 Let us say we will begin our instruction today. So, start with the review as usual. So, last day that means day before yesterday, what we did was to discuss first the Bayes theorem famously invented by Thomas Bayes some more than 200 years back and I have told you that application of Bayes theorem or the whole Bayesian domain of analysis in statistics and probabilities a lot more than what we had as Bayes theorem. And if you have checked your Moodle page of this course, then you might have seen that I have uploaded one or two links for additional readings for fun. You can get to know more about what Bayes theorem is, what are the different application of Bayes theorem is. It is a lot wide and it is amazing I mean where it has reached over the last 200 years, not actually last 200 years I mean probably within last 100 year or so, because after Bayes had developed the theorem, nobody talked about it for a while then it started again. So, we have gone through one or two examples on Bayes theorem. Then we discussed random variables, we discussed their definitions and then we discussed basic functions of random variables. For discrete random variables and continuous random variables, we have functions like the probability mass function, cumulative distribution function and probability density function and also discuss their properties with some examples. Finally, we just touched the joint distribution part with an example for joint probability mass function. Today's lecture starts with the definition of expected value of a random variable which is denoted by this notation E within square bracket x, x is the random variable. The definition given here is for the discrete one where you define it as the summation of this quantity x i times the this is the probability mass function. If you remember small p stands for mass function capital X in the subscript is the random variable under consideration and x i is the value that the random variable takes. So, if we sum it up over all the i values then we get the expected value of that random variable. This is how we define it for a discrete random variable and this is the definition for a continuous random variable where we have the same function x times instead of mass function now we have the density function and we instead of summing it up we integrate it over the whole range that means minus infinity to plus infinity. So, basically if you see their same formulas one applies for the discrete case, one applies for the continuous case. What it is basically if you look at it in detail is the weighted average of all the values that the random variable x can take. Where is the weight? This is the weight. So, this is basically nothing but the frequency it will more clear if you look at the discrete case. So, this is the frequency mass function is equivalent to frequency. So, we are weighting the value that x is taking x i by the frequency and you are taking a weighted average. So, the expected value is nothing but the weighted average of all the values that the random variable x can take. It holds good for both discrete and continuous random variables. This quantity the expected value of x is also known as the first moment of x. We will see a little more about that later. As an example I cite a very simple one for a discrete case. When you roll a fair die of six side then you have the expected value of x, x is the outcome of rolling a die. So, this is the value 1. What is the likelihood of that? What is the PMF probability mass function of that? It is 1 over 6 because all the numbers from 1 to 6 are equally likely. So, 1 times is PMF, 2 times the PMF, 3 times the PMF that is wrong it should be 6 times the PMF. If you add them all up you get 3.5. So, 3.5 is the expected value of the variable x where x is the outcome of rolling a fair six sided die. So, the question is it same as the mean of the random variable? Is expected value is same as the mean of the random variable? If you look at this example of course, you have this six numbers 1, 2, 3 and 4, 5, 6 the range is from 1 to 6 all of them are equally likely. So, you can easily see that 3.5 is nothing but the mean. Also if you look at the previous definition that it is nothing but the weighted average of all the numbers. Then also you can say that this is the mean of the random variable. We have discussed about mean earlier. So, I am going to what mean is anyway. So, we have a new name for mean which we call now expected value when we are talking about random variables. We can also define expected value of a function of x. So, g x over here is a function of x and we can obtain the expected value of that function of x as g x multiplied by again the mass function that is the weight and integrate it over minus infinity plus infinity. So, this is the definition for a continuous random variable. Similarly, you can have also a definition for the discrete random variable and you can obtain g x i multiplied by the probability mass function. Sum it up and you get the expected value of a discrete random variables function. So, this should be g x i will make that correction. So, this is e g x just like this one and I told you about moment. We said that the expected value is also known as the first moment. Expected value of x is the first moment of x. Here we discuss about the nth moment of x which is defined as expected value of x raised to n and how do you obtain this? Of course, x raised to n is a function of x. So, we write that function of x times the pdf of x and we integrate it from minus infinity to plus infinity. That means the whole range and you obtain the expected value of x raised to n and this is known as the nth moment of x. So, automatically if you put n equal to 1, you get the first moment of x and if you put n equal to 2 as you have put over here, we get the second moment of x and so on. It is clear? Many of you are familiar with these things, but this definition of moment I want to clarify again. Many a times people assign these terms moment of x or so to various other functions. You need to understand the differences. Basically, moment of x is the expected value of x raised to n that is the nth moment of x. Sometimes people also assign this name to a function of expected value of x n which is not really true, but you can say that it is order, it is of that order of that order of the nth moment of x or so. I will come back to that. The other very important property of a random variable distribution is its variance and how do you define variance is denoted as var and x within a square bracket is defined as the expected value of the quantity x minus mu x. What is mu x? What does this symbol define mu x is? The mean of the random variable x. As you can also see that x in the subscript is written in capital letters which means that x is the random variable and mu denotes its mean. So, x minus mu x square, you are taking an expected value of it which we call the variance of x. You can also write that variance of x is nothing but expected value of x square minus square of the mean. We will see later on how these two things are same. So, expected value is same as the mean we have studied that earlier and we know that mean is a measure of central tendency of a distribution. The variance on the other hand is a measure of the dispersion or spread that you can have in a distribution or for a random variable. So, two measures first two measures one the first one that is very important is of the central tendency then we go for the measurement of dispersion or spread in the distribution of a random variable. Do you know what are the next measures of distributions? Have you heard any terms? Did you hear skewness? Did professor Condon talk to you about skewness and kurtosis? So, those are the next level or next higher order of defining the spread of a random variable or as the spread of a distribution. First one the central tendency of a random variable then the dispersion random variable then you go for those two skewness and kurtosis and each one we go we go one level higher in the moment of x. You have seen that the expected value is the first moment of x variance is here you can see is a function of the second moment of x and so on. So, that is what we say over here that variance is a function of second moment of x. Many a times we just wrongly say that the variance is the second moment of x it is not really true, but well you can say that variance is also a measure of the dispersion or spread in x. The same parameter has another notation it is sigma square variance is also denoted by sigma square and with a subscript x x written in capital to denote that it is the variance of the random variable x. And we will also see that the square root of this quantity this variance is sigma x where sigma x is nothing, but the standard deviation of the random variable x. So, although this symbol does not necessarily mean that it is square of this quantity, but it actually is, but we will better look at this symbol as sigma square not as square of sigma is that it is like just like any other symbol like variance and square brackets there you have a symbol of sigma square not necessarily square of sigma although is the same quantity. So, we come back to the two different definitions for variance we said that those two are equal and we will see how. So, you are trying to equate these two expected value of x minus mu whole square and expected value of x square minus mu square. So, I have just dropped the subscript x, but mu here stands for mu x for this case because there is no other random variable this is how it goes. So, what we do first we expand that parenthesis x minus mu square we get x square minus 2 mu x plus mu square. And then we find the expected value of each one separately. So, expected value of x square expected value of 2 mu x expected value of mu square. In the next step we have this then you take the mu out of the expectation based on what consideration the mean or the expected value same thing of a random variable is not a random variable the mean of the random variable is a constant or deterministic value. So, what is its expected value its value does not change. So, the expected value of mu is same as mu expected value of any constant which is not a variable is the same constant. So, you can take that constant out of the parenthesis here that is what you are doing. So, we take 2 and mu out of the parenthesis we take mu square out of the parenthesis because those two are deterministic quantities not random variables. So, then you simplify we get this which also means mu is nothing but the expected value of x. So, you can also write it in this way very simple proof, but it gives you the idea of what how the operator expected value works on a random variable how it works on a constant value. We will see more of that when you discuss about the properties of expected values and variances. Now, an example I will take the same one again I have put the same mistake over here rolling of fair die six sided die. So, we first find out see it involves finding out the expected value of the x square and this one of course, we know already. So, the second moment of x or expected value of x square we obtain by x square times the mass function x square times mass function. So, on again this should be 6 not 7 I am sorry for this this is what we obtain as the you can say mean of x square or expected value of x square. So, the variance of x is this value 91 over 6 minus the mean that you have already obtained. So, this is what you get as the variance of that. So, I gave a 2 examples of obtaining expected value and variance for discrete random variable. Can you repeat that for a continuous random variable? Very simply we just have to replace the probability mass function by probability density function and the summation by an integral minus infinity to plus infinity ok. So, process thus will come next and we will talk more about continuous random variables. So, I will try to give my examples more on the discrete side. Then we come to the properties of these two quantities the expected value and variance of random variable x. So, if we have these two constants a and b constants means they are not random variables they are deterministic quantities we know their value exactly what they are. So, for that if you want to evaluate the expected value of a x plus b what is this? This is the function of x right function of the random variable x and how can you obtain this? You can obtain it by the formulation of if it is a discrete random variable then you can write g x times mass function of x sum it up over all i values. If it is a continuous random variable then you can write the function times the pdf of x and integrate it over minus infinity to plus infinity range ok. So, what you get is a times expected value of x plus b why because the constant value has an expected value equal to the same value right. So, you can take it out and you can take this out ok. Just like we did for the previous example where we tried to equate the two defined two different definitions of the variance. And if you are trying to find out the expected value of two different random variables x and y then we do the same thing it will be expected value of x plus expected value of y ok. X minus y again expected value of x minus expected value of y we will see a more general definition for this. Just a simple proof of y e x plus y will be equal to e x plus e y. So, we write e x plus y as this. So, here you can see although I did not specifically define it over here how do you define the expected value for a function of two different random variables. We write the function like g x y times the pdf. Now, instead of a pdf we have a joint pdf. Remember we discussed joint distributions in the last class. So, we have a joint pdf evaluated at capital X equal to small x and capital O equal to small y. And we integrate it over both x and y for the whole range of x and y that means minus infinity to plus infinity. Is that definition do you see the similarity with the previous one? Very simple just replace the single pdf with a joint pdf. So, now we break it up x times the joint pdf and y times the joint pdf two different integrals. The first one gives us expected value of x the second one gives us expected value of y. How? If you integrate this with respect to y what do you have? If you integrate y from minus infinity to plus infinity do you remember that example that we discussed on the joint probability mass function that you if you integrate on the other one then you get the individual density or mass function for x. If you integrate over x for the whole range of x you get the density or mass function of y. So, just considering that this one will reduce to single integration x f x x. This one will be reduced to single integration y f y y. Finally, you will have expected value of x plus expected value of y and this is the more general definition. If we have a function of multiple random variables. So, y is a function of random variable not one, but many. What you have over here are random variables x 1 to x n and we have a linear combination. So, a is here is a constant plus a 1 another constant times random variable x 1 a 2 another constant time random variable x 2 and so on up to a n x n and the expected value for this is expected value of y equal to a 0 plus a 1 times expected value of x 1 and so on up to a n times expected value of x n. Does that make sense to you based on what you discussed just now and previously like expected value of x plus y is expected value of x plus expected value of y. We are using that also earlier we said that expected value of a x plus b where a and b are constants is x a times expected value of x plus b. So, you can take the constants out of the expectation and you have a linear combination of the expectations. So, this is what we have a i expected value of x i summed up from 1 to n. Now, if you look at the variance the variance of a x plus b is a square times variance of x. And the b the other constant the lone constant which was not associated with random variable does not appear in the variance. So, that lone constant does not change the variance why is it any idea. When we subtract the mean that b constant will also appear. So, it will get subtracted. Subtract the b from that in the expression for variance. In the definition for expression for variance you have a x square minus mu square. Mu square. So, there the b will cancel out. I would like you to have a more physical feeling of that. So, when we discuss the continuous distributions like normal and normal then you will see how that constant not associated with any random variable does not really affect the variance in the system. Again we take that general linear combination of random variables y equal to a 0 plus a 1 x to a n x 1. For that the variance we have is the summation of a i square variance of x i. So, variance is multiplied by a i square not a i mean was multiplied by a i and this constant a 0 does not appear in the variance. Same thing it is a second moment. So, that is why you have square and of course, this one cancels out just like your friend said. We have another measure of dispersion. It is a relative measure of dispersion. We use it often times where we define the coefficient of variation v with a subscript x again x in capital because it knows the random variable x define as the ratio of the standard deviation to mean of the random variable. And you can express it sometimes in terms of percentage and usually it gives a better idea of the spread usually why? Let me give you an example. Let us say you are you are you are you are considering two random variables comparing that is why you use it more often use this coefficient of variation more often. You are considering two random variables like age of the faculty members one random variable and age of the students both in IIT other random variable. Let us say the mean somewhere around for the first one 40 and for the second one 20. Now if you count standard deviations for the first one you have 5 for the second one you have 3. So, now if you consider absolutely the standard deviations you see that the standard deviation is more for the variable age of faculty in absolute term 5 greater than 3. But if you are considering in relative terms you have 5 over 40 for the faculty's age and 3 over 20 for the students age. And I would say that there is more spread in the variable students age. Do you see my point? Do you agree to what I am saying? Most spread in a relative sense. I will give you another example. Let us talk about population ok. Let us say a country has a population of 1 billion consider another country having a population of 1000. There are countries like that. Now if you consider the spread let us say a population of that country over some years is your random variable. The spread in the first one having a population of 1 billion is let us say 1000 that is the standard deviation. Now you consider the other case where the population mean is 1000 and you have a standard deviation of 1000 same standard deviation. Would you call that the spread is same for the two populations? Not really right. So, standard deviation is not really a good measure of spread all the time. Did I get my point clearly to you? I will repeat that example again. You have two populations one having a mean of 1 billion and a standard deviation of 1000 ok. Another having a population of 1000 and let us say another having a population of 10000 and a standard deviation of 1000 ok. So, in which you see more changes of course the one with a population 10000 right because changes can be more drastic in this. Once you will have a population of 5000, once you will have a population of 150000, 120000 and so on. While the population close to a billion will not change that much if the standard deviation is only 1000. So, it makes sense to divide that standard deviation by the mean to have the idea of equivalent spread ok. Now if you compare these two statistics in terms of the coefficient of variation in one case you have 1000 over 10000 ok 10 percent coefficient of variation. In the other case you have 1000 over a billion much much less ok. And it makes sense to compare these two in terms of the coefficient of variation rather than in terms of the standard deviation, but you cannot apply this thing all the time particularly when the mean is close to 0. If the mean is close to 0 then the coefficient of variation goes towards infinity and mean having close to 0 is not a big deal. Many a times you are dealing with random numbers which go towards the negative direction and positive direction both and you might have a mean close to 0 ok. Ideal value close to 0, expected value close to 0. There it makes no sense to divide the standard deviation by the mean and compare the coefficient of variation ok. So, you have to use a judgment to see which is a better measure of dispersion, standard deviation or coefficient of variation ok. There is another related parameter covariance which you define for two random variables when you have a joint distribution of x and y. Then instead of variance we use the term covariance which defines a joint spread between the two ok. And how do we define it? The definition you can see is very similar to the definition of a variance for a single random variable. For a single random variable x what we have is expected value of x minus mu x whole square. Now, since we have another random variable we keep the dimension same what we do? We replace 1 x minus mu x by y minus mu y. So, we have the measure of joint dispersion in a joint distribution covariance written like this covariance of x y is measured at the expected value of x minus mu x times y minus mu y ok. We will not go into the detail of this only when we go into again continuous distributions probably this will be more useful. Now, for the later part of the talk we are going to discuss about some special discrete random variables. There are various common ones which we use and apply in our daily calculations in science and engineering. The one most common is the Bernoulli and binomial distributions. When you have the Poisson distribution hyper geometric, geometric, negative binomial and so on. We will discuss the first two start with the Bernoulli and binomial. So, x will be a Bernoulli random variable if for an experiment the outcome of the trial is binary. That means you have either this or that no other only two options. So, it could be failure and success and you denote 1 by x equal to 1 the other by x equal to 0. That is the normal way of denoting a Bernoulli random variable or the result is yes or no ok. Does it occur or does it does it not occur that is what yes or no. You put a value equal to 1 for this you put a value equal to 0 for the other that is a binary outcome. For that if you can write that probability of x equal to 1 is small p automatically probability of x equal to 0 becomes 1 minus p because there are only two outcomes. If that is the case then x is a Bernoulli random variable. What is the expected value? We compute using our basic formula. It is a discrete random variable. So, expected value summation of all x i times the probability mass function. So, there are two values 1 and 0, 1 times the mass function which is probability at x equal to 1 and probability of x equal to 0. So, this is 1 times p and 0 times 1 minus p finally, you get p. So, this is the expected value. Does that make sense? Now the outcome is either heads or tails you keep on doing the that. Then the expected value if the probability of head for each outcome is p the expected value over several will be equal to p that is a Bernoulli random variable. Then you go to binomial which is an extension from the Bernoulli random variable is that the random variable y will be a binomial if y is the number of successes in an independent binary trials. Both are very important. Independent is very important that two trials should not have any relation to each other. They are all independent. If this has occurred it does not mean anything to what is going to happen next. So, for n independent binary trials the number of successes is y and y will be a binomial if for each trial the probability of success is p and automatically the probability of failure is 1 minus p. So, it is an extension from the Bernoulli. What did you say about Bernoulli is x will be a Bernoulli if for each trial it is a binary outcome 1 or 0. Now you are saying y is the number of ones or number of successes among n independent trials independent binary trials. Then y is a binomial random variable. We will see example and see how and y is denoted as a binomial with parameters n and p that is how we write it. So, now we are looking at the mass function of the binomial random variable. So, mass function as always with a small p suffix capital Y at y is n choose y p raise to y and 1 minus p raise to n minus y. I want you to spend some time on this and see if this is reasonable or not. Look at the two quantities p raise to y and 1 minus p raise to n minus y. So, we say what is the number of success y and the number of failure automatically is n minus y. Probability of success in each trial is p. So, p raise to y based on the basic principle of counting if you get back to two lectures earlier and probability of failure is 1 minus p multiplied so many times that means n minus y times number of failures. How many ways you can choose number of successes y out of n that is n choose y. So, you have this formula of the mass function that means y equal to y number of successes in n independent trials equal to y is n choose y times p raise to y times 1 minus p raise to n minus y. Of course, for y equal to 0 to n only whole numbers. So, binomial random variable of course, only whole numbers you cannot have anything else. So, many successes out of n has to be whole numbers and has to be between 0 to n can be negative can be anything else. And we can compute the c d f from this mass function c d f is how many times y is less than equal to y capital y the random variable less than equal to y number of successes less than equal to y. We can obtain that by summing this mass function from all values of y starting from 0 to small y that is what we are doing 0 to small y we are adding up this p m f values and we get the c d f. So, this formula gives you the probability that the number of successes is less than equal to y out of n independent binary trials. There are a lot of application of this it is a very common one if you look at any text book you will see a lot. Let us first look at the expected value of the binomial random variable and also the variance. And for this we will use a simple technique we will write y the binomial random variable as summation of x i i ranging from 1 to n where x i is 1 if the success of each independent trial is it is a success in each independent trial and 0 if it is a failure. So, is that making sense to you. So, let us say total you have 10 trials in 5 you have success. So, what is y 5 of course, and x i for the successful ones are 1 each. So, 5 times 1 you get 5 is that ok I will just go for a simple example show. So, and x i is 1 for success 0 for a failure. So, if we have let us say 10 independent trials. So, I will just write them down first one failure second one failure success failure success these are different x i values what is y summation of all this. So, we can write y as summation of x i is if x i denotes 1 for success 0 for a failure then y is summation of x i which is total number of success that is how we represent it. So, if we represent it that way then expected value for each x i is what we know p that is the probability of success and the probability of failure for each binary trial is 1 minus p. And then we can obtain the expected value of y as the summation of expected values of x i i going from 1 to n. So, the expected value of the binomial random variable y is n times p a simple way of looking at it. Similarly, we can also obtain the variance of the binomial random variable y. And again we use that same x i where y is the summation of x i x i denoting 1 for a success in an independent trial and 0 for a failure. And first you write that expected value of x i square is same as expected value of x i equal to p how is that let us go through that. So, expected value of x i square what are the two different values that x i can have 1 and 0. So, 1 square times its probability which is p plus 0 square times its probability which is p plus 0 square times its probability. So, finally, you have p which is same as expected value of x i. So, with that if we have e x i square equal to e x i equal to p then variance can be easily written as e x i square that is what we needed minus mu square mu is e x i which is p square. So, p minus p square is the variance of x i and from that you obtain the variance of y as the summation of variance of x i. You remember that formula that we discussed earlier I will go back yes variance of y which is a linear combination of different random variables is a i square variance of x i. And now for this y is summation of x i there are no a values. So, variance of y is summation of variances of x i values. So, n times this which is p times 1 minus p simple ways of calculating it. And now we go for an example of the binomial random variable. I cite one example again from a civil engineering background. So, what we have is for a construction company the record says that 70 percent of the projects complete on schedule. And the company wants to find out what is the average number of projects to be completed on schedule for the next 15 projects. This is a problem of a binomial random variable where the success is completion on schedule and failure is not completing on schedule. Now, we treat it as a binomial random variable how? We consider completion of each project to be independent. We said that for a binomial random variable y for n independent trials remember that independent word was very important. So, this is the first assumption that completion of each project is independent may not be always true in the real life, but let us say here that is what it is. So, x is a binomial with parameters n p where n is the total number of independent trials 15 and p is the probability of success that means completing on schedule which is 70 percent 0.7. And we want to find out the expected value. We said what is the average number of projects to be completed? What is the binomial random variable is the number of success over n independent trials? We have 15 independent trials what is the number of success that is the random variable x and we are trying to find out what is the average value expected value of x. So, the mass function we know the formula n choose x any random number x. So, 15 choose x times p raise to x 1 minus p raise to n minus x and if we have the mass function from that we can easily obtain the expected value. How exit value is x times the mass functions summed up over all the trials that means 1 to 15. So, you have this formula this should give you the average of the expected number of projects completed on schedule out of the next 15. So, you can compute that and get the result. Alternatively you can also use the formula that I gave just two slides back that expected value of the binomial random variable x is nothing, but n p where n is the total number of trials and p is the success in each independent binomial trial. So, 15 times 0.7 equal to 10.5. Now, what you can do you can check yourself is if this formula gives you 10.5 or not. So, similar concept you can apply for various examples you know completion or non completion occurrence or non occurrence for all independent trials. So, you can apply it for having a flood of this intensity occurring a year that is on independent trial. Non occurrence in a year will be the failure. So, occurrence in a year is probability p, non occurrence in a year is 1 minus p. Now, you can compute what is the likelihood of having such floods in next 50 years. That will be a binomial random variable and you can compute how many times you expected to occur in next 50 years. Another common discrete random variable is the Poisson random variable and you see that we are writing p in a capital letter because that is based on a name of the person who first defined it. So, x will be a Poisson random variable if x can take values like 0, 1, 2 and so on only whole and positive numbers starting from 0 and has the following mass function and you have a formula like this e raised to minus lambda or lambda raised to x divided by factorial of x where lambda is greater than 0 it can be it can be a fraction it can be any other number, but greater than 0. Then we call x to be a Poisson random variable with parameter lambda. Here you just have a formula I will explain how to use it. The fun thing about this random variable is that it has the same expected value and variance both equal to lambda. Does it mean it has a very high spread same expected value and variance? No spread is measured by variance but it is the square root of variance that is the standard deviation which we usually use for that. So, depends on what is the value of lambda that will tell you what is the spread. You can check any book like the textbook that is being referred all the time process book on introduction of probability and statistics to engineers and scientists I think that is the name anyway. So, you can check any book for the proof that both expected value and variance is equal to lambda. Now application a Poisson random variable can be used to model the possible occurrence of an event at any point in time or space. So, given an interval could be in time could be in space what if this event occurs here or there or how many times in this range that can be very successfully modeled using a Poisson process. So, this is what a Poisson process is very model occurrence of things like earthquake over time. What is the likelihood that this earthquake occurs over next 20 years or earthquake in a seismically active location? Given this region take the San Francisco Bay area what is the likelihood that it occurs here or there or there in space you can also distribute it in space or the other example for distributing in space is the occurrence of fatigue cracks at some location in a well. You know what welding is right where you correct two metals using welding and they are very likely to have fatigue kind of cracks fatigue means which you have for repeated loading and unloading. So, if you want to locate for this length of weld what is the likelihood of having it here, here, here and so on that kind of process can be modeled using a Poisson distribution. It is very important to use this term mean occurrence rate in a Poisson distribution which is the average number of occurrences per unit time for a given interval t. Time can be replaced by space also. So, mean occurrence rate is average occurrence per unit for a given interval and it is denoted by nu and nu is equal to lambda that parameter which defines the Poisson distribution divided by t is the interval even in space. And Poisson become close to binomial if lambda equal to n p and n is very large and p is very small. I will try to give you some more example later how it becomes like that. Let us go for an example. We have this information that there have been three large earthquakes large meaning magnitude having more than 6 degree magnitude. So, we had three large magnitude earthquakes in the last 50 years in a given region let us say Uttarakhand and this occurrence can be modeled as a Poisson process. Then the question is what is the probability of having any such earthquake in the next 10 years? Very common use of a Poisson distribution. So, x is a Poisson process with nu the mean occurrence rate equal to 3 that is divided by 50. So, it is 0.06. So, with t equal to 10 and lambda equal to nu t we have what is the probability that you are trying to find out that the earthquake happens. That means, it happens at least once the probability x greater than equal to 1 is of course, probability of 1 minus not having it at all 1 minus probability x equal to 0. So, you put it 1 probability x equal to 0 is the mass function at 0. So, we use the formula again I will show you the formula it was e raise to minus lambda lambda raise to x divided by x factorial x is 0. So, you put that. So, this is the probability 0.451 or 45.1 percent that there will be at least one earthquake of magnitude 6 and larger in the next 10 years. Is that ok? Did you understand this? Similarly, you can also find what is the probability of having no such earthquake in the next 10 years and that will be no such earthquake means x equal to 0. So, p m f again at 0 and this is the one again I mean that would be definitely 1 minus the probability that you found earlier. So, it is 54.9 percent. So, these are occurrences of Poisson distribution. There are many other look at any book you will see many other examples things like discharge of alpha particles in a radioactive material and so on. You can model this using a Poisson random variable that concludes today's lecture. So, looking back these what we have gone through. Expected values, variances, their properties, binomial distribution properties as example and Poisson distribution properties application example. Thanks.