 Dear students, I will be discussing with you the properties of the joint cumulative distribution function and I will start by defining the joint CDF. So let me take a simple case when we have only two random variables X and Y and in this case the joint CDF is given by capital F of the random vector X, Y at the point small x small y is equal to the probability that the random variable capital X is less than or equal to small x and the random variable capital Y is less than or equal to small y. Now let me give you the properties of this joint CDF. As you can see on the screen there are quite a number of properties of the joint CDF but at this point in time I would like to focus on the first three. So the very first property reads as follows capital F of the random variable X at the point small x is equal to capital F of the random vector X, Y at the point small x comma infinity. You see that we have the second property capital F of the random variable Y at the point small y is equal to capital F of the random vector X, Y at the point infinity comma Y. So let's talk about the third property. Definitely I would like to try to explain these two first two properties which as you can see are quite similar to each other. That is to say that in the first property we are talking about capital F of the random variable X and in the second property we are talking about capital F of the random variable Y. So the point is that if you pay attention to these two equations, the left hand sides here students they are the marginal CDFs. Are they not? Obviously, when you are just capital F of X or you are just capital F of Y, obviously you are not considering the joint thing and you are actually talking about the marginal CDF of X and the marginal CDF of Y. And now if we look at the right side of the equation, look at the right side of both equations, then you will see that there is also a pattern there. When our left side is written as F of X, then in the right side where the comma Y is written, the place of infinity is written and when our left hand side is written as F of Y, then at that time our right hand side where X is written, there is infinity. So if you pay attention to this a little bit, then the only thing is that since you are taking out the marginal of X, then the second variable Y, if you have a continuous variable on it, then you will integrate on the entire support of Y. Because you are interested in X and you are not interested in the other and you will have to integrate over that whole thing in order to get the marginal of X. Exactly the same kind of logic for the second one, that if you want to take out the marginal of Y, then you will integrate over the entire support of X. If it is a continuous variable, then it will be integrated. If it is a discrete variable, then it will be summation. So this is rational for the first two properties. Equally importantly, let me discuss with you the third property which reads as follows. Capital F of the random vector X, Y at the point infinity comma infinity is equal to 1. The thing is that if you integrate both the variables from minus infinity to infinity, only then you will get this expression capital F of X, Y, infinity, infinity. So when you integrate the whole thing, then you will get this expression. And obviously in that case, the answer has to be 1. Because if it is a two-dimensional case as is the one in front of you now, then integrating over both over the entire support of X and the entire support of Y means that you are computing the volume under that surface entirely. And you know that in this particular case, volume represents probability. If there is a univariate situation, then area under the curve represents probability. Here, volume under the surface is representing probability. And if it is done entirely, then obviously if the total probability has to be 1, then obviously this thing will be equal to 1.