 The Poisson distribution models the probability of an event that occurs a discrete number of times over a continuous interval. For example, you might look at the videos produced during a period of time. The number of videos is a whole number, but time is a continuum. Or we might look at defects in a wire over a length. The number of defects is a whole number, but the length is a continuous quantity. Now we could just write down the formula for the Poisson distribution, and we have in another video, but in order to maintain my mathematician card, I do have to do some derivations every now and then. So let's take a look at deriving the Poisson distribution. To derive the Poisson distribution, we'll assume that our interval is actually discrete, and then we'll take a limit. So suppose we expect our event to occur lambda times during an interval of observation. For example, suppose our YouTube superstar publishes an average of two videos a week. What's the probability that they produce k equals 0, 1, 2, 3, and so on videos in a single week? So suppose we break down the interval into n subintervals. If the event can occur or fail to occur during each subinterval, then by the frequentist interpretation, the probability the event occurs in any subinterval will be lambda divided by n. Again, that's because by the frequentist interpretation, an event that occurs p times in q trials has the probability of occurrence on any given trial of p-qths. For example, if the YouTuber produces an average of two videos in a week, then we'd expect to see an average of two sevenths videos in a day, an average of two divided by 7 times 24 videos in an hour, and two 7 times 24 times 60 videos in a minute, and so on. But this also means that the probability the event will occur k times will be a binomial probability. So for concreteness, let's try to find the probability that exactly one video is produced when the interval is broken down into n subintervals. So again, this is a binomial experiment with n trials and the probability of success lambda nths, and so the probability that there's x equal to one success will be, now we've broken down our interval into n subintervals, but in fact our interval should be continuous, and so what we'll want to do is we'll want to find the limit as n goes to infinity of our probability. So taking our limit as n goes to infinity, we get, which gives us the probability we have exactly one success. What about the probability the event occurs twice? So again, this is another binomial probability, and so the probability that there are x equal to successes will be, and again we have n discrete subintervals, but we should actually let n go to infinity, so taking our limit as n goes to infinity gives us, which gives us the probability that there are two successes in the interval. What about the probability the event occurs three times? Again, this will be a binomial probability, which we can compute, then take the limit and again, and we can now try to generalize this result. If we break the interval down into n subintervals, the probability the event occurs exactly k times will be the binomial probability, and if we take the limit as n goes to infinity, we get, which gives us the Poisson distribution.