 K students, I would like to present to you the properties of the joint cumulative distribution function. In the case of two random variables, capital X and capital Y, you already know that the joint CDF is given as follows. Capital F of the random vector xy at the point small x, small y is actually the probability of capital X being less than or equal to small x and capital Y being less than or equal to small y. Ye jo bunyadi definition hai, iske baad let us focus on the properties of the joint CDF. As you can see on the screen, there are six properties. The first one reads as follows. Marginal CDF of X, capital F of X at the point small x is equal to capital F of the random vector xy at the point x infinity. The second one, the marginal CDF of Y, capital F of Y at the point small y is equal to capital F of the random vector xy at the point infinity y and the third one, capital F of the random vector xy at the point infinity infinity is equal to one. Iske baad fourth, fifth or sixth properties Johan, I would like to discuss them with you at this moment in time in some detail. The fourth property reads as follows. Capital F of the random vector xy at the point minus infinity comma y is equal to zero and at the same time capital F of the random vector xy at the point x comma minus infinity is equal to zero. Ye dono jo hain, they are equal to zero aur agar aap iss pe horkane and also if you draw the three dimensional space one axis depicting the random variable x and other axis depicting the random variable y and the third axis depicting small f of xy. Uske baad if you consider the point minus infinity on the x axis you will then realize that this property is correct and that the probability of capital X being less than or equal to minus infinity has to be equal to zero. Similarly, if you after drawing that diagram consider the point minus infinity on the y axis again it will be easy for you to realize that the probability of y being less than or equal to minus infinity is equal to zero. Even if you do not draw any diagram even then you can understand so we cannot have x being less than or equal to minus infinity or y being less than or equal to minus infinity. If these are impossible events then obviously that probability and that overall thing has to be equal to zero. So this is property number four. Now let us focus on property number five. It says that the probability of capital X being greater than x1 and less than or equal to x2 along with the random variable y being greater than y1 and less than or equal to y2. This particular probability can be found by the following formula. The formula involving the joint cdf and what is that formula? Capital F of the random vector xy at the point x2 y2 minus capital F of the random vector xy at the point x1 y2 minus capital F of the random vector xy at the point x2 y1 plus capital F of the random vector xy at the point x1 y1. I would like to advise you that if you have any confusion then you will find that it is correct. Last but not the least I would like to focus on property number six and this one reads as follows. If capital X and capital Y are independent random variables then the joint cdf is equal to the product of the marginal cdfs. Any capital F of the random vector xy at the point xy will be equal to capital F of the random variable x at the point x multiplied by capital F of the random variable y at the point y. I repeat that this will happen if capital X and capital Y are independent random variables. So these are the properties of the joint cumulative distribution function.