If there were a competition for the most useful distribution, that's over. The normal distribution won it in a landslide. However, we could have a pretty active discussion about what deserves second place. The Poisson distribution would certainly be in the running. At any rate, the Poisson distribution is used to model counts. The Poisson mask function is lambda to the x, e to the negative lambda, all over x factorial where x is defined on the non-negative integers. Zero, one, two and so on. The mean of a Poisson random variable is lambda, this parameter here. And the variance of this distribution is also lambda. So that's an interesting thing to take into account when you model things is if they're a poisson. The mean and the variance have to be equal. Which of you have repeated poisson data, this is a checkable assumption. I'd like to get some instances where we use the poisson distribution. Any time you want to model count data, the Poisson distribution is a candidate. Especially if those counts are unbounded. There's also another set of data that's very common in the field of bio-statistics. So called event time or survival data. So for example, in cancer trials, you might be trying out a new therapy. You would model the time until people have a recurrence of some symptoms. Some people may not ever have that recurrence. And some people may, in the time of the study. You need special statistical techniques to deal with that so, that so, that problem of so-called censoring. And those techniques have a deep connection with the Poisson distribution. Whenever you take a sample of people and you classify them according to some characteristics, and just look at the counts of the people of various hair colors. That's called a contingency table. You can create plot, cross classified contingency table for example you were to count hair color by race, and each cell would be the specific race hair color combination. The count of the people in the sample that had that specific combination. The Poisson distribution is the devault, default distribution for modeling contingency table data. And it turns out that it has again a deep connection with some of the other models you might think to use such as multi nomials, binomials, and A final instance where the poisson is used, and this is done so commonly it's not even really stated as people, people are doing it. Is in instances where you have a binomial, but n is very large and p is very small. This, for example, happens very commonly in the field of epidemiology. My friend Roger studies air pollution. And he looks at, as air pollution rates rise and fall, the number of new cases of say, respiratory diseases. In large areas, like cities for example. Well the n is very large, the population of the city, but the number of events that he's looking at are often quite small. And so he models those as if they were poisson. And this is done so commonly in the field of epidemiology that people don't even really say when they're doing it. They just do it, and everyone knows what they're talking. The Poisson distribution is often used to model rates. So for example, let's let x be Poisson lambda t. It's important to note that lambda here has units. It's the average number of events per unit time, where t is expressed in that, that particular time unit. So lambda is the expected count per unit time, and t is the total monitoring time. So this is a very common use of the Poisson distribution for modeling rates. So, imagine if the number of people that show up to a bus stop is Poisson with a mean of 2.5 people per hour. We watch the bus stop for four hours. What's the probability that three or four, three or fewer people show up the whole time? So that's just the Poi, Poisson probability of three, three, two, one, and zero. And the rate we want at this point is not 2.5. Cause it's 2.5 events prior. But we watched it for four hours. So we want to put in a rate of 2.5 times four. And here we get the probability here of 1%. Let's talk about the Poisson approximation to the binomial. Particularly when n is large and p is small. The Poisson can be a quite accurate approximation of the binomial. To tie down notation, let's let x be binomial n, p. And define lambda as n times p. But here we're considering circumstances where n is very large, and p is very small. Then the proposal is that the probability distribution governing acts which is binomial can be well approximated by Poisson probabilities with this specific notation lambda as n times p. As our final example to this, to discussing the Poisson distribution. Let's go through a Poisson approximation of the binomial. So imagine if we were to flip a coin with success probability 0.01 500 times. What's the probability of two or fewer successes? This is simply to illustrate that I have a setting that's a binomial probability with a large n, 500, and a small p, 0.01. And now I want, but it's, it's still in the realm where the binomial calculations are possible. So here when I do pbinom, for two, because the question asked for two or fewer successes, size equals 500, 500 coin flips, probability equals .01, suggesting that the true success probability of the problem. Of the coin is .01. I get around 12%. When I do the same calculation with Poisson, two and then now instead of inputting the size and the probability, I put in lambda as n times p 500 times .01. I get 12, 12.5%. So, fairly close.