 This is a video about how to find the mode of a Poisson distribution. The mode will be the number of events which gives the highest probability. So the question is, which number of events gives the highest probability in a Poisson distribution? Let's look at an example to start with. Suppose that x has the Poisson distribution with parameter 4. So 4 is the expected number of events. We can work out the probabilities. The probability of having 0 events is 4 to the power of 0 times e to the power of minus 4 divided by 0 factorial, which is e to the power of minus 4 because 4 to the power of 0 and 0 factorial are both 1. The probability of having 1 event is 4 to the power of 1 times e to the power of minus 4 divided by 1 factorial, which can be written 4 over 1 times e to the power of minus 4 or 4 times e to the power of minus 4. The probability of having 2 events is 4 squared times e to the power of minus 4 divided by 2 factorial, which is 4 squared over 2 times 1 times e to the power of minus 4 or 8 times e to the power of minus 4. The chance of having three events is four cubed times e to the power of minus four over three factorial, which is four cubed over three times two times one times e to the power of minus four, which turns out to be 32 over three times e to the power of minus four, and so on. The probability that x is equal to four, if you do the calculation, also turns out to be 32 over three times e to the power of minus four. The probability of having five events is 128 over 15 times e to the power of minus four, and the probability of having six events is 512 over 90 times e to the power of minus four. Okay, well, you can see from this that there are two numbers of events which have the joint highest probabilities, and that's three events and four events. This is because 32 over three is a little bit more than ten, and that's higher than all the other coefficients over on the right-hand side. It's obviously higher than eight and four, but it's also higher than the others. 128 over 15 is just a little bit more than eight, and 512 over 90 is just a little bit less than six. So we can say that the Poisson distribution with parameter four is bimodal, and the two modes are three and four. But before we move on, there's a pattern here, and if you look at the pattern, we can work out a method for saying really quickly what the mode is. Look at how the probabilities change going up the number of events. Going from naught events to one event, we multiply by four over one. Going from one event to two events, we multiply by four over two. From two events to three events, we times by four over three. From three events to four events, you times by four over four. From four to five, you times by four over five. And from five to six, you times by four over six. Now this explains why three events and four events give the joint highest probability. From naught up to three events, the probability keeps increasing because when you times by four over one, or four over two, or four over three, your multiplying number that's bigger than one, and that causes the probability to increase. Going from three events to four events, we multiply by four over four, and of course that's multiplying by one. So the probability doesn't change. Going from four events to five events to six events, the probability decreases because we multiply by four over five, and by four over six, which are both fractions smaller than one. And whenever you multiply by a fraction smaller than one, the answer gets smaller. Now generalizing from this, we can say that the probability of having x events is lambda divided by x times the probability of having x minus one events. And if you think about it, this means that if x is less than lambda, in which case lambda divided by x will be greater than one, the probability of x events will be greater than the probability of x minus one events. So the new probability will be greater than the previous one. If x is equal to lambda, then lambda divided by x will be equal to one, and the probability of x events will be the same as the probability of x minus one events. Finally, if x is greater than lambda, then lambda divided by x will be less than one, and the probability of having x events will be less than the probability of having x minus one events. The new probability will be less than the old one. So when x is less than lambda, the probability will keep increasing. When x is equal to lambda, it will stay the same. And when x is greater than lambda, the probability will decrease. And this explains why when x is equal to lambda, you get two joint highest probabilities where x and x minus one are both modes. Let's return to our example to see how this worked. Here, lambda was equal to four, and three events and four events gave us the joint highest probabilities. Up until three events, the probability kept increasing. From three events to four events, it stayed the same. And from four events onwards, it kept going down. In general, we can say that if lambda is a whole number, then the Poisson distribution with parameter lambda will be bimodal. That means that there'll be two modes. And the modes are lambda minus one and lambda. If the expected number of events isn't a whole number, if lambda isn't a whole number, then the situation is a little bit different. Here are all the probabilities for the case where lambda is equal to 4.1. This time, if we look at how the probability is changing, this is what we get. Here you can see that the probability will increase from naught events to one event, from one event to two events, from two events to three events, and from three events to four events. Because 4.1 over one, 4.1 over two, 4.1 over three, and 4.1 over four are all greater than one. After that, the probability decreases. If we go from four events to five events or from five events to six events, we're multiplying by a number that's less than one, because 4.1 over five and 4.1 over six are both less than one. So this shows that in this situation, the mode is four, because that's the number of events with the highest probability. In general, if lambda is not a whole number, then the Poisson distribution with parameter lambda is unimodal. That means that there's only one mode. And the mode is the greatest whole number that's less than lambda. Let's look at some examples to finish with. We can use the rules that we've worked out to answer these extremely easily. First of all, suppose that x has the Poisson distribution with parameter 9. This is a whole number, and when lambda is a whole number, we know that we will have a bimodal distribution where the modes are lambda minus one and lambda. So in this case, there'll be two answers, the numbers eight and nine. Secondly, suppose that x has the Poisson distribution with parameter 6.13. 6.13 is not a whole number, and we know that when lambda isn't a whole number, there'll be one mode which will be the greatest whole number less than lambda. So here we're looking for the biggest whole number that's less than 6.13, and that's 6. Finally, suppose that x has the Poisson distribution with parameter 0.45. Well again, we're looking for the greatest whole number that's less than 0.45, and that's zero. Okay, I hope this helps you to understand how to find the mode of a Poisson distribution. Thank you for watching.