[MUSIC] Previously, we discussed random variables, now, we will consider systems of random variables. It means that for a particular random experiment we will consider several random variables that are associated with this experiment. And we'll study the relations between these random variables. To begin with, let us consider example, let us return to an example with casino. As before, Alice bets $100 on red and at the same time, another player, Bob, bets $100 on black. Now, we have Alice's payout, and Bob's payout and these are two different random variables. Which values are defined by the result of the same experiment, spinning of the wheel of this roulette. We also see that these random variables are related to each other. For example, if we know that Alice won $100, then it means that the outcome of our experiment that gives us a red number. And it means that Bob cannot win and he loses $100 and so his payout is negative 100. So we see that there is a relation between these two random variables. Now, consider the third player, Claudia, Claudia bets $100 on the number 32. So she wins only if the result of our experiment is exactly number 32. We have new random variable that is related to, for example, Alice's payout. If Alice wins, then it means that the outcome that we have is red. And it increases chances of Claudia to win, because 32 is red. Now, we see that there is some kind of relationship between this random variable. But this relationship is not deterministic, we cannot answer exactly what will be Claudia's payout even if we know Alice's payout. Finally, let us consider Dan, he bets $100 on red, but on different roulette. Because two roulettes are independent to each others. We cannot use any information about Alice's payout to say something new about Dan's payout. So these two are the variables, are independent of each other. Now, let us look at how a similar reasoning about random variables can be applied to problem of machine learning. In machine learning, we have data that looks like this table. And every column of this data have some value, but we can assume that these columns are random variables. Because, for example, if we look at the first row of this table, we think about some particular person that gave these answers to some questions in questionnaire. And we understand that this particular person was chosen randomly from some population. And the same thing holds for every row of this table. So we can think about every row of this table as a realization of some system of random variables. And we can try to find some relations between these random variables. For example, if we see a person with large age, we probably expect that their income will be also large compared to some person with low age. Just because very young persons usually do not have large incomes. In any case, we'll think about these variables as a kind of random variables. For example, because we don't know exactly what is the income of a particular person of age 21. Now, let us discuss our plans for today. First of all, we'll discuss how what kind of relations between random variables exist. And we'll create new random variables from the existing ones with different transformations. Then we will discuss how to describe system of random variables. This can be done with so called joint distribution. Then we will discuss what independence mean in context of random variables. And why it is sometimes very useful to have independent random variables. Finally, we will discuss notion of covariance and correlation which are important for practical applications. [MUSIC]