The binomial distribution consists of Bernoulli trials, hence success and failure is defined in order to calculate the probability. However, the Poisson distribution, known as a type of a binomial distribution, does not require the user to define success or failure. It is applied when determining the probability of a situation where we can expect something to happen for some amount of times during a specified interval. For example, if Ken usually receives five pieces of mail in his mailbox per day then he would be expecting to receive five mails every day. However, this does not mean that he will never receive more than or less than five mails, since the five mails that he has been expecting per day is just an expectation. In these kinds of situations, there will always be a certain spread, where Ken may only receive four mails, more than five or even none. Through applying the Poisson distribution, it is possible to find out the probability of a situation where we expect something to happen. However, the expected rate, also known as the average rate, and a certain time interval needs to be given in the question. The formula for the Poisson distribution is:
As shown, the Poisson distribution will generate the probability of how likely it is to get a certain value during a period of observation. In other words, it predicts the spread around a known average rate (expected rate) of occurrence. Even though the Poisson distribution helps solve certain problems, there are also limitations to this type of distribution. Firstly, the events must be repeated and independent so an outcome will not affect the result of the next outcome. Secondly, the fact that this is a discrete probability distribution also suggests that events need to be counted in whole numbers. Furthermore, as mentioned, the average rate, or the expected rate, and a specific time period needs to be known (the lambda value) in order to utilize the equation. Moreover, the formula can be applied when the question asks the probability of an event occurring, but, unlike binomial distributions, it cannot be applied to situations where it is asking for the probability of an event that will not occur. Last but not least, the Poisson distribution is a limited case of the binomial distribution. Although it does not require the user to define success, the p-value exists in problems that require the use of a Poisson distribution. When a Poisson distribution is applied, the probability of success (p-value) is a very small number. This is due to the fact that most of the time a person is expecting the expected outcome, so the probability of something that is not expected to happen occurring should be very low.
Chapter Thoughts MDM4U 