 Hi, I'm Zor. Welcome to Unizor Education. This lecture is about continuous random variables, continuous distributions. This is part of relatively comprehensive course of advanced mathematics for high school students, presented on Unizor.com. And that's where I suggest you to watch this particular lecture because the lecture has notes, notes are presented on the website Unizor.com, and it's always beneficial to read the notes before or after the lecture. It's like a textbook basically, so I'm explaining during the lecture whatever is written in the notes. In addition, Unizor.com actually presents you the opportunity to enroll to take exams, etc. The site is free. It's basically a relatively comprehensive course of self-studying of mathematics or maybe some studying in in the environment of a flipped classroom, for instance. Okay, anyway, let's get to business. Continuous probability distributions. I have a little difficulty here. To present it in a relatively rigorous manner, which I prefer to present any topic in this particular course. I really need some calculus and I did not really touch the topics of calculus yet, so whatever I will say right now has a character of approximation, so I will explain what continuous distributions are. I will not really be able to define it properly, but I will define it with this approximation approach relatively precise, so I'm sure you will have the idea of what this actually is. And everything which, you know, regards to a little bit more tricky properties of continuous distributions probably are outside of this course anyway. So this is kind of an introductory to continuous distributions. All right. So, first of all, let's describe some process which we can actually meet the continuous distribution. Now, before we were talking about rolling the dice and we have only six number of occurrences, different outcomes. So the whole sample space contained only six elementary events. Well, in the previous lecture about discrete or probability distributions, I presented an example of infinite, but countable number of elementary events. So the elementary events was picking a number n from 1 to any infinity, basically. And I have defined the probabilities to pick the number n as 1 over 2 to the power of n. So that means that I will pick the number 1 with a probability of 1 half. I will pick the number of, let's say, 5 with a probability of 1 over 2 to the fifth degree, which is 1 32nd, which is much rarer. So I will be more often choose the, I will be choosing more often the earlier numbers and less and less frequently I will choose the numbers which are larger and larger. But the sum of these throughout sum of these for all n from 1 to infinity is a geometric progression. It's a sum of geometric progression and it's very easy to prove that this is equal to 1. So it satisfies all the properties of probabilities. Now, this is still a discrete distribution because there is a very finite difference between any two different elementary events and difference in probabilities and difference in whatever the representation we can choose. Now, let's talk about different example, which actually leads us to a continuous distribution. For instance, we are dealing with manufacturing of the tennis balls. Now, the manufacturing process is such that the weight is always in this range from 50 to 60 gram for any tennis ball. However, the precise weight can be any number in between these two. Now, if we are talking about mathematical model of this particular process, we should really allow as an outcome of this experiment of measuring the weight of the tennis ball to be any number, any real number from 50 to 60. And there are uncountable number of those. This is an infinity. This is a continuum as they call it in set theory. Now, what is the probability of our weight to be equal to, let's say 55? Well, let's just think about it. First of all, I'm talking about precisely 55, a real number 55. And again, this is an abstraction. This is a mathematical model. And we assume that we can measure exactly the weight down to the infinite number of decimal points, which means infinitely precisely. So what's the probability of having a weight exactly equal to 55? Well, let's just think about what is the probability? We make a certain number of experiments. In this case, we are measuring the tennis balls. And we are measuring the number of the balls which have weight equal to 55, ratio of this number to the number of balls which we have measured. Now, as the manufacturing process goes and goes, and we are measuring and measuring new balls and new balls, the probability of this most likely will be going down to zero, because even if we will find the ball with the number 55, exactly 55, we will always find infinite number of times of all other weights in this range, like 55.77 or 54.45 or whatever else. So the ratio of the number of times when it's exactly equals to 55 will be going to zero as the number of experiments goes to infinity. Well, what does it mean? Well, it means this probability is equal to zero. And so is the probability of every other elementary event. So what's an interesting point in this case is that unlike the situation in a discrete case, when all elementary events had certain non-zero probability and some of them is still equal to one, in this case we have an infinite uncountable number of elementary events, and each one of them has the probability of zero. Then how come we have an entire interval to be basically having the probability of one? Well, here is a trick. Although this is true, this will be greater than zero. So the probability of every interval is positive. In this case, the probability of the interval from 50 to 60 is equal to one, because I was just saying that all balls are manufactured within these range of weights. So the narrower interval, like this one, 54 to 56, has less than one probability, but still some positive number. So basically what I'm saying is that although this is true for the probability of the weight to be equal exactly to any particular value, what is true as well is that probability of weight to be in between certain boundaries would be greater than zero. So this is my first introduction into continuous distribution. So this probability distribution is distributed in some way analogous to physical weight, physical characteristic of weight. Now, in physics we sometimes deal with so-called point mass. Point mass is an abstraction, actually. It's a geometric point which has a certain non-zero mass. And then we are dealing with certain physical characteristics of this, certain laws of physics like gravitation law or something like this. But at the same time, we all understand this is an abstraction. In theory, the weight can be, let's say, distributed to a rod, evenly distributed within the rod. Now, if you will take one particular cut of this rod, which has a zero width, it will have a zero weight, but any particular interval from A to B on this rod will have a certain weight. So we can always say that the relative weight of this interval relative to this one is positive, but relative weight of one particular cut would be equal to zero. So this is an example of a continuous distribution of the mass in as much as this is an example of a continuous distribution of probabilities, because probability is a measure which is no different than the weight. Okay. Now, after I have introduced this concept in this non-rigorous, but I hope a descriptive enough form, I will go again into a particular example to basically move forward within this concept. Let's say you have a sharpshooter competition. So you have a target, a radius, let's say, 0.5 meter. And let's assume that all our participants in our competition are professional enough not to shoot outside of the target. So what I can say is that everybody shoots with the distance of a bullet from zero to 0.5 meters. So this is a distance from a center. Now, obviously, those people who shoot zero are better than those who shoot further from zero, but in any case, all of them are shooting within the 0.5 meter distance. So the probability of this is equal to 1, where c is the random variable which characterizes the distance from a bullet to a center. Now, let's assume we have a particular participant in this competition. Now, let's also assume that he is shooting and shooting we are repeating this experiment infinite number of times. Well, the number of times which is increasing towards infinity. Now, we can basically collect the statistics, and these statistics will tell something about the probabilities of certain things. So let's assume that 60% of his shots are within 0.1 meter of center. So it's somewhere here. So this is 60%. And obviously, the rest 40% are from 0.1 to 0.5 second. Now, does it characterize the distribution of probabilities? Well, very, very approximately, I would say. I mean, it's something it does not really allow us to, let's say, find the probability of this, because this is somewhere within this particular interval. We know that this particular interval occurs in 0.4 cases, but this is smaller. And I really have no idea about this, right? So I need to represent this distribution a little finer than these two, basically, this small one and the rest. So this is very rough, I would say, description of this distribution. This is obviously a continuous distribution, because any real number can be a distance from the center to the bullet, right? So this is an approximation, and that's exactly the approach I would like to take with continuous distribution. I would like to approximate it somehow with a little bit more familiar and, well, discrete actually values. Now, what we can do, we can do a little bit finer distribution. Let's say that instead of two intervals, from 0 to 0.1 and from 0.1 to 0.5, I have a little bit more. So this is still probability of C to be in this interval is 0.6. But the others, I will also put some concentric circles and I will put them with, you know what, let me just use the less than and greater than symbol. So we know this. Okay, now we will take the next one, next 0.1. Okay, and let's say this is 0.2. And then the next one, from 0.2 to 0.3, let it be 0.1. So it's 10% are going within this circle. So this is 60%. This is 20%. This is 10%. And this is from 0.3 to 0.4. This is let's say 0.07, which is 7%, and 3% is left. Some are supposed to be 160, 20, 80, 10, 90, and 10, 10. Okay. And the rest is 3%. Okay, now this is a little bit more precise distribution of probabilities. I'm still not specifying the everything about this particular distribution. I did not use anything from calculus because that's what's needed actually to specify more infinitely precisely, if you wish. But I'm approximating this continuous distribution with, in this case, 1, 2, 3, 4, 5 elementary events. Each one has certain probabilities which are based on statistical frequency of occurrence of particular event. Now, what does it allow me to do? Well, it's actually better than it used to be before, because now I can calculate that the probability to be between 0.1 and 0.4 from 0.1 to 0.4. It's 2 and 1 and 0.7. Its probability of this equals to 0.37 or 37%. So, what was my purpose? My purpose was to approximate the continuous probability with certain discrete model, which will allow me to do lots of different calculations of this type, but not everything. I cannot, for instance, calculate from probability from 0.25 to 0.37 distance, right? Because I don't have elementary events which can fit into this. But think about this. What if I will break it instead of 5? I will break it in 500 pieces, 500 concentric circles. Now, if I will break 0.5 meters to 500 pieces, I will have 0.001 meter intervals, right? So, the distance between these concentric circles will be one thousandth of a meter, which is one millimeter. That's much more precise. Now, I can calculate, let's say I just wanted before, I said from 0.25 to 0.37. Yes, that I can calculate in this particular case, because I will have the probability for each concentric circle with a step of one millimeter, and indeed that would fit. Certain number of these concentric circles, exact number, will be from 0.25 to 0.37, but precisely whatever number of 120 or whatever concentric circles or intervals, if you wish. So, the more precisely I built this discrete approximation of the continuous distribution, the more I can basically say about this distribution. Because what the purpose of this thing? Well, the purpose is to evaluate any possible event. And if I can break the event into a set of elementary events, I can just sum up the probabilities of elementary events to get the probability of the whole event. So, in this case, I cannot do the event from 0.25 to 0.37. But with this distribution, with this fine division of my interval, with this fine approximation, I can do it. So, this is basically the whole concept. We are approximating the continuous distribution with the discrete one. And the discrete one can be computerized and do all the calculations, et cetera, et cetera. Now, just as an example, let's consider, instead of this particular distribution, let's consider another distribution for another person who is also participating in the same sharpshooter's competition. Let's say he has different probabilities. So, his name is not Xie, but let's say Etta. And he is less skillful. So, the probability of hitting a very small circle around the center instead of 60%, it's 50%. So, it's 0.5. And then, after that, I have 0.25, 0.2. That's what? 95. So, 0.03 and 0.02. So, it's different distribution of probabilities, which basically is an approximation of the second guy's continuous distribution of his results, of the outcomes of his experiments, of his shooting. Or, again, we can do it at finer division, the finer breaking of this interval from 0 to 0.5. And then, we will have different numbers. Again, this is yet another distribution for another person. So, the first person has the distribution which looks one way in this tabular form. And for the second one, it's another. It's two different distributions of probabilities. Two different continuous distributions of probabilities for two different people. Well, considering during the competition, their skills are the same and they don't get tired. All right. So, this is basically a description of approximate description of the continuous distribution using the discrete one. Now, if you remember, in a discrete case, I was using the graphical representation of the probabilities. And that's what I'm going to do in this case as well. As soon as I have approximated my continuous distribution with discrete presented in this particular form, I can use the graphical representation of discrete probabilities to again demonstrate approximately, again approximately, the continuous distribution. So, for instance, we had this simple case when we had intervals from 0 to, let me now use the graphical form, from 0 to 0.1, 0.2, 0.3, 0.4, 0.5. So, in this case, this particular probability to hit the target, not further than 0.1 meter, was 60% for the first shooter. For the second interval, it was like 20%, right? 20%, then 10%, then 7%, and 3%, right? So, this is my probability, which is equal to 1. Now, on this particular base, I will put a rectangle of the height 0.6. Now, this one would be 0.2. On this interval, from 0.1 to 0.2, the probability is 0.2, 20%. Then I have 10%, then I have 7%, and then I have 3%. So, this staircase is a graphical representation of these probabilities for the first sharpshooter. Now, just as a comparison, let me just do it for the second one, that we have different numbers. So, we had 0, 0.1, 0.2, 0.3, 0.4, 0.5. Now, for the second one, we have 50%, so it's 0.5. So, it's this type of 0.5. Then we have 0.25, 0.2, and then 0.3, and 0.2. So, these are different graphical representations of two different discrete distributions, which are correspondingly approximating their respective continuous distributions. And again, my point was that if we will divide in a smaller pieces, this particular interval, we can have a better approximation of our continuous distribution. And again, the purpose of this is to be able to find out the probability of any event. So, the probability of any event with a precision of 0.1, we can do here. But with a precision of 0.001, for instance, we cannot. For this, we need a finer distribution. So, we need to accumulate more statistics. And based on our statistics, we can basically come up with a different cable of values for these distributions or different graphical representation for different distributions. Well, this is basically all I wanted to say. This is an introduction into continuous distribution, just to have a feel of what this actually is about. Thanks very much. Try to read the same thing at Unisor.com, the notes for this lecture. That's it. Thanks and good luck.