 these students, I'm going to present to you the concept of the joint cumulative distribution function. This concept arises when we are not dealing with a closed variable, but two, three, four or more. So we are actually dealing with a multivariate or a multi-variate situation. Given the random variables X1, X2 and so on that are defined on a probability space, the joint probability distribution for these variables is a probability distribution that gives us the probability of each of these variables falling in any particular range. May the variables be discrete or may they be continuous. Let us concentrate on that case when we have only two variables X1 and X2. In this particular case, we will say that we are dealing with a bivariate distribution because obviously, that stands for two and the joint distribution that can be expressed either in terms of a joint cumulative distribution function, which can also be called a joint CDF or it can be expressed in terms of a joint probability density function. Of course, if the variables are continuous or a joint probability mass function, if the variables are discrete, so students, let us focus on the joint cumulative distribution function and what is the formal definition if we have two variables. The definition is as follows. The capital F of the random variables capital X1 and capital X2 defined at the values small X1 and small X2, this capital F is equal to the probability that the random variable capital X1 assumes a value less than or equal to small X1 and the random variable capital X2 assumes a value less than or equal to small X2. For all values, all ordered pairs small X1, small X2 that are defined on the two dimensional Euclidean space are square. Actually, whatever I said, maybe it was very complicated for you, although it is very simple. When we say and, then when we express it mathematically, then you know that we use the intersection sign there. So that is why what you see on the screen, that carries the intersection sign whereas when I said it, I said and. The second thing is that at the time you are in a univariate situation and you are simply capital F of X, what happens is that the probability of the random variable capital X being less than or equal to the value small X, but also you say this sentence that for all X belonging to R, obviously when it is one, then we talk about R i.e. the real line, which we can also say X axis, but if there are two, then obviously ordered pair will be X1, X2, so it will not be just the line, it will be the two dimensional plane, which can also be called Euclidean space. So it's quite simple. It's just a natural extension of the concept of the cumulative distribution function in the case of one single variable. Now after this, you note that what we are saying is that capital X1 is less than or equal to small X1 and capital X2 is less than or equal to small X2. So we can say that this is a joint event, i.e. two events are different, so we only say that if we are going to deal with them separately, but if we are talking to them together, then we are talking about a joint event and after this, you have a very important result, which has been mathematically derived. I want to share about the cumulative distribution function and I'm talking about that case only when we have two variables, i.e. the simplest possible multivariate situation, when we do not have three, four, five, but only two variables. So what is this result, my dear students, that I would like to share with you? It is as follows. The probability that A1 is less than X1 and X1 is less than or equal to B1, where A1 and B1 are two real numbers and A2 being less than X2 and X2 being less than or equal to B2, where A2 and B2 are two other real numbers. This probability is given by capital F of X1, X2 at the point B1, B2 minus capital F of X1, X2 at the point A1, B2 minus capital F of X1, X2 at the point B1, A2 plus capital F of X1, X2 at the point A1, A2. This is basically the probability of your X1 lying in the range A1 to B1 and your X2 lying in the range A2 to B2. And I am saying that you can denote this by the intersection, but also students that can also be denoted by a comma. You have often seen that the comma is put in this context, which we are discussing It means and. Now these numbers A1 to B1, the limits given for the random variable X1 and the numbers A2, B2, which are for others, it is obvious that A1 is smaller than B1 and A2 is smaller than B2. So now how will we remember the formula given in the form of capital F? We will try to remember that the first one that I said, F of X1, X2 at the point B1, B2, that upper limit of B1 and B2, that's the one. Then when we are minusing it twice, once A1, B2 and the second time B1, A2. And then when we are plusing it in the end, now it is A1 and A2. That is, the limits defined for both the variables are for the lower limit of A1, X1 and lower limit for X2. I am just giving you a tip as to how to remember this particular formula. So this entire discussion that I have just done with you, this is the concept of the cumulative distribution punch.