 Hi, I'm Zor. Welcome to a new Zor education. I would like to talk about distribution of probabilities of discrete random variables. In particular, we will talk about probability mass distribution for discrete random variables. This lecture is part of the course of advanced mathematics for teenagers and high school students. It's presented on unizor.com. If you watch it from any other source, I do recommend you actually to use unizor.com because every lecture contains very detailed description of whatever the topic is at issue. And it also contains some additional educational functionality. You can enroll into a specific topic, you can take exam, etc. The site is completely free, so I do suggest you to watch it from unizor.com. Now, let's talk about this particular topic. First of all, we are talking only about discrete random variables. Random variables with discrete distribution of probabilities. Well, let me just remind you what it means. It means basically that the random variable C, which takes real values, takes only finite or infinite but countable, I put these three dots, real values and there is probability associated with each value it can take. Now, obviously, the sum of all these probabilities is equal to one. Now, even if it's an infinite, if it's an infinite, obviously the probabilities should decrease and it would be still summed up with 31. All right, so this is basically a full description of random variable which takes discrete values with certain probabilities. However, what kind of description this is? It's a table, basically, right? The value and the probability. Now, mathematicians don't really like to deal with tables which define certain things. They prefer formulas, for instance, like this or they prefer graphs, something like this. This is a more natural representation of functions in mathematics. So, to make this particular description of the probabilities a little bit more in tune with the main line of mathematical thinking, mathematicians have invented a function, basically, which describes exactly the same thing. There is absolutely no difference between description as the table and description as the function I'm going to talk about. This function is called probability mass distribution function. Let me just remind you a story from the physics course. I mean, in the physics, for instance, if we are talking about gravitation, we are talking about point mass, right? So, it's like the whole earth, actually, relative to a sun, for instance, is considered to be just a material point which has certain mass. It does not have any kind of geometrical dimensions in this particular theory. So, we will actually approach this particular distribution from exactly the same point. It's like a point mass. So, we are assuming that the probability is a mass which is concentrated in these points and these are the values, actually, where it's concentrated. And there is no mass which is concentrated everywhere else except these points. Like, in case of gravitation, for instance, vacuum has no mass associated with it, well, at least in classical physics. Right now, everything is completely different. So, we have material points and we have vacuum in between in the physics, right? Here as well, we have certain masses concentrated at these particular points and no masses, which means zero mass, if you wish, concentrated in all other points. So, we are considering a function, the, it's called mass distribution, probability mass distribution function, which is defined for all real numbers x. And it's equal to pi if x is equal to xi, where i is one or two or three, etc. And it's equal to zero if x is not equal to any of x. So, anything outside of this set of finite or infinite countable number of values, outside of this set, the function is equal to zero and for each point inside of this set, the corresponding value is pi. So, that's not exactly a formula, but at least it's a function which is defined on all real values because this table is not really defined everywhere, right? So, my question is where, what exactly the value of probability is if x is equal to none of these, something like, for instance, it's one half or something like this is one, if one half is not among these values. Well, you can say, okay, it's not defined, which is not really nice. People don't like something which is not defined. This is defined everywhere. And the probability is zero. And now let's recall what is the probability? Remember, the probability is some kind of a limit of the frequency of occurrence. Now, if the value x is not one of these x i's, then it does not occur. So, the frequency of occurrence is equal to zero. It's a perfectly valid definition. So, this is a function. It's defined on all real values of argument x. The values of these functions are somewhere between zero and one, zero in all these cases and some probabilities which are less or equal to one in these cases. So, the function is defined. Now, graphically, this function might look something like this. For instance, this is x1, x2, x10, et cetera. So, the function is equal to zero here. Then it's equal to p1 in this case, then zero here, p2 in this case, zero here, something p, three, four, five, ten, and then zero everywhere else. So, function is equal either zero or some positive number from zero to one. So, sometimes we add the vertical bars because the point is not really visible if you are printing it or displaying it, whatever. So, sometimes it displayed with these vertical bars, which means that the probability is zero and then jumps up to this value of p1. Then again, zero then jumps to p2, et cetera. So, we have a function and this function is called mass distribution function, probability mass distribution function. Okay. Now, let me just go to a few examples. Okay, example number one, we shoot a target. We have one, two, three. We associate this with ten points. Now, if a person goes into this area, shoots into this area, it's five points and two here and zero outside. So, we are talking about, let's say, sharp shooting competition and every person has certain level of skills. Now, based on this level of skills, we know that this particular person has the probability of hitting ten, five, two, or zero points, right? So, let's assume that our sharpshooter competition participant A has the following probabilities for hitting targets. This is 0.17, 0.23, 0.28, and 0.32. Some of them is equal to one, of course. Okay. Now, how does this particular distribution of probabilities can be expressed in terms of mass distribution? Well, we can say the following. This is zero, this is two, this is five, and this is ten. Now, zero would be 0.32, two would be 0.28, five would be 0.23, and ten probability would be 0.17. So, this is our mass distribution function. So, this, this, this, and this are vertical jumps at zero, at two, at five, and ten. The probability is decreasing, obviously, with the, with the precision of shooting increases. And some of these probabilities is equal to one. So, this is basically the representation. And the function, let's call it X, would be equal to 0.32 for X is equal to zero, 0.28 for X is equal to two, 0.23 at X is equal to five, and 0.17 at X is equal to ten. And it's zero for X not equal to zero, two, five, and ten for all other values. So, our random variable can take these values with these probabilities. Now, if you consider some other maybe less skillful participant B, he can have 0.5, 0.15, 0.35, and 0.45. Now, why is this guy's not as good a sharpshooter? Because the probability to hit ten is significantly less than this guy. And on the other hand, probability of completely missing the target is greater. Well, again, this particular guy represents basically another random variable which takes exactly the same values, ten, five, two, and zero, but with different probabilities. In this case, the probabilities will be higher here, higher here, lower here, and lower here. So, it will be something like this, this, this, this, and this. So, we are distributing the same total mess, which is one, the total probability is one, or a hundred percent. We are distributing differently among the possible values of the random variable. In one case, it's distributed as this, this, this, this, another this, this, this, and this. But in any case, we are still distributing the same probability one among, in this case, four different values which our discrete variable can take. All right? Now, another couple of examples, very simple. What's the distribution of probabilities and mess distribution function if you are rolling a die? Well, if you are rolling a die, you have one, two, three, four, five, six different values our random variable can take, right? Now, what's the probability? Well, if it's a right die, it's one, six, here, one, six, here, one, six, here, one, six, here, one, six, here, one, six, here. Graphically, it would look like this. So, this is one, two, three, four, five, and six. This is one, six. So, you have this, this, this, this, this, and this. So, this is a graph. Zero until it's one, then jump up, immediately goes down. Zero until two jumps up to one, six value, down, et cetera, up to six. So, that's the probability distribution for this particular variable and this is a graph of the mess distribution function, probability mess distribution function. So, probability is, you remember probability was like a measure. It can be like a length, for instance, or area, or weight, or mass, or whatever. So, because it has this additive properties. So, in this particular case, we are distributing the total mass, probability mass of one among six points, one, six for each. And the last program, slightly more complex. What if you have two dies and your variable, your random variable is some of these two dies? Now, what values this sum can take? Well, the smallest value is if you have two ones, right? So, one plus one, which is two. And the biggest value is six plus six, which is twelve. Now, in between, you can have one plus two or two plus one, which is three. You can have one plus three or two plus two or three plus, or three plus one, which is four. Now, how can we get five? One is one and another is four. Or two and three. Or three and two. Or four and one. That's our five. Then six, six, seven. The first dies on one, and the second on six. Or two and five. Or three and four. Or four and three. Or five and two. Or six and one. That's seven. That's the seven. Okay. Now, eight. How can eight be done? Well, there should be no ones, right? Because if you have one die to show the number one, another maximum is six. One is six is seven, which means eight we will never get. So the minimum is two. Minimum is two, and then another to get eight should be six. Or three plus five. Or four plus four. Or five plus three. Or six plus two, which is eight. Now, how can we get nine? With nine, we can get the minimum number of three, right? One of them should be three, and another should be greater, which is six in this case. Or four and five. Or five and four. Or six and three, which is equal to nine. Now, to get ten, we have to have one of them at least four, and another will be six. Or five and five. Or six and four. And that will be ten. Now, for eleven, we should have minimum five, and then it can be either five plus six or six plus five, which is eleven. And for twelve, there is no other way but to get this, right? So, how about our probabilities? Well, probabilities is, as you understand, the number of combinations of these two dies which gives us a concrete number, divided by the total number of combinations. Now, total number of combinations is obviously six times six, because we have two dies, and each one of them has six different values, so it's six times six. Now, how many combinations are when our sum is equal to two? One combination out of thirty-six. So, the probability is one thirty-six. In this case, we have two combinations, so it's two thirty-six. In this case, it's three thirty-six, four thirty-six, five thirty-six, six thirty-six. That's the maximum, right? One, two, three, four, five, six. It goes down again. Five thirty-six, four thirty-six, three thirty-six, two thirty-six, and one thirty-six. So, our distribution of probabilities is the following. So, we have one, two, three, four, five, six, seven, eight, nine, ten, eleven different values our sum can take. Sum of two dies, two dies. Sum of two dies can take eleven different values from two to twelve, right? So, it's two, three, four, five, six, seven, eight, nine, ten, eleven, twelve, thirteen, fourteen. Now, the probability is concentrated only in points from two to twelve, inclusive, and only on integer numbers. Now, the probability for this one is one thirty-six, two thirty-six, three thirty-six, four thirty-six, five thirty-six, and six thirty-six, right? That's on seven. Then it goes down again. Five, four, three, two, and one. So, these are points where our mass is concentrated. These from two to twelve, and this is the value of the concentration. So, the probability mass, again, the total number is one. We can memorize them together. One and two, three, six, ten, fifteen, twenty-one, twenty-six, thirty, thirty-three, thirty-five, thirty-six. Thirty-six, thirty-six, obviously, sum is supposed to be one, and it is one. So, we have distributed our total probability mass of one among these eleven points on the x-axis from two to twelve, all integer points. And this is the value we put into the value of this function. This is the mass distribution function for our random variable, which is the sum of two dice. All right? Now, basically, that's all I wanted to say about what is the mass distribution function. Well, a couple of warnings, actually. People like to have some nice rules about everything. Like, for instance, probability of the sum of two random variables of, you know, is equal, let's say, not sum, but not the probability, but the mathematical expectation. Mathematical expectation of sum of two variables, random variables, is equal to sum of their mathematical expectations, right? Very nice rule. Well, product is not always like that, only for independent random variables. So, mathematical expectation of the product almost always, except when they're dependent on each other, equals to the product of their mathematical expectations. So, it's nice rule. Now, how about the probability distribution? Well, if you think that the probability mass distribution function of sum of two random variables equals to sum of their mass distribution functions, well, that's not true, obviously. Well, there are many philosophical reasons for this, but very simple mathematical reason is that sum of these values should always be equal to one, right? So, if you add one and another, then obviously your sum would be equal to two. So, that's not true, obviously. So, I would like to warn you that probability mass distribution function is not additive relative to the random variables it's related to. So, let me just express it in a formula. So, you will always be kind of on guard if this is not true. So, if you have a variable c and your mass distribution function is this, and you have variable eta with mass distribution function like this, then if you have this, and you have mass distribution function this, this is not equal to, this is absolutely not equal. Well, I remember somebody was just adding two things, like one two plus one four is equal to one six, something like this. They just added the denominator. So, you do not really do this type of things. It's just wrong and for many different reasons, and this is wrong as well. So, I want you to basically be warned that all these nice things are not always true. Now, same thing about the multiplication by a constant. If you multiply the random variable by the constant, it doesn't mean that its mass distribution function is multiplied by constant. That's absolutely not true. It will be different obviously, but it's definitely not the result of multiplication. Let me just give you a very simple example. For instance, this value, it takes, let's say, one half when x is equal to, I mean, not c, f. One half when x is equal to, let's say, one and one half when x is equal to two. So, the whole distribution is concentrated in two values, one and two, one half and one half. Now, what is, let's say, function random variable 2x? Random variable eta equals 2 times c. What is this distribution? Well, which values 2c will take? Well, if c takes one and two, 2c will take two and four. With what probability? Exactly the same. Probability will be one half and one half for x is equal to two and x is equal to four. So, what will be the graph? It will be new graph, which is two and four, and it will also be one half and one half. So, from this function, we go to this function. We are not multiplying function by two, because it's a completely different variable, completely different distribution. So, I'm just warning you that these nice rules are not necessarily kind of blindly applicable. But in any case, I just wanted to prepare you for functional and graphical description of the distribution of probabilities. Because before, we were actually thinking mostly in terms of tables. Like, okay, this is variable, it takes value one and two with probability one half and one half. For instance, or x1, x2, xn with probability p1, p2p. So, this is a table kind of view. And this is, instead of the stabular view, we are working with function, which basically contains exactly the same information as this table. But it's a function. And again, I was just saying before that mathematicians like to deal with functions. All right, that's it for today. Thank you very much. Please go to theunisor.com to basically go through the textual description notes for this lecture. It's very useful. And, well, that's it. Thanks and good luck.