 So let's go into looking at Poisson distribution also with Poisson Poisson distribution gives you the area of opportunity and that area will always be will come from a continuous unit of an interval. So this is part of the discrete distribution, but that discrete, the discrete variable will produce a continuous value of some sort because we use an average. And when we convert it to an average, remember when the minute it takes up a decimal point, therefore it means it's something that you measure, it's something that is continuous. So the area of opportunity for a Poisson, it will be a continuous because of the interval time that it will have. Like the number of scratches on the car, on average, they might be on average 1,2. The number of mosquito bites on average in a night, they can be five or they can be 3.5 something like that. So you use the discrete probability, but at the end it will also because of the number of trials that you get, it will create a continuous value out of it. So let's see what are the other characteristics that you need to know. You need to know that for a Poisson distribution, the expected mean is the same as your average, which is the same as your mu, your mean, and for the variance, your variance is the same as your mean. So you must also remember this, your variance is the same as your expected mean, is the same as your lambda, which is the average, is the same as your average, they mean one and the same thing. And your standard deviation is always going to be the square root of your variance. So your standard deviation is the square root of your variance. Since you're also expected to calculate this, you are expected to know how to calculate them. So in your question, they will say on average 3.5, emergency calls are recorded per hour for this police station and these calls are following a Poisson distribution. Sometimes they will give you this hint, they will tell you that it's a Poisson distribution and then you will notice that you are using a Poisson or you need to calculate the Poisson characteristics. So for example, if I need to calculate the mean of a Poisson, I know that the mean is the same as the average, so my mean is 3.5. So my mean will be 3.5. And if I need to find my variance, I know that my variance is the same as the mean, therefore my variance will be 3.5. And my standard deviation is the square root of your variance. So my standard deviation will be the square root of 3.5, which gives me 1,87. 1,87 and that is the Poisson distribution. Calculate the variance of the statement above. What will be the variance? You can just say it out loud. What is the variance? The variance is the same as the average, or is the same as your mean? This is the same as your average. So you must know all this. Is the same as your expected value? Is the same as your lambda? Is the same as the mean? Your variance will be the same as the average, will be the same as the mean, will be the same as your expected. So what is the average? What is the answer? Your question is asking. 1.5. That's how easy it is with Poisson. Which one is the correct answer? This follows a Poisson distributed with a mean of 0.2 per day. Use the information to answer the following question. What is the value of the mean? 0.2. The information that they gave you. What is the value of your standard deviation? Square root of 0.2. And which one will be the correct answer? 0.4. So your mean is 0.2. And your variance is 0.2. Which means the square root of your variance? 0.4. Which is equals to 0.4. And you can choose which one is the correct answer from there. And that's all what they're asking you. So if I put this in this manner. And that's how easy it is. Any question before we go into the probabilities? No. Okay. If there are no questions then let's look at how we calculate the probability of a Poisson distribution. This is the formula that you need to also know. That calculate the Poisson distribution. Which gives you the probability of a Poisson. Also taking into consideration. Remember when they say it's greater than or less than. Or it is at least or more than. We didn't do that when we did the binomial. You need to also remember all that. So if they say it's greater than or equal. You will need to add all the probability. The exact, it's just that one probability. But the minute they have the less than. And they also have the greater than. You need to add all probability. So for example. With this formula if. Let's say our N equals 4. And they're asking us to calculate the probability that X. Is less than. Or equals to three. What they're asking you is to calculate this formula. Three times if. With actually four times because you have to calculate X is equals to zero. Plus X is equals to one. Plus. The probability that X is equals to two. Plus the probability that X is equals to three. Because our N is four and. Our probability is less than or equals to three. Therefore it must also increase three. Then it means you have to calculate all of them. So it means you have to take this probability formula. And calculate for X is equals to zero. And calculate for X is equals to one. Calculate for X is equals to two. And calculate for X is equals to three. And add all of them up. Where you know that they will also give you the value of your lambda. And your X will be represented by those values that you have there. So. Easy. Yes, it's easy to substitute the values into the formula. But it's also easy to calculate the probability without even looking at this. And. By looking at the table, the Poisson table. Which I think it's table E seven or E eight. We can look at the correct table number just now. The Poisson table looks different to the binomial table with the Poisson table. It's broken down by the average is. So you will look at the top. There will be average is your lambda values. And going down there will be your X values. So for every lambda table that you will have. Has its own corresponding X values. So for example, on this one, the X might end up at seven. The next one, the X might end up at eight. The next one might. It's different. So you need to be very careful when you use the tables and not assume. That they are the same for all of them. So. Let's say. Yeah, we. We want to calculate that the probability of X is equal to two. So let's say. Our lambda. Which is our expected many make our expected. Value is zero comma five. So we go to the table. We go look for zero comma five and we go to the side. We look for X is equal to two. And when they meet, that would be the problem. The probability. Or you can use your formula, your X. Your X is two and. Your lambda is zero comma five. So you just substitute the values and you calculate. Your Victoria. On your calculator, you do have an n factorial. That is what we use to calculate this value. So you can use that to say n and then you go second function and. Find your n factorial. All calculators have that. So you must look for. That button on your own calculate. The e to the power. All calculators have it. It's called e to the power of X. So you can find it on your calculator. And you will have to press the second function and then go find that e to the power. So on the financial calculator for those who have the financial calculator. You press second function and you press the plus or minus button, but. Calculators are different, so you need to find it on your calculator. To use that button. Otherwise you don't even have to worry about that. Because you can use the table to answer the same question. Let's say the question was what is the probability that X. The same question. So find the probability. That X is less than or equals to two. So therefore they're asking us to calculate the probability that X is equals to zero because this table starts from zero. Plus the probability that X is equals to one. Plus the probability that X is equals to. Is a question to because it says less than or equals to. So therefore it means I must say. Zero cover. Zero comma. Six zero. Six five plus zero comma. I can see the values here. I think it's zero comma. Three. I'm not sure. And the last one is zero comma. Zero seven. Five eight. And you add all of them up and that will give you your probability of less than or equals to. Which is point. Six zero. Six five plus point. Three zero. Three three plus point zero seven. Five eight. Which gives us. Oh, sorry. I didn't calculate it right. Six oh six five. Plus point. Three zero three three plus. Point zero seven. Five eight. Which gives us zero point. Six. And therefore the answer for this probability will be zero comma. Nine eight. Five six. So let's look at the table. I'm just going to out and show you the real actual table that you need to be using. So you're going to go to and this is the table. So since I'm using. This online. I must tell it. And that's. And that is the table. And it's called table. Seven from your book. So let's say we want to answer the same question, not the same question, but let's say this is the question that we want to answer. Okay. Police station receive on average 3.5 calls. What is the probability that it will get at least. Six calls. So we go there. The probability that the station will get six calls. That is the exact probability. So it means. We know that our lambda. Is it cost to 3.5. So we need to look for the lambda 3.5 table, which is that this next table. So we go to this table. So we know that we look for lambda is equals to 3.5. And we are told that we need to find the probability. The probability that X is equals to six. That is receive six calls. So we go yeah we look for the lambda and then we go and look for six and where they meet. That is the probability and that probability will be zero comma 0771. That's one way of doing it. What is the probability that the station will get at most four calls per hour. So this one says at most four calls per hour. And since it says at most what is at most if you can remember what we spoke about earlier. What is at most at most means less than or equal. But since we're looking for the less than or equal so the lambda still the same. So yeah, the question asked us what is the probability that X is at most. For calls that are six so at most less than or equals to four. Therefore they're asking us to calculate the probability that X is equals to zero plus the probability X is equals to one plus the probability X is equals to two plus the probability X is equals to three plus the probability that X is equals to four. So we need to go and add all of them cannot use the I must erase. It means I must do all of them. So if I go to the table so I must add all these venues. So therefore I must say 0.0 302 plus for X is one is point one zero five seven plus point one eight five zero plus two point two one five eight plus point one eight eight eight equal and that gives us the probability that X is at least or at most it gives us zero comma. Seven two five five and that's how you find the probabilities if it says at least then you will add all of them going down. Equal you do the probability of exactly. And with that, then you have your exercise unless if you have a question. Let's do one simple one. I will do this one with you again as well. So let's say we want to find the average. We told the average number of adults with ASD consulting with the neuropsychologist per day follows a poison distribution with the lean of 1.5. So they give us here a lambda which is average because we know that is a poison. Use the information to answer the question. What is the probability that on a given day a neuropsychologist will consult only one adult. The question says only one adult. So if I look at this as the probability but it also gives me something funny here which is looks like a formula. Make sure that I choose the correct answer for this because they're not asking me which one is the correct and which one is the incorrect. But since because they are not telling me if I must find which one is the correct answer. So I must assume one of the questions here is correct and the rest are not and since I do not know which one is correct and which one is not. I'm going to calculate the probability as if like I am calculating the probability. So to start with we can start with the formula can say we know what the formula is. So we know that the formula to calculate the probability that x is equals to 1 which is the most complex one than the rest. If I can eliminate the formula then I can start calculating the probabilities. So we know that the formula states the probability is e to the power of negative lambda times lambda to the power of x divided by x factorial. So I know what my x is and I know what my lambda is then I can substitute into the formula e to the power minus lambda of 1.5 lambda. Oh sorry my lambda is 1.5 so I say 1.5 to the power of x and I know that my x is 1 and divide by my x factorial which is 1 factorial. And if I look at this yay that was supposed to be my answer. So I could have started first with the probabilities and then come to the question and answer this but that is not the case. So this is one way of finding out if this is the right way of doing it. So the other side is so let's say let's assume that that question this was not the correct answer that we're looking for. Then it means I must go find the probabilities instead of me calculating the probabilities by using this table this whole thing I can just go to the table. So I'll go to the table and we go to the table we go look for you know that we're looking for lambda 1.5 so we go to the lambda table. So we know that we're looking for the probability that x is equals to 1 and our lambda is 1.5 so we come here we look for lambda 1.5 and our x. And it says the answer is 0 comma 0 comma 3 3 4 7 0 comma 0 comma 3 3 4 7 so you will go to back to your question and check if that answer is there. And if you look here that answer is not there before the only answer that we were looking for was just number 4. And that is what you're going to do also in the exam when you are given the question don't panic you just need to apply step by step method of elimination for every question that you get. So let's see if you are able to answer this one question. Using the same information, your poison, your meme, which is your lambda is 1.5. So what is the probability that at least seven. So yeah they're asking you to calculate the probability that x is greater than or equals to seven. So if you go to the table, you go to the table. And we look for lambda 1.5 again and we look for the probability that x is greater than or equals to seven. Therefore it means you need to calculate all of them only those ones in the blogging. And it should be easy. So you just need to go and find the probability that x is equals to seven plus the probability x is equals to eight plus the probability x is equals to nine because there are only three values on there that are left x is equals to nine. Do you have the answer to be easy to become number two. Yes, because x is equals to seven was zero comma zero zero one x is something like that so let's go there. x is equals to seven is zero comma zero comma zero eight and x is equals to eight is zero comma zero zero one and x equals to nine is zero comma zero zero zero. Then the answer will be just to add all the probabilities zero comma zero zero zero eight plus zero comma zero zero zero one plus zero comma zero zero zero which gives you zero comma zero zero nine. Okay. I'm not going to do this one for you you need to do it yourself. Suppose that the number of daily fake news is a poison with the mean of zero comma two per day use the information to answer the question which one of the following statement is incorrect. The old story. So you need to go find all the probabilities so the first one. One fake, two fake, at least three, at least four. So the first one will be the probability of x is equals to one. The second one you need to calculate is the probability that it equals to two. And the third one is the probability that x is greater than or equals to three and can continue. So when you're done adjust to gauge how many people are done just say done done done so that I can know how many are done with the question. Done. Also done. Those who are done try and answer this one as well. I think it will be the same as yes. No, it's not going to be the same as anyone. Try and answer number four as well. Are you still busy? Yes. Remember number four, they are asking you what is the probability that there are no fake news. The probability of x is equals to zero. That's all what they're asking you. Are you still busy? No, done. I'm done. Okay, so let's go back to the first one, which is exercise three. We know what the average is, our lambda is 0.2. The first question they asked, what is the probability that x is one? What is the probability that x is two? What is the probability that at least three? What is the probability that at least five? So we're going to go. And number five, it was asking what the probability or number four was asking what is the probability that it equals to zero. So I'm going to go to the table. We will work from the table. So I need to go through the probability. It was zero point two, not two, but zero point two. So I must go to the top of the table and go look for zero point two, which is this one. So number one, we were looking for the probability that x is equals to one. And that probability of x is equals to one. Zero comma one, six, three, seven. So you must remember all those values that you have. One, six, three, seven. What will be the probability that x is equals to two? Zero comma zero, one, six, four. Zero comma zero, one, six, four. And what is the probability that at least three? It will be the probability that at least it's greater than or equals to three. Therefore, now here is the thing that I didn't say to you guys. You see, if I start from three and I go down, it means I must add one, two, three, four, five. I can also find the same answer by just saying one minus the probability of x less than three, which means I will be finding the probability of one minus into bracket. The probability that x is equals to two plus the probability that x is equals to one. Plus the probability that x is equals to zero. So now I don't, instead of adding one, two, three, four, five values, I'm just only going to add one, two, three values, which will give me one minus the probability of point eight. One, eight, seven, plus point zero, point one, six, three, seven, plus point zero, one, six, four, plus. And there, which gives me zero point nine, nine, eight, eight. One minus zero point nine, nine, eight, eight, because I need to add all of them. And that gives me zero point zero, zero, one, two. If you have done this, if you have added all of these ones, they should give you the same as that. So you can do the complement of the question that they gave you. So the next one, which was option four, it asked what is the probability that x is greater than or is five. And this one is easy because I'm also going to only add three values so I can just go and add the probability that x is equals to five. Plus the probability that x is equals to six. Plus the probability that x is equals to seven, because it makes it easier. I'm only adding three values. And since I'm adding the three values, x is equals, x is greater than or equals to five is equals to zero because zero plus zero plus zero plus zero cannot change from where it is. So if we go back to our question and see which one is the correct one or the incorrect one. So we can assume which one is the incorrect one because if I put this side by side, if I put it side by side, number one, the probability that there will be one phase x is equals to one. So we can see that that was the correct one. The probability that x is equals to two is zero comma zero, one, six, four, that is the correct one. The probability that at least three will be fake, that is the correct one. The probability that there will be at least five phase per day, which is equals to zero. We calculated that and we found that it's equals to zero. And which one is the incorrect one. One of the above is incorrect. And that will be your answer for exercise three. Then I also gave you exercise two. What is the probability that x is equals to zero? What would be that probability that x is equals to zero? It should give you that value there. That should be your value. So what did you get? What is the probability that x is equals to zero? It also shows there. What is your answer? Sorry. What is the probability that there will be no fake news on any given day? None of the above? None of the above because no fake news. It means x is zero. Therefore, when x is zero, it should give you zero comma eight. And that's how you answer the Poisson distribution as well. And that completes today's session. Thank you for joining what we have learned. If I need to recap, you have learned how to do like normal distribution and how to do the Poisson distribution. Now, in terms of your assignment, you should be able, since I'm on your tutorial letter one one. I must go to your second semester. Where are we now? I should reduce the size. So on the second semester, you should be able to do exercise question number one. We've done this yesterday. You should be able to do question number two, question number three. And after today's session, you should be able to answer question number four, question number five, question number six, question number seven. And question number eight. And question number nine. And question number 10. And on Friday, we will do question number 11. And so this will only cover from Friday. So Friday, I don't have it open. So but Friday, we will cover normal distribution. And I think on Saturday, we cover the sampling distribution. I am not sure. I might have split it into two. I will have to look at the topic that we're covering on Friday and Saturday. But you should be able to do up until question, up until question number 10. Don't over stretch yourself. But if you are way ahead of the pecs and you want to try with your little understanding that you have on some of the questions, go ahead and do them. Don't submit your assignment as yet. Just do them. And then when we do the exercises and do the activities after we've done, you can go and double check your answers. But up until today, you should have all the questions up until question number 10. If you are struggling to answer any of the questions from one up until 10, don't hesitate to ask. But it does not also stop you from asking any question if it relates to any of the questions up until question number 25. Up until question number 25, you can go ahead and try and answer all of them. But don't submit as yet. Just do them and then ask if you are lost. And that concludes today's session. Any question you need to raise? Any questions?