A simple overview of Poisson distribution

Imagine you heard of a place somewhere remote that has a natural geyser. The geyser sprays randomly and there is a popular belief that if you happen to watch 3 geyser spouts in an hour, it will bring you good fortune. Okay, you don’t believe in all this stuff but still would want to experience this and then brag to your friends that you indeed happened to see 3 geyser spouts in one hour. So, you decide to head towards the spot. You drive a long way and then you finally reach the spot and you are eagerly waiting to see the first geyser spout. Once, you watch that, you are going to wait to see if you are lucky enough to see two more in a single hour. Suddenly, you get a call from your mom. She wants you to immediately return home as some guests are arriving and that she badly needs you to be home. Now, you are in a dilemma! You drove all the way to this spot to experience this natural phenomena and see if you are lucky enough to catch a glimpse of this occurring three times in an hour. You really want to wait and see but is it worth waiting?

So, you walk to a small neighbourhood in the vicinity and talk to one of the local people. He says with past experiences that the geyser spouts 4 times per hour on an average. But this is not enough. You want to know the chance of you seeing 3 geyser spouts within one hour. Suddenly, you remember about your data scientist friend and you dial him up and you tell him your plight. The moment he hears about what the local person has told you regarding the average spouts, he gets excited! He says that that is the lambda value and then he messages you the following formula and asks you to substitute the 𝜆 with 4 and X with 3.

You look at the formula and are awestruck! You are in no mood to do any calculation and you ask your data scientist friend to do the necessary math and let you finally know whether it is worth for you to stay or leave. Then, he messages you the following screenshots of the code that he executed through Python.

He tells you that the probability is just 0.195 which means there is just 19.5% chance that you will be able to see three geyser spouts in an hour and so it is better for you to leave the spot and head back home so that you can at least escape your mother’s rants.

And now you ask him how exactly he came up with this probability and he tells you that was through a kind of discrete probability distribution known as the Poisson distribution.

Poisson distribution is named after the French mathematician Denis Poisson and is the probability distribution of a number of events that occur at fixed intervals of time or space. To calculate this probability, you need to know the mean rate beforehand which is the average number of successes in a specified interval of time or space and the happening of the events are independent of each other. In the story above, this was nothing but the average number of spouts per hour based on the past data that the local person in the neighbourhood provided and this value is often represented by the Greek symbol 𝜆 which is pronounced as ‘lambda’. The probability distribution for 𝜆 = 4 will look as follows and from the plot, you can see the probability of seeing 3 occurrences is 0.195 as seen via the Python code.

Now that you have understood the Poisson distribution and that there is a very little chance to experience what you came to experience, you head back home and decide to come to the same spot some other day when you really have a lot of time in hand!