 Okay, welcome to your session 8 of your tutorial sessions for STA 1610. We will continue with what we did on Wednesday with today's session. We'll look at more activities. Before we start with today's session, I just want to find out if you have any comments, query or question you want to ask before we start. I'm good, thank you. So there are no comments or queries. Please remember to mute your microphone. Okay, in the absence of comments or queries, we can continue and recap on what we did. So the time we met, we discussed the probability distribution of a discrete variable. We looked at how to calculate the mean and the variance and standard deviation of a discrete variable distribution. And we also looked at how we calculate the probabilities of those discrete variables as well. And what we've learned was that a discrete variable, because we know from chapter one that a discrete variable is a quantitative numerical variable. And because it's a discrete variable, it is a variable that you get from a counting process. And we also said when we look at the probability distribution of a discrete variable, we said all possible outcomes are mutually exclusive because when the events happen, they do not include the other events or they do not have any influence on the other events as well. They should be also independent. And for every outcome, it's going to be associated also with the probability of its occurrence as well. Like we have this table that looks at the interruptions per day in a computer network, in X variable, which has zero interruption, one interruption, two interruption or five interruptions. And for every interruptions, there are corresponding probabilities. Therefore, it repeats the number of cases. This probability gives us the number of cases that occur across the total sample size. And that is where we calculate the probability. And we said also we are able to calculate the mean of a discrete variable. And to calculate the mean of a discrete variable, we use the sum of your observed values multiplied by its corresponding probability. And because it's the summation, therefore it means when you multiply your observed variable with its corresponding probability, you have to add all of them in order for you to calculate the mean, which is the population parameter mu, or we can also call it the expected value, or we can call it the mean. And we looked at the examples of how we calculate the mean. And we also said we are able to calculate the variance. And we said the variance is the sum of your observed minus your expected value squared multiplied by the corresponding probability. We also said we are able to calculate the standard deviation, which is the square root of your variance, which you just put the square root on the variance, which is the sum of your observation minus the expected value squared multiplied by its corresponding probability. And we looked at the example of how we calculated the standard deviation. And we said once we calculated the standard deviation, before you take the square root, the value you get, it's the variance. And when you take the square root, you get the standard deviation. And we did an example of the standard deviation. Then we went on and looked at the probability. And we said when we calculate the probability for a discrete probability distribution, going to always remember, if they give us the weight phrases, we can tie the weight phrases back to the symbol sign. Or we can use the symbol sign to make sense of it, in terms of what really is it that we need to be calculating in terms of those probabilities. And you need to remember all this from now on until you go right there, except because from now on, we're going to be using, they will give you either the symbol, or they will give you either the weight phrases for all the probability questions and the hypothesis questions that we're going to be looking at going forward. So you need to know this. By heart, what do they mean? What do they mean when they say more than? What do they mean when they say at most? What do they mean when they say at least? What do they say when they mean between and between inclusive and between exclusive? So you need to know all those in terms of the symbol or in terms of what the meaning of that meaning, of that weight phrase or symbol mean, in order for you to answer the probabilities. And we looked at some of the examples we did, this example. And we also looked at some other examples of how we calculate the probabilities and the mean. And we looked at this example as well. So we're not going to go through this exercise. And also we did this exercise, we're not going to go through it. And today, we're going to continue on that. Today, we're going to continue on looking at the exercises or doing more activities on the discrete probabilities. If you have any questions, feel free to ask in between. Likewise, we're going to use the chat to put our answers. And then we're also going to discuss the options as well. So without wasting any time, the first one, we are all going to do it together. And then the next one, I will give you some few minutes for you to go through the questions and answer it. And then we'll come back and do a feedback together. So for number one, we forgot about what we did Wednesday. Let me just refresh your minds and let's do it together in a team. So it means you guys are going to work with me and answer the questions and do the calculations. And I will write it on the presentation. So speech therapist, before we continue, any questions? If there are no questions, let me go to the chat as well. Just to see what you guys are saying on the chat for now. Okay. So a speech therapist knows that the number of children that consult with her on a given day is given by the following discrete probability distribution. Let X be the number of children consulting with the speech therapist on any given day. And probability of X will be given by the value of X and it will be associated, be the associated probability for that X variable. And we are given the table with the X values, which are the number of children consulting a speech therapist. And we also have their postponing probability. Which of the following statement is incorrect? And that is what we are looking for. Let me get to the pen. Which of the following statement is incorrect? The probability that the speech therapist will consult with at least one child. How do we write that probability? We say because it's at least remember from the table, the probability that X at least is greater than or equals to, and they say at least one. So therefore it means one greater than or equals to one. And that probability will be all the probabilities. So it will be the probability. So you will have to add the probability that X is equals to one plus the probability that X is equals to two plus the probability that X is equals to three plus the probability that X is equals to four plus the probability that X is equals to five. So it means you're going to add all these probabilities together and the sum of all probabilities is always equals to one. Next, the probability that the speech therapist will consult with one child on a given day because it says with one child therefore it's exactly so that will be the probability that x will be equals to one and that is 0 comma one because it is only that probability there. Where x is one, it's 0 comma one. Sorry, I think that dot should have been here so this should be 0 comma one. The probability that the speech therapist will consult with no children at any given time, the probability that x will be equals to zero because it says no children so no children means zero. So the probability that the speech therapist will consult with no children. The table starts with one and with five so therefore that probability will be zero because we know if an event is uncertain that probability will be equals to zero. Okay, the probability that the speech therapist will consult with more than five children that will be the probability that x is greater than five because it says more than a more than it's every value bigger than five but does not include five so what will be that probability like with the previous one the table starts with one and with five there is no other value or no other outcome after weight after value five therefore that probability will be equals to zero because it's an uncertain event and that will be equals to zero. The expected value so the expected number of children consulting the speech therapist so since they are asking about the expected value we know that our expected formula says it is the sum of your observation times its corresponding probability. Therefore going to say one multiplied by 0.1 plus two multiplied by 0.15 plus three multiplied by 0.2 plus four multiplied by multiplied by 0.3 plus five multiplied by 0.5 25 0.25 and then we're going to add all the values after we have calculated them so 0.1 times 0.1 is 0.1 two times 0.13 two times 0.13 is equals to 0.26. I think it's two times 0.15 not 1.3. 0.15 okay sorry it's me I wrote here 0.15 but it didn't come out it looks like 1.3. Okay thank you so it's two times 0.15 which is equals to 0.3 0.30 and three times 0.2 which is plus 0.6 plus four times 0.3 you must also do the calculation so that if I go wrong on one of them you can correct it as well and five times 0.25 which is 1.25 plus 1.25 equals and we can add all of them 0.1 plus 0.3 plus 0.6 plus 1.2 plus 1.25 equals 3 3.45 so our expected in some writing right at the end of the page doesn't allow me to write well so I'll just make an error so our expected value which is the sum of your observation times your XPS is equals to 3.45 so we know that that is correct it was equals to 1 we know that that was equals to 0.1 we know that this was equals to 0 and that was equals to 0 and that is the incorrect one and I hope now you know how to answer the questions so with that in mind you have your five minutes let me say five minutes to answer the following question what is the standard deviation of this answer of this question that we have we already is the same question as you can see we already calculated the mean so don't go ahead and calculate the mean again because we already calculated the mean so on this question we know that our expected value it's what it's sure it's 3.45 it's 3.45 so now go ahead and calculate the standard deviation which is the square root of the sum of your observed value minus your expected value squared multiplied by the corresponding probability remember you can also do the things inside the bracket the square one and then going multiply with the the value in the bracket so you can say one minus so you can also start there by saying one minus 3.45 squared times 0.1 and write the answer for each and write the answer for that one then say plus and then you go to the next one two minus 0.15 wait and multiply that with zero comma one five and then write the answer plus and then you continue three minus oh sorry four five three four five i don't know why i call it 0.45 0.45 three minus 0.3 0.45 squared and multiply that with 0.2 and continue and write all the answers for all the questions and add them together when you're done i'll give you some time to do that when you are done and you have an answer you can post it on the chat i'm also wicking it out okay let's see have you got the answer okay yes just one more minute okay the probability that i'm getting the answer right is 50 but we not calculating the probability we calculating the standard deviation yeah yeah no no i'm saying the probability that i am getting it right is 50 percent oh okay remember if you if you agree with the answers given you can just like the answers so that i can also see how many people are done are we done i'm done we already stuck it with a couple of things so i just need you to give me the answer for every statement that is there so one minus three comma four five squared multiplied by zero comma one what is the answer you get zero comma six zero zero two five yeah at least leave your answer to three to four decimals so in zero comma six zero zero zero two five so it will be zero comma six zero zero three the next one two minus three comma four five squared times zero comma one five zero three one five four three one five four we all agree three minus three comma four five squared times zero comma two zero comma zero four zero five okay four minus three comma four five squared times zero comma three zero comma zero nine zero eight and the last one five minus three comma four five squared times zero comma two five zero comma six zero zero six okay so all this what we've done we have done what is underneath the bracket but i'm sorry underneath the root this is the we have solved the summation we didn't do the summation we have solved all these values we just need to do summation so it means we need to add all of them together so if we add that plus that plus that plus that and get the answer we will be doing the summation underneath the square root so let's add all of them when you add all of them what do you get zero comma six zero three plus zero comma three one five four plus zero comma zero four zero four plus zero comma zero nine zero eight plus zero comma six zero zero six you get one comma six six four seven five seven six if you get seven five it means you were leaving your digits or you were rounding off too quickly and that is why you're not getting it or you were using your whole calculator to calculate the sums but it should be that and our standard if we take the square root of this standard deviation the answer is the square root of one comma two eight three five one comma two eight three five which is the same as which we can estimate is the same as one comma two eight which is option number five any question there are no questions we can move to the next one we also have five more minutes to work it out and then we will do it together based on the past experience a researcher knows that the probability distribution for x is equals to the number of students who come to her office on Wednesday in is given by this distribution table which of the following statement is correct we're looking for the correct answer but i want you to go through all the statement regardless of whether you know you found the correct answer is number one or number two just go through the rest of them so that you can see how we answer the questions uh Lizzie yes i'm not sure that i'm slowly all right can you link it pick up to the exercise number two exercise two please uh oh okay i think i've kept this under number four zero point two instead of zero point three okay no sorted now thanks a lot are we winning are we winning are we done one more minute please okay okay that one more minute are we done yes thank you i'm done okay so let's go and answer the questions now okay the first one says the probability that x is greater than two what does that mean it refers to the probability that x is x zero x is equals to two plus the probability that x is equals to three three plus the probability that x is equals to four four therefore it means you're going to add zero comma five plus zero comma one five plus zero comma zero five which is equals to zero point eight zero point seven zero point seven zero point seven yeah zero point seven the next one hey i just literally edited now okay i have zero point eight probability that x is less than two how do we represent it so it's where x is equal to one and x is equal to zero plus x is equal to zero plus and x is equals to one which is zero comma one and zero comma two which is equals to zero comma three zero comma three the next one the expected value and we know that the expected value is the sum of your observation times the corresponding probability which means we're going to say zero zero times zero comma one plus one times zero comma two plus two times zero comma five plus three times zero comma one five plus four times zero comma zero comma five which is the same as i'm just going to write it at the bottom zero times zero comma one is zero one times zero comma two is zero comma two plus two times zero comma five it will be one comma zero plus three times zero comma one five three times fifteen is 45 so it will be zero comma four five plus four times five and zero comma zero five four times five is 20 so it will be zero comma zero and if you add all of them the answer will be one comma eight five eight five the next one the probability that x lies between one and four where x is less than four but it is sorry x is greater than four but it's less than or equals to four what am i saying x is less than one but it's sure wait my time is x is greater than one but it's less than or equals to four four and what does that mean means the problem we're looking for starting by i mean see that x equals to two because two is bigger than one plus the probability that x is equals to three and because it's less than or equals to so therefore it means also it's equal to four so it means you're going to add all those probabilities which is zero comma five plus zero comma one five plus zero comma zero five and your answer is zero comma seven the last one says you need to find the probability that x lies between two and four exclusives so it means we'll be looking for x is greater than two but it's less than four and that will be x is equals to three any number bigger than n less than four will only be three the answer here is zero comma one five one five thanks and that's how you answer the questions okay next question halfway from the following table that contains the probability distribution for the number of traffic accidents per day in a town what is the mean accidents what is the average number of accidents that happens in a day in that town remember the mean is your expected value which is the sum of your observations times its corresponding probability are we done in state of writing options write the actual number because i am confused now with all your fours and the truth whether is it the four isn't the answer or two isn't the option so maybe it's clear yeah because i was working at it not three times i'm thinking where am i going wrong where they're getting four they're getting two that's me i'm thinking of them coming through and i'm thinking oh yeah is this the option or the answer to the question so please let's number four are we all done yeah so we're doing the expected means so you're going to give me the answers so i'm just going to do x multiply by the probability there and write the answers so what is zero times zero comma one zero is zero one times zero comma two point two it's point two two times zero comma four point nine it's point nine three times zero point five one five and four five point four five four times zero comma zero five point two point two five times zero comma zero five point two five point two five so therefore it means zero plus zero point two plus zero point nine plus zero point four five plus zero point two plus zero point two five is the question all of them the answer is any question anyone who is still lost anyone who still needs help or are we all on the same page now it means we are all writing on the same cloud we are all on cloud nine together so let's move on next question um Lizzie yes you will excuse me please i have another i have another class i need to attend to if it's okay with you i will catch up the rest in the recording no problem thank you okay next question are we done let's so number one i think we're missing at most number one the probability that the speech therapist will consult with four children that is exactly so it will be the probability that x is equals to zero zero point three zero point three therefore that is correct the probability that the speech therapist will consult with eight most four children most at most is less than or equal me remember that so the probability that x will be less than or equals to four let me write the number the actual number it will be the probability that x is equals to one plus the probability that x is equals to two plus the probability that x is equals to three because it's at most plus the probability that x is equals to four i'm going to do the long way and then i'm going to show you another shortcut so that will be equals to adding up all those probabilities zero point one zero point one five zero point two and zero point three which will give you zero point alternatively we know that we're looking for those probabilities we can also say the probability of x less than or equals to four is the same as one minus the complement probability which is the probability that x is equals to five because that's the only one that is left on this side if the way two we're going to add it will be a cumulative of those one x is equals to five you can do it this way as well you will see that you will get the same answer because the sum of all probabilities are equals to one and therefore this will be equals to zero comma two five which in turn will give you zero comma seven five so there is no right or wrong answer in the way you're doing it nobody's going to look at how you calculated it but instead of adding all the values you can just use the shorter value and find and use it as a complement of other events so you can either do this the the long way or the short way it's up to you but is the probability that the speech therapist will consult with between two children and the biggest way they is exclusive therefore how we write that is the probability that x lies between because it's exclusive therefore it will be x is greater than two but it's less than four so x is greater than two but it's less than four therefore it is at that point so that will be the probability that x is equals to three then it is zero comma agree yes okay the next one the probability that the speech therapist will consult with two between two or four children inclusive and that will be because it's inclusive therefore x will include two and it will also equal to four and that some of them might not and that probability will be the probability that x is equal to two please mute your microphone plus the probability x is equals to three plus x is equals to four then it means we need to add zero comma one five plus zero comma two plus zero comma three zero comma one five plus zero comma two plus zero comma three three and that probability will be equals to zero comma six five zero comma six five last one the probability that the speech therapist will meet with at least four children then we're looking for what is at least equal or more bigger than or equal right yes so that will be the probability that x it's more than or equal to four which then will include the probability that x is equals to four plus the probability that x is equals to five which is zero comma three plus zero comma two five which is zero comma five five five five five that is the incorrect answer that you are looking for any questions happy yes happiness happiness question number six can go ahead and do it are we done are we winning are you are you winning have we answered only so that it is confirming what I'm also confirming um yes I think there is there was an error I've probably sent through with this question if it was an assignment question or an exam paper then people who write those like that in the paper they they got three for max as well there are two correct answers there are two incorrect answers yes that's what I think that's why we were not responding yeah let's let's answer the questions do this so the first one is asking what is the probability of p three so now p three is the value there so therefore we can say that is one minus the sum of all the other completed probabilities that we have on the table which will be the complement of p three so we'll say one minus all of those values which is zero comma three plus zero comma one plus zero comma two four plus zero comma one plus zero comma one two which is equals to one minus zero comma eight six all of them you get eighty six zero comma eight six did you all get that he is okay and that will be one minus zero comma eight zero comma one four so now we do this probability one four now we can answer the question about two what is the probability that it's zero and three exclusive this will say the probability that it's greater than zero will include x is equals to one plus the probability that x is equals to two but it does not include three so then it's only those two because it does not include three and it also does not include zero so then it will be zero comma one plus zero comma two four which is equals to zero comma three what is the probability that at least at least means the probability that x is greater than or equals to four and that will be x is equals to four last the probability that x is equals to five one which is zero comma one one plus zero comma one two which is equals to zero comma two two so when they were probably when they did this question and I think it comes from 2018 tutorial letter 101 so when they did that question we ever wrote that uh I assume that at least it was the greatest what was the same as at most because that answer would have given you the at most answer but yeah so they had assumed that incorrectly the probability that at most two students so this is at most two students which will be the probability that x is less than or equals to two then it means any value including two so it will be the probability that x is equals to zero plus the probability that x is equals to one plus the probability that x is equals to two and that will be zero comma three plus zero comma one plus zero comma two four correct yes please and that will be zero comma six and the last question the probability probability that all students will attend the class is the probability of all students therefore we can also collect the probability that it's greater than or equals to one because it's referring to all students when it's referring to all students zero does not refer to all students so the number of students who missed class zero means zero students oh who missed class sorry wait the number of students who missed class are those ones so it is equals to all students who missed class we need to be very careful with this so this will be all students who missed class I need to write it correctly inside here all missed class students or all students who missed class something like that so we need to be very careful when we answer this question the probability that all students miss class the probability that not miss class the probability that all students will attend oh gosh the probability that all students attend the class so that is the same as the zero so zero means all students attended the class they didn't miss because this says our x is the number of students who miss a class so zero will mean they all attended the class one will mean one student missed the class two students missed the class three students missed the class four students missed the class and five students missed the class zero will mean no students missed the class so that will be the probability that all students attended will be equals to zero and that will be equals to zero comma three zero therefore option number three is the incorrect one so you need to think what i'm saying given easy yes what i'm saying can you why i'm confused with the zero and yeah you say x is right okay so x is the number of students who miss a class okay remember that okay is the number of students who are miss or who missed a class if it's zero it means no student missed a class right if it's one it means one student missed a class if it's two two students missed a class if it's three three students missed a class so now if we say zero is no student missed a class therefore it's the same thing as saying all students attended the class and that day because one will mean not all students attended a class two will mean not all students attended a class three will mean not all students attended a class because three students would have missed a class the question we were asked was that all students attended the class if they would have said all students did not attend the class then it's a different question it's a different total question altogether please yes i am confused let's go back to question four it says the probability that at most two students will miss a class on friday yeah then when we calculated we included the zero yes yes we would include zero because at most at most the first two either all students came to class at least oh at most one student came to class at most two students came to class it includes all of them it includes an instance where everybody was in was in class because if you say at most when you say at most it will include even those who on those days where everybody came to class okay the tricky part would have been if they would have said at most two students attended the class we have we don't have that the we don't have information about that where there were only two students in class we don't have that we the only thing we know is either everybody in the class came or we know that one student missed the class we also know that two students missed the class that's all what we know and that's what we can calculate and if no student missed the class in a way is the same thing as everybody has come to class on that day because nobody missed the class so when they give you sentences you need to think long and hard in terms of understanding and unpacking the statement the the question in relation to the statement and you will find in most cases you will always do that you will always have to go back and and think about what you are given in relation to what you are okay right moving forward if there are no other questions or clarity on that based on the past experience a researcher knows that the probability distribution of X is equals to that G is supposed to be equals is equals to the number of students who come to her office on a Monday we did a similar question before but now we have different statements which of the following statement is correct so remember that X represents the number of students who come to her office on a Monday they can be zero students so it means no student came to to visit her office one student came two student came three students and so forth number one what is the probability of at least number two what is the probability of at least two calculate the expected value of X and calculate the variance and calculate the probability that X lies between two and four let's do the working we're looking for the correct answer now are we winning you have an answer nobody has posted anything are you still struck by the all student question okay so let's do it together since I see nobody has answered the question messlers we're so busy can you give us a piece of but I'm asking and you're not answering so I'm you need to talk to me okay I'll give you two minutes please I'll give you five more minutes it's been five two we can look at it thank you thank you go make myself coffee when I come back when look at it okay are we done so let's silence means you are done the first question probability that at least one student so to answer that will be the probability that X is greater than one also the same thing we can do here because instead of adding all the values going there because it's all of them we can say one minus the probability that X is equals to zero which is one minus zero comma one which will give us zero comma nine happy no please please please instead of eating all of the values because the same thing as zero comma five plus zero comma two which gives us zero comma seven plus zero comma one five will give us zero comma eight five plus zero comma zero five will give us zero nine in state of doing it that way the probability of X greater than one will be equals to zero comma five plus zero comma three plus zero comma one five zero comma zero five which is the same as zero comma nine we can take a shortcut and say a compliment because there is only one in this slide you can say it's one minus that one okay because the sum of all probabilities are equals to one instead of spending more time calculating and adding so many numbers you can just use the one that is left out of the peg to calculate the rest of the other probability so the probability that at most two students will be the probability that X it's less than or equals to two and it means it's all of them going there we can also say it is the same as the problem one minus you can do it this way as well one minus the sum of all the greater than probabilities one minus you can say it will be one minus the probability of X greater than two which is the compliment so this is a compliment of less than or equals to which is the same as one minus the probability of X equals to three plus the probability that X is equals to four I know you did it the other way around I just wanted to give you an alternative which will be one minus zero comma one five plus zero comma zero five which should give you um zero comma eight so if you have done it the other way around you would have said it is zero comma two zero plus zero comma five zero plus zero comma one zero it will give you the same answer you would have gotten the probability that X is less than or equals to two will give you zero comma one plus zero comma two plus zero comma five will give you zero comma eight eight zero which is the same thing as doing the compliment probability so you can use either one method either one of them okay so the next one was to find the expected value so the finding the expect finding the expected value you can use the table and multiply so zero times zero comma one will be zero one time zero comma zero five will be zero comma five two times zero comma two zero will be zero comma four three times zero comma one five will be zero comma four five four times zero comma zero five will give us zero comma two zero and this is x times px and if we want to put the summation then we just need to add all of these values and that will give us if we add all of them together the sum of all of them will give will give us 0.05 plus 0.04 is 0.09 plus 0.045 plus 0.20 gives us 1.45 1.55 am I in the right place 1.55 yes 1.55 this would be equals to 1.55 and we know that that is wrong this is going to be wrong as well the variance with the variance we need to subtract so I'm going to say the variance I'm going to do the x minus some x minus the expected which is 1.15 squared times the probability which is 0.10 let me know if you're not getting the right the answers as I am getting which is 0.2403 plus 1 minus 1.55 squared times 0.05 I get 0.1513 2 minus 1.55 squared times 0.3 0 I get 0.0405 do you all get the same plus 3 minus 1.55 squared times 0.15 I get 0.3154 plus 4 minus 1.55 squared times 0.05 I get 0.3001 do you all get the same yes yes by edge all of them 0.2403 plus 0.1531 plus 0.0405 plus 0.3154 plus 0.3001 equals I get 1 comma and I'm going to round it off 1.05 do you all get the same yes thank you yes so that means that is also incorrect the last one it says find the probability that x lies between 2 2 and 4 so if I go to my x lies between 2 and 4 so x is less than oh x is greater than 2 so it goes there but it is less than or equals to 4 so it starts from 4 and it goes there but it does not include 2 so therefore it only goes 2 so this will be the same as finding the probability that x this will be x is equals to 3 plus the probability that x is equals to 4 which is 0 comma 1 5 plus 0 comma 0 5 which is equals to 0 comma 2 which means also that one is incorrect and the correct answer is number 2 in the exam you don't have to go through all the statements the students will get the answer as 2 just take 2 and kill on so yeah it's for practice paper so that you can get used to how we solve the probability solution questions we left with 15 minutes I think I have only three more questions left so let me just double check two I have four questions five four five some things some way went wrong with my number of counting all five questions left we can do the next one we can do it together now without doing with um looking at it I just want to do it for you the probability distribution of this discrete variable x what is the probability that x is greater than one so we're looking for x greater than one that's not include one so those are the probabilities so it will be the probability that x is equals to 2 plus the probability that x is equals to 3 so it will be 0 comma 2 plus 0 comma 1 5 which is 0 comma 3 5 happiness yes yes correct I'm going to do 8 and then you do 9 you will do this no we will skip 9 you will do 10 let's see 10 yes you will do 10 let me just do 8 all together so the probability that x is less than or equals to 2 or x is at most 2 is given by the probability probability that x is equals to 0 because it is less than or equals to 2 so it will include 0 x is equals to 1 plus the probability that x is equals to 2 and that will be 0 comma 2 5 plus 0 comma 4 0 plus 0 comma 2 which is 0 comma 2 4 6 5 the next one it says it is between so we're going to look at the between it says x is greater than one so it's at one and it's greater than one so it goes that way but it is less than or equals to 2 so it's up right here and 2 and it goes there so it only goes to so it means it's the probability that x is equals to 1 plus the probability that x is equals to 2 happiness which then is 0 comma 4 plus 0 comma 2 which is equals to 0 comma 6 next one says the probability that x is less than 1 so with less than 1 it does not include 1 so it is any value less than so therefore it's the same as the probability that x is equals to 0 and that will be 0 comma 2 5 which means also that it is correct the last one says x lies between 0 and 1 so says x is greater than 0 so it does not start at 0 so any of the values greater than 0 so greater than 0 it means starting from there not including one not including zero but it is also less than one so therefore it means it starts from less than or equals to 1 it starts from there and it goes and it gets dropped from there so therefore it's the same as the probability that x is equals to 1 and that probability is 0 comma 4 and that is the incorrect one and this is your homework you can go and do this they have calculated already the mean so it means you can just use the mean to answer the question because they want you to calculate the variance which is sigma squared or vx which is the sum of your observation minus the expected squared times the corresponding probability they already gave you the expected value all you just need to do is substitute and calculate the rest of the variance that is your homework your homework and this one this you can do in uh also homework I want to give you the one that you can do in five minutes you can calculate number 11 what is the expected value of the value of effective world that you can do now are we winning okay let's calculate that so zero times 0.6 is zero? zero yes 0.2 and two times 0.1 zero point two zero point two three times 0.05 zero point one five and four times zero point zero three zero point one two and 5 times 0.01 and 6 times 0.01 0.06 and we can just add them 0 plus 0.2 plus 0.2 plus 0.15 plus 0.12 plus 0.05 plus 0.06 and the answer we get is 0.78 which is option number three. This you can also do as homework if you want to go through it. I think this comes from STA 1610 exam paper of 2019. May, June 2019 is the only exam paper on your site as well so you can go through it and please be reminded when you when you get hold of past exam papers from before 2020 any past exam paper before 2020 because they ask and papers you will notice that there will be missing dots. I just replaced the dots here but on most of the documents you will realize that they are missing dots. You when you see a space you must know that there is a dot there. Okay so this you can also do as homework and this should be 12 not 11 and that concludes today's session. Any question? So we are on the route or on the way to completing our study unit five in order for you to be able to do your assignment too. By next next week Wednesday we would have we will cover the rest of study units five which includes bilomia and also poison distribution and then on Saturday we are going to do all the activities. I might have a challenge on Saturday because I'm going to a place where the network is not that stable but I will let you know if we are unable to connect I will record the video and post it on my UNISA for you to be able to continue and be able to do your assignment. So I will record the session for Saturday but if we are able to connect on Saturday and we are able to continue then we will continue so but let's see how it goes on Saturday. If the the connection is a challenge then I will record the session when I get time and then post it or post it for you so that you are able to continue. Otherwise I will see you on Wednesday and if there is any question or query feel free to ask. Thank you for joining the class today. Any comment or question in the absence of comments or questions or queries?