in this short animated video we will deep dive into the fascinating world of statistics and probability and we'll talk about the second most common distribution after the normal distribution and that is poison distribution we'll also look at how it can be applied to real life situations with help of the relevant examples so please stay tuned don't go anywhere else just sit back relax and watch this entire video till the end so poison distribution is derived as a limiting case of binomial distribution that help us understand the likelihood of different number of events happening when we know the average rate of those events the description was discovered by French mathematician Simon poison one of the earliest application of poison distribution was in analyzing the number of death caused by horse kick in the Persian army poison use the distribution to model the rare event of fatal accidents caused by horse kick and studied the pattern and the probabilities associated with such rare occurrences it was named after him to honor his contribution in the field of mathematics and statistics now imagine that you have a jar filled with color papers and you want to know that if we take out five marbles from the jar every minute so using the probability distribution we can figure out the chance of taking out a specific number of marbles in that minute well let's say you're interested in the number of cars passing through the road in a given time that is where the poison distribution can help us predict the probabilities of different number of cars passed by based on the average rate of car arrivals the poisson distribution and the normal distribution are two most commonly used probability distribution in the statistics I think it would have heard that these two are the most common probability distributions distribution applies to accident rates arrival times the defect rate the occurrence of bacteria of fungus in the air and many other areas in the day-to-day life now when we talk about the shape of the poison distribution it depends upon the parameter Lambda which represents the average rate of event occurring in a fixed interval when Lambda is small say for example Lambda equal to 1 the distribution is skewed to the right with long tail on the right side as Lambda increases from 5 to 10 the distribution becomes more symmetrical and bell shaped resembling the shape of a normal distribution conditions for Poise distribution so the poisson distribution is derived as a limiting case of binomial distribution under certain conditions that the number of trials in the binomial distribution n is large and the probability of success in each trial p is small the resulting distribution approaches the poison distribution to apply poisson distribution certain conditions should be met like the fixed interval the event of Interest must occur within fixed interval of time space and other discrete unit for example counting the number of phone calls is saved and are and the number of accidents at particular insertion in a month Independence the occurrence of event must be independent of each other this means that the happening of one event should not influence the likelihood of another event occurring for example the arrival of phone calls at call center should be random and not influenced by the previous call constant average rate the average rate at which the event must occur remains constant over the entire period of time this means that the probability of an event happening in a given sub interval should be proportional to the length of that sub interval for example if an average rate is 3 accident per day it should remain constant throughout the day rare events the events of Interest should be relatively real and have a low probability of occurrence this condition ensures that the poison distribution is a suitable approximation as it assumes that the probability of more than one event occurring in very small sub interval is negligible and last is the event count the variable of interest is a count or the number of events that occur within the fixed interval the poison distribution provides the probability of obtaining a specific count of events it is important to note that the poisson distribution is an approximation and may not perfectly fit all the situation however it oftens provide a good approximation for events that satisfy above conditions now let us see how do we calculate mean and variance for the poisson distribution for that let's first look at the poisson distribution formula where X denotes the count per second for the probability that X is equal to K is given by e k power minus Lambda into Lambda K power K by K factorial where Lambda is the mean number of success K denotes the number of success and e is a constant which is 2.718 when we come back to what is mean mean is given mathematically mu is equal to Lambda so mean represents the average or the expected number of events that occur within the fixed interval now what is variance so variance of a population distribution is also given by Lambda so the variance represents the measure of spread or dispersion of distribution around the mean standard deviation standards is nothing but the under root of variance or Lambda so for poison distribution mean and variance are equal and are both determined by parameter Lambda so this means that if you know the average rate of events occurring you can determine the expected mean and the variance of the distribution now we can see this using some examples let's look at some real life examples of the person distribution the number of phone calls received at the call center within the specified time interval the number of accidents at a particular location within a given time period the number of earthquakes in the particular region number of emails received per unit of time number of defects found in a batch of product produced in a manufacturing process the number of customers arriving at retails to bank or restaurant the number of patients admitted to hospital or emergency room within the certain time period these are some of the real life examples so let's take the first example on an average lightning kills three people every year in U.S now we need to calculate what is the probability that only one person is killed this year assuming the Lambda equal to 3. so here we are given that three people are killed on average in U.S but we need to calculate that only one person is killed this year not more than one so in this case assuming that they are independent random events as in what how many people are killed last year how many people will be killed this year are totally independent and that is why it follows the poison distribution so we'll use the formula p is x equal to K E power minus Lambda into Lambda to power x by X factorial so we are assuming Lambda equal to 3 here which is given in the question so we'll put the values where Lambda is equal to expected number of people killed X is the number of people killed in a air and we put the values for x equal to 1 that is one person is killed for e k power minus 3 into 3 power 1 by 1 factorial we calculate we get 0.15 so the probability is fifteen percent that only one person will be killed this year now let's take a little complicated example five percent of the population are left-handed what is the probability that a random sample of 100 people contain zero dependent people second case at most left-handed people third case more than three left-handed people we start with first case here we have the formula for poison distribution where X is a random variable that represents the number of left-handed people in a sample K is the number of left-handed people who we are interested in Lambda is the mean of poison distribution which is equal to n into P where N is a sample size and P is the probability of success that is the proportion of left-handed people now in our case k equal to 0 so Lambda is equal to n into p so that n is number of people that is 100 probability of success here is five percent that is 0.05 so we get Lambda is n into p as 5. we put this value in the table or formula for person distribution for x equal to 0 we get e k power minus 5 into 5 Cube over x square by 0 factorial so X is nothing but 0 here uh we put the values we get 0.0067 so the probability that comes out is 0.0067 we convert into percentage we get 0.067 percentage let's take the second case at most three left-handed people here we apply the same formula for person distribution in this case we need to calculate for k equal to 1 2 1 3 because we the case talks about at most three left-handed persons so this includes 0 500 people one definite people two and three as well Lambda as usual will have same value 5 so this is what we need to calculate we will calculate for x equal to 0 1 2 and 3 and add all at them all we already done for x equal to 0 in the previous example we start with x equal to 1 which get 0.0337 for x equal to 2 we get 0.0842 for x equal to 3 we get 0.1404 we put this here and we sum it up we get 0.2650 the probability that the population contains at least at mostly left handed is 0.2650 if you convert into percentage it comes to 26.5 let's take the third example where we have more than three left-handed people we again apply the same formula for poisson distribution but in this case we need to calculate 4 K greater than or equal to 3 not equal to but K greater than 3. so we have the same Lambda as NP we get five so this is what we need to calculate for X greater than equal to 3 so we need to calculate for four five six and so on till n which is little difficult to calculate the other way is we know that the sum of probability is always equal to one so we for x equal to 3 will subtract it from 1 Which is less than equal to 3. so that p x less than equal to 3 that we have calculated in the previous case two you can look at that and we subtract from one we get p all the values which is more than three so we subtract we get as P greater than x 3 equal to 0.7350 so probability that comes up is 73.5 so if you are still watching this video don't forget to hit the Subscribe button and do press the Bell icon for all the notifications from digital e-learning and if you like this video don't forget to hit the like button as well share this video with all your friends and colleagues and in case if you have any suggestions or comment do let me know in the comment box below now is the quiz time on this topic first question the mean of poison distribution is equal to Lambda Lambda Square 2 Lambda or EK power Lambda second question the poison distribution is characterized by which of the following parameters mean and the standard deviation mean and variance mean and median and mean and mode question three which of the following is true about poison distribution it is a continuous probability distribution it is used to model continuous random variable it is used to model the number of events occurring in picks interval of time or space it is assumed a normal distribution you can leave your answers in the comments section below [Music] [Applause] [Music] thank you