Shishir Ahmed Prima

Difference between Binomial Distribution and Poisson distribution The main difference between the Binomial distribution and Poisson distribution is that they both measure the amount of specific random events within a certain frame, the Binomial distribution is based on events that are discrete. On the other hand, the Poisson distribution is based on mainly continuous events. While in case of random variables that are discrete the allocation of the whole anticipation is assigned to various mass points. And diversely, when the goods are random and continuous in nature than based on the different class intervals, we distribute the whole probability. In a binomial distribution, the outcomes available are 2. On the contrary, the number of outcomes in a Poisson distribution is infinite. Meaning there can be numerous outcomes that cannot be determined beforehand. Both of these probability distributions are discrete. Comparison Table between Binomial Distribution and Poisson distribution Parameter of Comparison Binomial Distribution Poisson Distribution Definition When the probability of redone trials is examined then we call it binomial distribution. The Poisson distribution gives the measurement of events that are independent and occurs randomly in a specific duration. Genre Biparametric as the outcomes are two in this regard Uniparametric as the outcomes of the Poisson distribution is infinite or countless. Trials in the experiment Certain and fixed beforehand Trials are not finite in case of this part of the discrete distribution. Achievement The probability to achieve a positive outcome is constant. The probability of achieving any positive outcome is very little. Results Success and failure are the only two outcomes come out from binomial distribution. The number of outcomes in the case of Poisson distribution is boundless. What is Binomial Distribution? Binomial Distribution is thought out to be the possible passed or failed conclusion that has been copied so many times in a review or experiment. In a binomial distribution, the mean or average of all the events is greater than its variance or Whole Square of the sigma. Sigma is also known as the standard deviation in this purpose. Which shows how much a particular event deviates from the mean or average of the whole events. For example, in the event of coin tossing, we use the binomial distribution where the outcomes are fixed. In other words, we know about the success and failure of the coin tossing. Each of the outcomes in this regard carries a 50 percent probability. As discussed earlier the distribution is Biparametric as the outcomes will always be 2. The number of trials to design the experiment is fixed. To get the idea about the distribution we repeat the same thing over and over until we reach the point we fixed before getting into it. Then based on the success and failure of the events we decide what result we have reached from this statistical experiment. The mean of the binomial distribution is µ = np and the variance is denoted as σ^2 = npq. The function is: P (X = x) = nCx px qn-x, x = 0, 1, 2, 3…n The trial is something that happens when a particular outcome is formed that is not fixed is called a trial. These trials are fixed and positive integers also they are independent. What is a Poisson distribution? The definition of Poisson distribution is an individual distribution of frequency that generates the chance of a number of independent events happening in a certain amount of time. We use the formula shown below for Poisson distribution: In a Poisson distribution, the mean is equal to its variance. For example, we use Poisson distribution to find how many of the total printing errors or typos are there in a whole book. Please note that the focus to find out the errors or typos can be any possible given positive integers. The number of errors cannot be fixed beforehand as the mistakes by analyzing the whole book can be 0 or even if there are some, we don’t know how many of them we will find after completing our analysis to find out those mistakes. As we discussed earlier it is Uniparametric in nature as the outcome is only 1. In our example, the focus was just to look out for the number of mistakes in that book. We are not looking for anything else. It expresses in a fixed time interval the probability of happening of a certain event. We use “λ” to express both mean and variance in a Poisson distribution. In the above function, we see an “e” which carries an approximate value of 2.71828. It denotes the transcendental quantity. Meanwhile if “N=number of events” is high and “p=occurrence of events” is low then we use this distribution. Above we can see the pages numbers of the book are represented in terms of N and the errors we look out for are the occurrence of that event which should be very low in case of finding mistakes in a printed book. Main Differences between Binomial Distribution and Poisson distribution Mean is greater than the variance in a binomial distribution while in Poisson distribution it is equal to the distribution’s variance. Number of outcomes are fixed in binomial distribution but in Poisson distribution, the outcomes can be infinite Each outcome carries 50 50 probability but in the case of Poisson distribution, the occurrence of a certain event is very low. The trials in binomial distribution are fixed which means the n is not supposed to be that much big. On the other hand, for Poisson distribution “N= number of events” in our example of finding the printing mistakes or typos can be really big though the outcome might be near to zero or something which we cannot assume before getting into the experiment. Video 1. https://www.youtube.com/watch?v=InLiQ3rnCH0 2. https://www.youtube.com/watch?v=u9onO78hDlw Conclusion The Binomial and Poisson distributions may seem similar in some ways but they have differences too as we have discussed earlier. Sometimes one may think that they are the same as the known fact is both of them are discrete. While practicing, under specific circumstances the Poisson distribution can be used to approximate the binomial distribution. For example- we can use the Normal curve in place of a Binomial distribution under the exact conditions. References 1. https://www.researchoptimus.com/article/normal-binomial-poisson-distribution.php 2. https://www.researchgate.net/publication/-_Poisson_and_Binomial_Distribution 3. https://library2.lincoln.ac.nz/documents/Normal-Binomial-Poisson.pdf