 Dear students, I'm now going to present to you the concept of the continuous random vector and its joint PDF, the joint probability density function. So what is the definition of a random vector if we are dealing with two variables X1 and X2 and if they are of the continuous type well obviously it will be represented as an ordered pair and we can write inside the bracket X1, X2 and we can say that it is defined on the space capital D if its cumulative distribution function capital F of X1, X2 at the point small X1, small X2 if this cumulative distribution function is a continuous function meaning that again basically that the variables are continuous variables then we say that it is a random vector of a continuous type. Because continuity exists right now, capital F of the random variables X1, X2 at the point small X1, small X2 will be given by the integral from minus infinity to X1 and then the second integral from minus infinity to X2 of small F of capital X1, X2 at the point W1, W2, to TW1, TW2 for all ordered pairs small X1, small X2 belonging to the two dimensional Euclidean space or in other words the two dimensional plane denoted by capital R square. Now what I said it is actually quite simple, it is again simply the extension of what we have in the case of a single continuous random variable that usually confuses people, however it is so simple, if you keep the concept of dummy variable in your mind then you will not have any problem, you write W1, W2 there or L1, L2 or anything it is okay actually in the subscript the capital X1 or capital X2 we have written that tells us that basically we are dealing with the random variables capital X1 and capital X2, we are writing W1, W2 or anything when we apply the limits, then look carefully at the limits that are written there we have written X1 and X2, so we have to go from minus infinity to X1 as far as the random variable X1 is concerned minus infinity to X2 as far as the random variable X2 is concerned. This is the definition of the joint cumulative distribution function for the continuous case is the joint PDF, the joint probability density function of the ordered pair capital X1, capital X2, again small f of capital X1, X2 at the point W1, W2, again that is a dummy variable. If we are actually writing inside then we will write W1, W2, now if we are talking about Alexia then we will not write W1, W2 there at that time, now we will write X1, X2 at this time. You are feeling this tricky, it is not so tricky, the thing is that when it is in the integral, it is in the definite integral, so the limits are required and the variable is written in it, it does not matter, but when you are talking about Alexia then you will write X1, X2, so this is how it is, small f of X1, X2 at the point small x1, small x2, this is the joint PDF of the random vector capital X1, capital X2, now I will tell you another very important thing and that is that if you take the second derivative because right now we are talking about two variables, so you will take the second derivative which is the one which you now have on the screen, curl square of capital F of X1, X2 at the point small x1, small x2 over or by curl x1, curl x2, that is capital F's second derivative you have to take is this one, otherwise you know from calculus that you can have another kind as well, if we do this differentiation, one derivative with respect to X1 or two derivative with respect to X2, so what will we get? Students, we will get our small f of X1, X2 at the point X1, X2, so this is a very important relationship between the cumulative distribution function and the joint probability density function. Here again we will see that it is simply an extension of what you have in the univariate case, you will be sure that if capital F of X is for a single variable X, so its d by dx if we take it, so that is small f of X. After this I will talk about the fundamental properties of the joint probability mass function and the properties are as follows, number one small f of X1, X2 at the point X1, X2 has to be greater than or equal to zero and number two the double integral of small f of X1, X2 at the point X1, X2 with respect to X1, X2 this double integral is has to be equal to one. Now note that our random vector is X1, X2, X1, X2 as I said earlier it is defined on a space capital D but usually we do not write capital D, we try to simplify matters, we try to extend the definition of the PDF over the entire two dimensional Euclidean space or in other words the entire plane and how do we do this? We do this by writing zero elsewhere. You know that when univariate case is written, then how do we write PDF? Small f of X is equal to something and with that X's range work is given. For example if it is a uniform distribution where X is going from zero to one then what do we write? We write that f of X is equal to one where X itself goes from zero to one but we do not finish here. Just below that do we not write that f of X is equal to zero elsewhere. The same thing that is happening in univariate is applied here as well. So when we write zero elsewhere with whatever else we are writing then we are actually extending the definition of the PDF to the entire two dimensional plane and when we do this then we do not need to write capital D and we can for example, the property that the integral has to be one over that entire space we will write it like this. Integral from minus infinity to plus infinity and then the second integral also from minus infinity to plus infinity. Small f of X1, X2, DX1, DX2 this is equal to one. So this simplifies matters for us. Last but not the least my dear students, I would like to talk about the concept of an event for this type of random vector. Again it is an extension of the univariate ideas. That is defined from X from zero to one. So if we are interested in this particular event, what is the probability that X will not go from zero to one but it will be less than 0.5 meaning that X goes from zero to 0.5. So how do we do that? Do we not simply take the integral from zero to 0.5 and compute it and that probability is there. So this is exactly what we will do here for an event capital A belonging to the space capital D. The probability of the ordered pair capital X1 comma capital X2 belonging to this capital A which is a subspace of capital D. We will simply integrate our joint PDF f of X1, X2 at the point small x1, small x2. We will integrate it over that subspace capital A. One more thing that I would like to share with you is that the probability that you will compute it is the volume under the surface that is given by this bivariate PDF small f of capital X1, capital X2 at the point X1, X2. Geometrically it is a surface in the two-dimensional space and the floor, XY floor in this case X1, X2 floor it is a volume. In the univariate case we always say area under the curve, area under the curve they think any. Now because there are two variables we are talking about the volume under the surface which is above that floor which can be called in this case that we have in front of us now the X1, X2 floor. Of course agar main invariables ka naam X or Y rakthiti then I would have said the floor, XY floor or uske upar wa surface uske under not the area but the volume and the total volume has to be equal to 1 because of that same reason that the total probability would always be equal to 1.