Data Science & Statistics Tutorial: The Poisson Distribution

Key Takeaways

hello again in this lecture we are going to discuss the Poisson distribution and its main characteristics for starters we denote a Poisson distribution with the letters P o and a single value parameter lambda we read the statement below as variable Y follows a Poisson distribution with lambda equal to 4 okay the Poisson distribution deals with the frequency with which an event occurs within a specific interval instead of the probability of an event the Poisson distribution requires knowing how often it occurs for a specific period of time or distance for example a Firefly might light up three times in 10 seconds on average we should use a Poisson distribution if we want to determine the likelihood of it lighting up eight times in 20 seconds the graph of the Poisson distribution plots the number of instances the event occurs in a standard interval of time and the probability for each one thus our graph would always start from 0 since no event can happen a negative amount of times however there is no cap to the amount of times it could occur over the time interval ok let us explore an example imagine you created an online course on probability usually your students ask you around 4 questions per day but yesterday they asked 7 surprised by this sudden spike in interest from your students you wonder how likely it was that they would ask exactly 7 questions in this example the average number of questions you anticipate is 4 so lambda equals 4 the time interval is one entire workday and the singular instance you are interested in is 7 therefore Y is 7 to answer this question we need to explore the probability function for this type of distribution alright as you already saw the Poisson distribution is wildly different from any other we have gone over so far it comes without much surprise that its probability function is far different from anything we have examined so far the formula looks as follows P of y equals lambda to the power of Y times the Euler's number to the power of negative lambda over y factorial before we plug in the values for our course creation example we need to make sure that you understand the entire formula let's refresh your knowledge of the various parts of the formula first the e you see on your screen is known as Euler's number or Napier's constant as the second name suggests it's a fixed value approximately equal to two point seven two we commonly observe it in physics mathematics in nature but for the purpose of this example you only need to know its value secondly a number to the power of negative n is the same as dividing one by that number to the power of n in this case e to the power of negative lambda is just one over e to the power of lambda right going back to our example the probability of receiving seven questions is equal to four raised to the seventh degree multiplied by e raised to the negative four over 7 factorial that approximately equals 16384 times point zero one eight three over five thousand forty or point zero six therefore there was only a six percent chance of receiving exactly seven questions so far so good knowing the probability function we can calculate the expected value by definition the expected value of y equals the sum of all the products of a distinct value in the sample space and it's probability by plugging in we get this complicated expression eventually we find that the value is simply lambda similarly by applying the formulas we already know the variance also ends up being equal to lambda both the mean and the variance being equal to lambda serve as yet another example of the elegant statistics these distributions possess and why we can take advantage of them great job everyone now if we wish to compute the probability of an interval of a Poisson distribution we will take the same steps we usually do for discrete distributions we find the joint probability of all individual elements within it if you found this video interesting and want to gain an edge in your career make sure to LIKE comment and subscribe and don't forget to check out some of our other videos for another quick win in the data science skills department