[MUSIC] If you have two random variables defined on the same probability space, these random variables can interact with each other. It means that if you know something about the value of one random variable, then you have some new information about the value of another random variable. Let us consider this example. In this case, if we don't know the value of X and are interested in the value of Y, we know that Y can take values 0 or 1 with equal probabilities. But if we know that X is equal to 0, we also know that Y cannot be equal to 0 as well, and it have to be equal to 1 with probability 1. So the distribution of Y when you know that X is equal to 0, is different from the distribution of Y that we can use when we don't know that X is equal to 0 from this distribution. Now, let us look at the pair of variables X and Z. Here, the distribution of Z is the same. But if you know that X equals to zero, then again, you know that Z can be as a 0 or 1, again with equal probabilities. So, if you know that X equals to 0, you don't know anything more about the distribution of Z, then information that you had before you know that X equals to 0. You again know that Z can be either 0 or 1 with equal probabilities. In this case, we can say that X and Z are independent variables. Let us go to mathematical statement of this definition. Let us consider two random variables X and Y. Now, let us consider the following conditional probability. Let us assume that we know that X is equal to one of these values. What is the probability that Y is equal to one of these values? Mathematically, I want to know something about this conditional probability This is probability that Y is equal to yj and the condition that X is equal to xi. If this probability Is equal to just simple probability of that Y is equal to yj. And this equality holds for any i and j Then we can say that random variables X and Y are independent In other terms, X and Y as a random variable are independent. If all errands of this kind are independent, for all possible i and j, we can also state the same definition in different terms. As you probably remember, two events are independent if probability of their intersection is equal to product of probabilities, so we can rewrite this condition in the following way. Probability that X equals to xi, and at the same time Y is equals to yj is equal to product of probabilities that X equals two xi times probability that Y equals to yj. Again, this equality should halves for all possible values of i and j. This is alternative and are well known definition of independence of two random variables. Let us return to this example. In this case, this 1/4 is equal to product of this 1/2 and this 1/2, and the same halves for all other cells. So in this case, if we know that two random variables are independent, then we can find joint probability by multiplication of the corresponding values of marginal probabilities. But in this case, we see that these two variables are not independent, for example, because this 0 is not equal to product of this 1/2 and this 1/2. It is sufficient to find just one cell that violates this condition to say that two variables are not independent of each other. In this example, we discussed two extreme cases. Here, if we know the value of one variable, we know the value of another variable exactly. And here, two variables are independent of each other. But this is are just two extremes of all possible spectrum of dependence between two variables. Now, let us consider example that is in the middle of the spectrum. In this examples, the two variables will not be independent of each other, but knowledge of the value of one of these variables will not give us exact information about the value of another variable. We will do it in the next fragment. [MUSIC]