 One of the more important probability distributions is known as the Poisson Distribution. Before introducing the Poisson Distribution, we have to talk a little bit about an important distinction in the data that between discrete and continuous. A discrete quantity can only take on certain specific values within a range. You can have five people in a room, or six, but never 5.3. Or lightning can hit a building five times, or 500, but never 3.14159. On the other hand, we can have a continuous quantity which can take on, in principle, any value in a range. You can be 1.83294 meters tall, or you can wait 10.3498751 minutes. We can describe random experiments in terms of whether the event of interest is discrete and whether the interval of interest is discrete. It's useful to think of this interval of interest as opportunities of occurrence for the event of interest, when the event of interest can occur. For example, the number of heads when you flip a coin repeatedly. The number of heads is discrete. Moreover, since the coin can only land heads or tails when you actually flip it, then the opportunities of occurrence for the event is another discrete quantity, the number of flips. On the other hand, consider the number of times lightning strikes a building in a year. Here, again, the number of times lightning strikes a building is a discrete quantity. On the other hand, lightning can strike a building at any time, and time is continuous. This leads us to the idea of a Poisson experiment. A Poisson experiment is one that can be described as the occurrences of a discrete quantity over a continuous interval, where the event of interest can occur an unlimited number of times within an interval. For example, a certain type of vehicle suffers a major breakdown about once every 50,000 miles. So is the number of breakdowns in 50,000 miles a Poisson experiment? So definitions are the whole of mathematics. All else is commentary. And so we might remember our definition for a Poisson experiment. It's one that can be described as the occurrences of a discrete quantity over a continuous interval, where the event of interest can occur an unlimited number of times within the interval. And it's useful to think about the interval of interest as the opportunities of occurrence for the event of interest. And so we might make the following observations. The number of breakdowns is discrete. You can have 0 or 20 breakdowns, but never 1.54. Meanwhile, the breakdowns have continuous opportunities of occurrence. They can occur at any distance. And finally, there is no theoretical limit to the number of breakdowns that can occur. You could have any number of breakdowns during that 50,000 miles. And so we're looking at the occurrences of a discrete quantity over a continuous interval, where the event of interest can occur an unlimited number of times in the interval. And so this is a Poisson experiment. Or how about this? A type of genetic defect occurs in one in every 100,000 births. Can this be described as a Poisson experiment? So we note that the number of occurrences of the genetic defect is discrete, but the genetic defect has a discrete number of opportunities of occurrence. It can only occur once per birth. And so this cannot be described as a Poisson experiment. Now, if we have a Poisson experiment, we can calculate the probability using a Poisson distribution. Suppose an event can be expected to occur lambda times in some interval. The probability x equals k is given by this formula, where k factorial is defined as follows, let k be a whole number, then k factorial is the product of the whole numbers from 1 through k, where we also define 0 factorial as 1. And again, for the most part, we're not going to calculate the Poisson probability by hand. Most spreadsheet statistical programs and calculators have something called a Poisson dist function. The syntax varies, but Poisson dist k lambda is common. So let's take a look at that vehicle. A certain type of vehicle suffers a major breakdown once every 50,000 miles. How many breakdowns would you expect to occur during the first 100,000 miles? And what's the probability of having this number of breakdowns? And what's the probability of being lucky and having zero breakdowns in 100,000 miles? So we've determined that a Poisson distribution is appropriate. In order to calculate the Poisson distribution, we need to know the number of times the event can be expected to occur. And so we might reason as follows. If breakdowns typically occur once every 50,000 miles, then in 100,000 miles we'd expect to see lambda equals two breakdowns. So again, most spreadsheet statistical programs and calculators have a Poisson dist function. So we need to know the number of times the event is supposed to occur and the mean, the expected number of times the event is supposed to occur. So here we want to know if the event occurs two times while the event is expected to occur two times. So the number is two and the mean is two. And so our probability is approximately 0.2707. How about zero breakdowns? Well, in this case, the number of breakdowns we want to see is k equals zero, but the number we expect to see is still lambda equals two. So we want to find the probability of seeing zero when we expect to see two. And so the probability of zero breakdowns in 100,000 miles is about 0.1353. Or maybe we have a manufactured wire with an average of three defects per 100 meters. What's the probability that a 10-meter length of wire has more than one defect? So we note that the number of defects is discrete, but they can occur at any point along the length. So a Poisson distribution is appropriate. In order to use the Poisson distribution, we need to know lambda, the expected number of defects. Since there are an average of three defects per 100 meters, we can set up the proportionality. Three defects per 100 meters is lambda defects per 10 meters. Solving this equation gives us lambda equal to 0.3 defects, and so we can use a Poisson distribution with lambda equals 0.3. Now we're interested in the event more than one defect, and this is the same as the event two defects, or three defects, or four defects, or five defects, or, well, there's an infinite number of cases, and we don't want to spend the rest of infinity calculating. So what can we do? We note that it's the complement of the event zero defects or one defect. And so we can compute the probability of having zero defects or one defect. Again, most spreadsheet statistical programs and calculators have a Poisson dist function. So first, the probability of having zero defects when you expect to see 0.3 is 0.7408, and the probability of having one defect when you expect to see 0.3 is 0.2222. So the probability of having zero or one defects is, which means the probability of having more than one defect will be 0.0370.