number of customers? What if you want to find the probability 5 to 10 customers looking at tops?

We can do this by setting an upper limit and a lower limit.

Refer to sheet ‘PD with Upper and Lower Limits’ in the workbook. 1. The probability of 5 to 10 customers looking at tops inclusive of 5 and 10.

(Poisson cumulative probability of x = 10) – (Poisson cumulative probability of x = 4)

1. The probability of 5 to 10 customers looking at tops excluding of 5 and 10.

(Poisson cumulative probability of x = 9) – (Poisson cumulative probability of x = 5)

1. The probability of the number of customers is greater than 5 and less than or equal to 10

(Poisson cumulative probability of x = 10) – (Poisson cumulative probability of x = 5)

1. The probability of the number of customers is greater than or equal to 5 and less than 10

(Poisson cumulative probability of x = 9) – (Poisson cumulative probability of x = 4)

As this simple example case illustrates, the Poisson distribution has several applications in business. During a particular time-interval, if you have the rate at which an event occurs that has discrete and independent outcomes, you can use Poisson distribution to calculate the probability of the event occurring.

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