 Hello and welcome to the course on dealing with materials data. In the present sessions we are working with random variables and its expectation. And today what we are going to work on is little different from that. Anybody who has a data or is dealing with a situation or an event always has a question as to what proportion of data would be larger than certain value or what proportion of data would lie between these two bounds or what proportion of value data will be more than or less than certain value. So these kind of questions are being answered in statistics by several inequalities. In today's session we are going to talk about two such inequality and one such law which is called a weak law of large numbers. So in outline let us see, we would like to introduce the estimation of proportion of data larger than the given quantity. We would like to do this through Markov inequality, secondly with Shebyshev inequality these are all improvement on one or the other. And finally we will introduce what is known as weak law of large numbers and let us start. So Markov inequality is defined as let X, if X is an random variable such that it is a positive random variable and there is any number a which is positive then probability that a random variable takes value larger than a that probability itself is smaller than expected value of X divided by a. The proof is very simple expected value of X is minus infinity to infinity X fx dx. So you can break the integral from minus infinity to a and a to infinity which is fine. Now because all the quantities are positive here each of these summation is positive therefore if you drop one of the quantities it will be smaller than the given quantity expected value of X. So you get expected value of X is larger than integral of a to infinity X f of X dx. Now here you are considering all values of X which are larger than a and therefore if you replace X by a the quantity becomes further small and therefore from minus infinity I think I have made a mistake here this has to be a to infinity. So please make the correction I will make it right here let us do that this should have been a to infinity just as this case and therefore and therefore we have expected value of a is equal to probability that or is equal to a multiplied by probability of X greater than or equal to a I hope I said expected value of X and not expected value of a. So it is expected value of X which is greater than a times probability of X greater than or equal to a and which proves the inequality. So what it really says is that how much of data can be expected above a given value a this bound is given by this it cannot be larger than expected value of X or the it cannot be larger than the ratio of expected value mean value of X divided by a quantity a this is called Markov's inequality. We move to another inequality which might be known to you in other cases also which is called Shebyshev's inequality again X is a random variable but now we are not assuming that it is a positive random variable so we are generalizing the case it has a mean mu and a variance sigma square then for any value k we are again not putting a restriction of k being greater than 0 as in the case of Markov inequality this is for any value k probability that the absolute difference between the random variable and its mean value is greater than or equal to k is smaller than the variance of the random variable divided by k square. This can easily be derived as shown here from Markov's inequality because X minus mu absolute value is a positive value you assume k to be greater than 0 because otherwise if this inequality does not make any sense and therefore for any value of k you have this. So probability of X minus mu square is greater than or equal to k square and then you put that expected value of this quantity and k square and which is this so you derived the Shebyshev's inequality I think here instead of saying any value of k, k is implicitly coming out to be positive so that may be noted. So both these inequalities provide bound on the probability when only mean or mean and variance are known to you for any random variable. In Markov's case it is only a positive random variable in the case of Shebyshev inequality we do not put such a restriction. Please note that no distribution assumptions have been made it is a normal distribution of course these distributions are to be introduced later but no distribution assumptions are made but we should know that if you know the distribution these bounds can be further refined and can be exactly computed. And this we move because this inequalities will be useful for us to prove a weak law of large number. What does a weak law of large number says? It says that if X1, X2, Xn are independently and identically distributed random variables we call it IID. So they are independently identically distributed random variable with expected value of Xi is mu for all the Is and for some epsilon which is greater than 0 probability the absolute difference between the arithmetic mean of or the average of the random variables X, X1, X2, X3, Xn and the mean value mu the absolute difference is greater than a small value epsilon tends to 0 as n tends to infinity this probability tends to 0 as n tends to infinity it means that the difference between the average of these random variables X1, X2, X3, Xn and its common mean value almost vanishes as n becomes large and large as your sample in future you are going to X1, X call X1, X2, X3, Xn or sample of size n. So from a random variable X you take a sample of size Xn which is your data. So as your data becomes larger and larger the arithmetic average the average of these sample is going to be very close the difference between this average and the mean common mean value is going to diminish as n becomes large and large. The proof is rather simple we have to make one assumption that there is this random variable variables have a variance sigma square which is finite. See there are cases in which you have infinite variance is also. So here we are defining that assume that variance of all X is sigma square which is finite then we apply the Shebyshev's inequality to this and then if you apply the Shebyshev's inequality it is obvious that as n tends to infinity this quantity tends to 0. So this once again this is a weak law of large number. Let me tell you why it is called a weak law because it is the probability which tends to 0 and not actually the difference which tends to 0. If therefore you call it a weak law of large number we are going to have a central limit theorem which is a stronger law of large number we will come across when we move further in this course. So what is the implication? What it says is that as I said before the sample average under the stated condition by stated condition I mean that the common variance of X1, X2, X3, Xn is exist and it is finite then it differs from its mean value by more than epsilon or rather the difference between the sample average and the mean value as n tends to infinity it eventually diminishes. So if it is more than epsilon that probability goes to 0 as the sample becomes large and large. So let us summarize what we have learned today. We have introduced some measures to find bound on the proportion of data being larger than given quantities. The first inequality we introduced was Markov's inequality in which we assume that the random variables are positive. In the Chebyshev's inequality we removed that restriction of random variable being positive. We said that it could be any random variable with having a mean value mu and a common variance sigma square and then we gave an inequality in which we showed that the difference between the mean and the arithmetic average actually the random variable X actually is smaller than the ratio of variance and the quantity k above which you are looking for it. Looking for the value to be larger or let me clarify very specifically it means it tries to say that the random variable X minus its mean value the absolute difference is larger than a value k. This proportion is smaller than the variance of X divided by k square. This is the bound it gives and finally we introduce a weak law of large number which indicates that under the assumption of finite variance the sample average differs from its mean value more than any small value epsilon it tends to 0 as sample size becomes sufficiently large. So, as sample size is large the difference between the sample average and the mean value diminishes. With this we cover up the complete session on the random variable and its expectations and next we will move on to the special distributions. Thank you.