 Hi, I'm Zor. Welcome to Unisor education. I would like to present a simple statistical problem right now and basically discuss certain approaches which we can take to solve this problem. Well, let me start with the purpose of mathematical statistics. Well, the purpose is having certain data from the past behavior of random variable, let's say. We would like to obtain certain evaluation of the distribution of probabilities of this random variable in order to be able to, let's say, predict the behavior in the future. So that's the whole line, from mathematical statistics to probabilities, not exact probabilities, obviously, evaluation, approximation of probabilities, and then using the apparatus of theory of probabilities, we can predict certain future results. All right, so I would like to talk about the simplest possible random variable. We're talking about Bernoulli random variable, Bernoulli experiment. Let's say we're tossing the coin or doing something similar. So it's an experiment which has two results. One of them has certain probability and another the opposite probability. So one is probability, let's say, p from 0 to 1 and the opposite event is 1 minus p. And we would like to basically evaluate what is the distribution of probabilities of this random variable based on certain results in the past. Now, the whole distribution of probability is such a big term. It's actually one number, the probability of taking one of the two events, let's say, p. And what we are doing is the following. We will have some random variable, c, and we assume that c is equal to 1 or 0, 1 with a probability p and 0 probability 1 minus p. So that's an assumption. And we want to evaluate the distribution of probabilities of this random variable, Bernoulli random variable, which basically comes up with a variation of one single number p. So what is the probability p which we would like to evaluate? Well, let's go back to the foundation of the theory of probabilities and think about what is the probability. Well, there are many different approaches to define the probability. And the one which I have chosen, which I believe is the most intuitively understandable, is that the probability of certain event is a limit of the frequency of occurrence of this event as we continue experimentation to infinity. Okay, that actually brings us to a very important approach to evaluate this p. Well, let's make experiments with this random variable and let's take the frequency of occurrence of this particular event when it's equal to 1. And maybe, since we are saying that the probability is a limit of this frequency as number of experiments goes to infinity. So maybe if we will go just to certain limit, like n, for instance, experiments, and we will use the results of these n experiments, the frequency of occurrence of this event among these n experiments, as an approximation to a probability. Sounds reasonable, because that actually corresponds exactly to the definition. And if our definition of the probability is, well, reasonable, which means, yes, we are defining as a limit, but it means that if we will provide a substantial number, a really large number of experiments, it will be more or less close to that limit, right? Obviously, we don't know how many experiments we have to conduct to achieve certain degree of closeness. But in theory, this is something which seems to be reasonable to do. So I would like to formulate mathematically this particular task. Alright, so first of all, we have to conduct a certain number of experiments. Let's say we have number n, which is number of experiments. Now, if it's a coin tossing, for instance, so we can toss the coin n times and see how many times the heads actually came up. If this is the heads, when our variable is equal to one, when the coin shows the heads. Or alternatively, we can have n coins and just toss them together at the same time. There is no difference between one coin tossed n times or n coins tossed once. What's very important is that they are completely independent from each other. So these experiments are supposed to be independent and under the same conditions, which means temperature in the room should be the same or everything else. So, now, what we do is, we make this, I would say, combined experiment which consists of n individual coin tossing or n coins tossed at the same time. So, we have this one combined experiment. So, we have basically a single experiment when we toss one coin once and we have a combined experiment when we have n coins tossed once or one coin tossed n times. So, combined experiment gives us some result. So, let's say results are x1, x2, xn. So, each one of them is either 1 or 0, because these are experiments with our random variable. Now, let's think about it. If you do this combined experiment and get this set of numbers, let's say I'm doing another combined experiment. Well, I will have another set of numbers, right? So, they're all 1s and 0s, but at different times different results will happen. Now, I would like to make this experiment not many times, just once. I mean, combined experiment. So, I make one combined experiment because usually all we do is we have a series of results of experiment and that's it. I mean, yes, somebody else can repeat the experiment. That would be a different story. I will use these numbers which I have received as a result of the experiment as a foundation to find an approximation for p. But what's interesting is that if somebody else does similar experiment, he will have different numbers and then somebody else will have still different numbers. What does it mean? Well, it actually means that the combined result of our n experiments is not a set of constants. It's actually, in theory, is a set of random variables. So, this is combined experiment. We are dealing with a set of random variables. Each one of them is exactly the same as this one. They're all independent and identically distributed as this one. So, we are dealing with n random variables. And based on these n variables, regardless of which value this series of n random variables takes, this or this, we would like to make a variation of this p. And the variation must be good, otherwise the whole thing is not really working. So, my task right now is to find out under what condition and how I can use just one single series of the result to get the good evaluation of the p. Which means I have to prove that if I will deal with random variables, and any value, whatever the set of random variables will take, would be relatively close to p. But I have to obviously define what does it mean close, what is the good approximation. Obviously, it depends on the number of experiments, etc., etc. So, I'm trying to formulate the task right now. Now, the typical way, as I was saying before, since we are evaluating the probability is to evaluate the frequency of occurrence of this particular event. Now, what is the frequency of occurrence, let's say, of c equal to 1 in this series of 1s and 0s? Well, that's sum of these, because sum of these is actually the number of 1s divided by the number of experiments. So, I have a new random variable, and this is the frequency, which is also a random variable. Because one person conducting this combined experiment will get one value of this, another person will get another value of eta. So, eta is a random variable, which is the result of the combined experiment on n tossing or n some other experiments. And now, what's very interesting is that we would like to approximate a constant, p is a constant, unknown constant, but still constant. We would like to approximate a constant with a random value, and that approximation must be good, otherwise, as I was saying, the whole approach is not good. So, that's the very interesting, I would say, philosophical dilemma, because what we are doing, just think about it once more. You are approximating an unknown constant with a random variable, which might take different values. But again, regardless of the values which it takes, it still must be relatively close to our constant p, which we would like to evaluate. If we will prove that this variable, eta, under certain conditions which relate to n and some other properties, if this eta would be relatively close and we have to define what does it mean relatively close, if we will prove that it will be relatively close to p under some circumstances, then the whole approach works, then the whole mathematical statistics as a subject make sense. So, I would like again to formulate the task. I would like to prove that under certain conditions, this particular random variable will be close to the constant which we don't really know. So, that's the task. The solution to this task will be in the next lecture. But again, let me just once more to emphasize the philosophy of the mathematical statistics. We are using random variable and it's random because different people conducting this combined experiment will have different values of eta, right? But still, regardless of that, we would like to prove that this eta will be close to p. And if we will, then we can say that mathematical statistics is a valid mathematical subject which makes sense. But we have to really define what does it mean to approximate, what's the quality of the approximation. We have to quantify it and under what conditions this approximation is good or bad and then we have to basically limit our expectation of quality of this approximation to a certain degree. Because if you will think about it, can this particular value be 0 for instance? Well, yes, it can because if every result is 0, then the sum will be 0. Can this result will be 1? Well, yes, because if these are all equal to 1, if we will have coin tossed again and again and it's always head, head, head and head. Then there will be n variables, each one of them is equal to 1 divided by n is equal to 1. So this sum can be 0, this sum can be 1. And I know actually a priori that p is a probability which means it can be anything from 0 to 1, right? So I can get 0, I can get 1, I can get anything. So how can I say that a random variable which can take any value from 0 to 1, can it be a good approximation of some constant? Well, given this problem in this particular way, the answer is, well, basically the approximation is not good. Since we as a result of the experiment can get any value from 0 to 1 and I know that p is from 0 to 1 anyway, so what good does it make? Well, it does because we know that 0 would be very rarely, it means that all n experiments should be the same and the probability of this is very low. So in our evaluation of the quality of our approximation, we should really get involved in the probability. We can say that this thing would be close to this thing within certain probabilistic terms. Like within, let's say, with the probability of 0.9 it will be within such and such vicinity of this particular point. So that's actually the definition of our task. We have to define exactly all these numerical quantitative terms and to define what is the approximation and what's the good approximation, what's the bad approximation and how to measure the quality of this evaluation of our constant p. Okay, so this is basically the presentation of the task at hand and the next lecture will be devoted to solution of this. Thank you very much and good luck.