 Hi, I'm Zor. Welcome to Unizor education. I would like to start a new topic, mathematical statistics. It's presented on website Unizor.com. That's where I suggest you to watch this lecture from because there are very important notes for every lecture which is presented on this website. So it's basically a course. It's a course of advanced mathematics for teenagers, primarily high school, and well this is just a new topic which I would like to initiate. First of all, there is absolutely mandatory to study theory of probabilities as it is presented on this website, Unizor.com, before you attempt to listen to these lectures. It's mandatory because I mean everything, whatever mathematical statistic does, it's based on theory of probabilities. And there is very important connection between these two subjects. In some way, they are inverse to each other. Let's just think about it. What theory of probabilities actually tries to accomplish? Well, it deals with certain random variables. For instance, we have a random variable and we know everything about this random variable. We know it's probabilistic characteristics. We know it's distribution of probabilities, whatever you want to say about it. So we know the behavior of this variable. Now, based on this behavior, sorry, based on this knowledge, based on these characteristics and probability distributions, etc., we can actually say something about the values of this particular variable as experiments with this variable go through. So, for instance, if our random variable is the result of rolling the dice, and we definitely know that the probability of any number from 1 to 6 on a dice is equal to 1, 6, then we can say with relative certainty that if we perform, let's say, 1,000 rolls of a dice and we are interested in number 1, well, we can say that approximately 1,6 of 1,000 times, which is what? Like 1,6,7 or something like this, we will probably get number 1 close to this number. Not exactly this number, but close to this number. And we can even evaluate how close we can say that with certain probability, like 95%, for instance, the number 1 will appear, let's say, from 150 to 180 times or something like this. So, again, knowing the probability, theory of probabilities gives us tools to evaluate the results of the experiment. Now, the mathematical statistics is in somewhat inverse, as I said. We do know the results of the experiments in the past and based on these results, based on the values which we have obtained, we can make some kind of a judgment about probabilistic characteristics of our random variable, which we do not know in advance. So, again, theory of probability is from the distribution of probabilities, we predict certain values which it can take within certain boundaries with certain probabilities. Mathematical statistics goes from data to the probabilistic distribution and probabilistic characteristics of our random variable. So, in this way, they are inverse to each other. What actually constitutes a complete theory is the following. Again, considering we do not know our priori in advance the characteristic of this variable, so what we do, we do a certain number of experiments which result in certain judgment about characteristics, probabilistic characteristics of our variable. So, this is the mathematical statistics and then based on this information which we have obtained about our random variable, we make a judgment about future results. So, this is past and this is future and this is zero probabilities. So, that's a complete picture as I would like to present it to you. So, you will always have in mind what's the purpose of this and what's the purpose of that. So, again, the mathematical statistics serves to define in certain, well, with certain precision to define the probabilistic qualities, characteristics of random variable based on certain data which we have obtained from the past experiment and then based on this we can predict the future. So, theory of probability predicts the future, but the basis of this is the probability distribution which we, if we don't have it, we need to somehow obtain derive from the results and that's what the role of mathematical statistics actually is. All right. So, this is some kind of an I don't know, introduction to what exactly we will be dealing with. Now, based on this, now for instance, we don't know that one-sixth is the probability of number one if we roll the dice. What do we do? Well, we roll the dice one thousand times, right? And let's say one hundred and sixty times we have one. What can we say based on these observations? Well, we can say that approximately one sixty over one thousand which is zero point sixteen is the probability of having number one. We cannot make any judgment about any other unless we do know some results of any other like probability of number two, probability of number three, etc. But that's what we can see. So, if our experiment shows that out from a thousand rows we have one hundred and sixty times rolled number one, then we can say, okay, the probability of our random variable equal to one approximately is equal to zero point sixteen. Absolutely not exactly because experiment is definitely not completely conclusive, but it gives us a certain point of judgment. And then we can say that for the future, maybe we will make another one thousand experiments in the future, we might expect the number of times when we have rolled one be more or less around number one hundred and sixty. That's a judgment for the future. That's how statistics and theory of probabilities are working hand in hand starting from the data in the past to certain judgment about probabilistic characteristics and to predicting the data for the future. All right, next. Now, why actually can we say that if we have out of a thousand times, we have one hundred and sixty times number one rolled, then we can say that probability maybe is around zero point sixteen. Well, this is all based on the law of large numbers. So, if you will go back to the theory of probabilities and study the law of large numbers, you will understand that that might be a good foundation for this. So, that's basically where we are. That's the main apparatus, the main foundation for the mathematical statistics, which is at the heart of it. Well, obviously, there is another very important theorem, the central limit theorem. And that will also be very important in our predictions in the future. Okay. Now, let me say the following. Unfortunately, many people are abusing mathematical statistics. There are even some saying about this. There are lies, some kind of other lies in the resist statistics. Well, this abuse is basically nothing more than just using something in a completely unintended way. And instead of approaching with a good theoretical foundation, we approach something in a completely un-theoretical, completely spontaneous and, well, let me tell you, wrong way. Example. I mean, for instance, some agencies would like to predict the results of elections in the United States. Now, there are two major parties, Republicans and Democrats. So, one agency pulled 100 people and found that there are 60 goals for Republicans and 40 goals for Democrats. And that agency said, you know what, based on my information, I predict that Republicans will win. Now, another agency, or maybe the same agency in some next day or something like this, also asked 100 people and they had slightly different picture. They have 40 Republicans and 60 for Democrats and they say, okay, we predict that Democrats will win. Now, they cannot be both, right? So, somebody is wrong, which probably means that nobody is actually right. Both of these statements, like prediction of this win or that win, they're absolutely wrong. There cannot be a statement like that. What might be a statement is that with certain probability, the Republicans or Democrats will win. And that is usually completely missed in many cases. Also, what's important is this number. This number is supposed to be significant and we will talk about why. The greater this number, the more precise our evaluation is. And there are many other factors which are participating in this thing. You will see that the best results can be get if we want to evaluate the random variable, which is basically the experiment with which can be repeated again and again without any modifications. Now, in this case, 100 people, that's 100 different people, which probably supposed to represent a much larger number, but how well it represents, nobody really knows. So it's all very much unscientific. It's very much basically kind of analysis which maybe pursues some political purpose or something like this. And it's far from being mathematically correct. What I would like to talk about is what is the theoretical foundation of mathematical statistics to avoid confuses like this. And let me start with one simple example. Let's consider you have one random variable and you have only one experiment of this random variable which gives the result X. Now, what can you say about this particular random variable? Quite frankly, nothing much. If you have only one result, probably the best you can say is that the mathematical expectation of X of C is somewhere around X. So you're evaluating you're evaluating the mathematical expectation of your random variable using one particular one particular result. Now, how good is this? Well, not very good, obviously. Can you improve it? And here is the very important part of it. Excuse this noise, somebody is just doing something. All right, so how can we improve the result of the variation of the mathematical expectation of random variable C with one and only one value X? Here is what we can do. Let's consider that we can repeat this experiment of basically checking the value of random variable X of C n times. And we have values X1, X2, etc., Xn as a result of this. Now, the typical way people are approaching this problem is, okay, let's just have an average of these n results of experiment. And that would be our evaluation of the mathematical expectation of X. Is it good or bad? Well, it's very easy to mathematically analyze. And here is how I would like to do it. Let's consider a new variable. Now, C is our random variable. So let's assume that we have n different random variables which are independent from each other. Again, the word independent is assumed in a probabilistic sense. And you have to really go back to the theory of probabilities if you forgot about this. So independent variables which have exactly the same distribution as C. Now, what is this particular new random variable? Well, let's analyze it. Now, obviously, if I will take m equals to X1 plus, etc., plus Xn divided by n, where X are these now, these results of the experiment of C, I will have a one single value of variable eta. Now, can I do exactly the same as before? If I have a single value for variable, random variable C, I'm saying that, okay, X is probably as much as we can do to evaluate the mathematical expectation of C. But let's talk about this in this particular way. Now, the mathematical expectation of eta is equal to, well, again, from theory of probabilities, you know that the factor can be moved out of the expectation. And expectation of sum is sum of expectations. So it would be expectation of C1 plus, etc., plus expectation of Xn. Now, what is this? Now, we are saying that these variables all have the same distribution as C, which means they all have the same expectation as expectation of C. So this is equal to 1n. And then I have expectation of C, 1, 2, 3, 4, n times, right? Which is expectation of C. So all I have proven right now is the expectation of this average is exactly the same as expectation of my initial random variable C, which is good, which means that this can serve as an expectation, as an evaluation of expectation of eta. And since expectation of eta and expectation of C are the same, so this is also a good evaluation of expectation of C. Now, that's okay, but what about the quality of this expectation? Now, let's think about what is the quality of expectation of any random variable if you have only one particular value of this random variable. Now, random variable has certain distribution of probabilities. Now, these distribution of probabilities can be different and they might be arranged in such a way that our values are very, very close to the mathematical expectation of the random variable. Now, it can be something like, let's talk about, for instance, a normal random variable, which has a distribution, something like this. And this would be the mathematical expectation. Now, you can have this type of a distribution. Now, you also can have this type of distribution, which means values are much further distributed with relatively large probabilities. Or you can have this type of a distribution where values are much narrower. Now, the measure of this span around the mathematical expectation is usually the variance, right? The smaller the variance, the closer values are to the mathematical expectation. And, therefore, if we can reduce the variance, that would improve the quality of our evaluation. Now, let's talk about variance now. We were talking about a random variable eta, which is C1 plus, etc. plus Cn divided by n. What is its variance? Now, variance of, first of all, factor can be moved out of the variance in square. And again, if you don't remember, go back to the theory of probability, part of this course, so it will be 1 over n square. Now, variance of independent, now the independence is very important, variance of independent, where the sum of independent random variables is sum of their variables, right? Variances. So, I will have n variances of each one of them, which is the same as C, which is one n of variance of C. So, what happens with our result like this? What we have is, we have a variable eta, which has exactly the same mathematical expectation as C, but its variance is n times smaller. And, therefore, standard deviation is 1 divided by square root of n smaller, right? So, if my standard deviation is smaller, that means my single value evaluation and my single value evaluation of eta is x1 plus x2 plus, etc. plus xn divided by n. That's my results of my experiment. So, my n results of the experiments, I have basically converted into a single result of a different variable, which is basically, I'm just talking about the same, the same thing in different ways. So, yes, you can consider x1, etc. xn as n different results of experiment of n experiments with one variable, C. Or, you can consider as one single experiment, result of experiment, with variable eta. Okay. So, basically, what we have done, we have significantly reduced the variance and standard deviation of our initial variable, and therefore, we can increase the quality of our evaluation of the mathematical expectation of C with increasing number of experiments. That's what's very important. And that's what actually is all about in mathematical statistics. It replaces the quality with quantity. And in this particular case, the result is that the quality is growing as the quantity is growing. So, the more experiments we have provided for this particular variable, and the more the greater number n, obviously, is in this particular case, therefore, what we can say is that evaluation of our mathematical expectation with this number is greater. So, that's the foundation for providing more and more experiments. But, however, what's very important is that all these individual random variables should have exactly the same distribution as C, and they should all be completely independent from each other. Otherwise, these mathematics would not work. So, if you can roll the dice in such a way that every roll is completely independent from another roll, and you roll it, and you roll exactly the same dice within exactly the same environment, and nothing actually affects your experiments, like you don't really change the table where you're rolling, you don't shake it or whatever else, in this particular case, the greater number of experiments you are making, the more precision you can have in your evaluation of every result of every number of this, like mathematical expectation or some other probability, whatever you're analyzing. In this case, it's analysis of mathematical expectation, but at the same time, you can analyze any characteristic of a random variable. So, that's basically everything which I wanted to say in this introductory lecture about mathematical statistics. It's basically the purpose and the tools. The purpose is, as I was saying, kind of an opposite to probability, to theory of probability. So, not from probability to future data results, but from the past data results to probability. That's what mathematical statistic does. And the second thing is, what's very important is, all these conditions, we have to provide as much data as possible. That's the quantity, that's this n, and the conditions of the experiment should be exactly the same. Then we can really approach this mathematically correct. If conditions are changed or something else is changing, then it's not working that well. And we will consider some experiments like climate change, for instance, or whatever else. All right. So, I do suggest you, if you didn't do it before, look seriously at theory probabilities if you don't remember exactly this particular part of the course. And then, obviously, we can go into theoretical and practical aspects of mathematical statistics. Well, thanks very much and good luck.