 This video is an introduction to the Poisson distribution. Now Poisson is the French word for a fish. And so you might think that this distribution has something to do with fish. But actually it doesn't. It's named after the French mathematician Simeon Poisson, who invented it. So try to put fish out of your mind. OK, so what is the Poisson distribution? Well, the Poisson distribution is a probability distribution that's used when a random event happens at a constant average rate. It gives the probability for every possible number of events in a fixed period of time or a particular area of space. And there's a formula for calculating these probabilities. So for example suppose that a bear goes fishing and he catches on average 5 fish per hour. We might be interested in the probability that he catches 3 fish in a particular hour or 10 fish over a period of 2 hours. And we could use the Poisson distribution to find both of these and other probabilities. Also suppose that we're interested in the distribution of pandas within a forest. It might be that on average there's one panda per square kilometre and we're interested in the probability of finding 2 pandas in a particular square kilometre or 5 pandas over an area of 2 square kilometres. And again, the Poisson distribution would enable us to calculate these probabilities. There are some things to bear in mind though when we want to use the Poisson distribution. One is that the events must happen one at a time and the other is that they must occur independently. So for example when the bear goes fishing we need to assume that he only catches 1 fish at a time. And also that catching a fish doesn't increase or decrease the probability of catching another fish. Likewise with the pandas we need to assume that they live alone and so you only find 1 panda at a time. And also that having a panda doesn't increase or decrease the probability of having another panda in the vicinity. Something that probably isn't true because actually pandas are territorial. So how do you calculate probabilities in a Poisson distribution? Well the first thing is that we use the letter lambda to talk about the mean number of events that happen in the interval. So lambda is the number of events that you would expect to happen on average. And then if we say that the random variable x is the actual number of events that occur in the interval. We write that x follows the Poisson distribution with parameter lambda and it can be shown that the probability of having x events is lambda to the power of x times e to the power of minus lambda divided by x factorial. In this formula e is the base of natural logarithms and is approximately equal to 2.718. So this is a formula that we can use to find the probability of having a particular number of events in an interval of time or space when we know the average number of events that will happen in that interval. Let's look at some examples of using this formula. First of all, if x has the Poisson distribution where 3 is the average number of events what's the probability of having 5 events? Well remember that the probability of x events is lambda to the power of x times e to the power of minus lambda over x factorial. And in this situation you can see that lambda is equal to 3. So the probability of having 5 events is 3 to the power of 5 times e to the power of minus 3 divided by 5 factorial. And if you work it out that's equal to 0.101 to 3 significant figures. My next example is about koala bears. Suppose that on average tourists see 2 koala bears per day. What's the probability that on a particular day a tourist sees at least 2 koala bears? Well let's say that x is the number of koala bears seen on this particular day. Then x will have the Poisson distribution where 2 is the expected number of sightings. The question is asking us for the probability that x is greater than or equal to 2. And one way of working this out would be to add up the probability of having 2 koalas, 3 koalas, 4 koalas and so on. But actually this would involve adding up an infinite number of different probabilities. Something that might take you quite a long time. So a better method is to subtract from 1 the probability of getting no koalas or 1 koala. Because the opposite of having 2 or more koalas is having 0 or 1 koalas. Now according to the formula the probability of having no koalas is 2 to the power of 0 times e to the power of minus 2 divided by nought factorial. But 2 to the power of nought and nought factorial are both equal to 1. So this can be simplified to e to the power of minus 2. Likewise according to the formula the probability of having 1 koala is 2 to the power of 1 times e to the power of minus 2 divided by 1 factorial. But 2 to the power of 1 is just 2 and 1 factorial is 1. So this simplifies to 2 times e to the power of minus 2. So now we can say that the probability of having 2 or more koalas is 1 minus e to the power of minus 2 plus 2 e to the power of minus 2. Which is 1 minus 3 e to the power of minus 2. And if you type that into your calculator you'll see that the answer is 0.594 to 3 significant figures. Before I move on I should point out that this example about koalas only works. If we assume that koalas occur one at a time and they occur independently. So that the presence of one koala doesn't increase or decrease the probability of finding another koala nearby. My third example is about polar bears. Most of whom live in Svalbard in the Arctic Circle. Which in the summer looks something like this. Suppose that on average there are 0.05 polar bears per hectare on Svalbard. What's the probability that there are four polar bears in a given area of 100 hectares? Well the first thing we need to do is to work out the expected number of polar bears in our area of 100 hectares. And that's going to be 100 times 0.05. Which is 5. So this means that lambda is equal to 5. Let's say that x is the actual number of polar bears. Then x will have the Poisson distribution where 5 is the average number of events. Note that as with the koala bear example. We need to assume that the polar bears occur one at a time. And that the presence of one polar bear doesn't increase or decrease the probability of having another polar bear in the region. I think that these things are both roughly true. Because polar bears are solitary animals at least in the summer. Now the question is asking us for the probability that x is equal to 4. The probability of having four polar bears in this region of 100 hectares. And the formula tells us that this will be equal to 5 to the power of 4 times e to the power of minus 5 divided by 4 factorial. And if you do this on your calculator you'll get the answer 0.175 to three significant figures. My last example is about the North Atlantic right whale. Here's a picture that shows them to scale. Unfortunately the North Atlantic right whale is critically endangered. And they're estimated to be only about 400 individuals alive. They mostly live off the east coast of America. And in the summer months they mostly swim to the Gulf of Maine in order to feed. Suppose that we're interested in the probability of finding a whale in Cape Cod Bay. So suppose that the 400 Atlantic right whales are distributed evenly throughout the 180,000 hectare Gulf of Maine. We also need to assume that the right whales occur one at a time and independently. But again I think this is roughly true because they are solitary creatures. The area of Cape Cod Bay is 1500 hectares. So what's the probability that there's at least one whale in Cape Cod Bay? Okay well the first thing to do is to calculate the expected number of whales in Cape Cod Bay. And so we need to see what fraction of the 400 whales we can expect to find there. Well the answer is that the expected number of whales, lambda, is 1500 over 180,000 times 400 which is equal to 10 thirds. 1500 over 180,000 because that's what fraction of the Gulf of Maine, Cape Cod Bay takes up. So now we can say that the number of right whales will have the Poisson distribution where the average number of events is 10 thirds. The question is asking us for the probability that x is greater than or equal to one, the probability of having at least one right whale. And we know that that's going to be equal to one take away the probability of having no right whales. Using the formula we can say that that's one take away, 10 thirds to the power of zero times e to the power of minus 10 thirds divided by nought factorial. But 10 thirds to the power of nought and nought factorial are both equal to one. So this can be simplified to one take away e to the power of minus 10 thirds. And if you type that into your calculator you'll get the answer 0.964 to three significant figures. So this is the probability of finding at least one North Atlantic right whale in Cape Cod Bay. Okay, this is the end of my video about the Poisson distribution. I hope you found it useful. Thank you for watching.