 Dear students, I would like to present to you the concept of the marginal probability density function and I will be presenting it with the help of an example. If we have two continuous random variables X1 and X2 then the joint PDF is written as small f of capital X1, capital X2 at the point small x1, small x2. And of course, we will need to take the double integral of this function with respect to both X1 and X2 in order to obtain the total probability that the one that is equal to one. This is the basic thing regarding the joint PDF. Now that we have this thing, now how do we define the marginal PDF of X1 and the marginal PDF of X2? Well, it is very simple for finding in order to find the marginal PDF of X1. All we have to do students is to integrate joint PDF with respect to X2. Let me say it again. If you want the marginal PDF of X1 then integrate the joint PDF with respect to X2. Or if you want the marginal PDF of X2 then integrate the joint PDF with respect to X1. Let me present to you the algebraic formula that can be written as follows. Small f of the random variable X1 at the point small x1 is equal to the integral from minus infinity to infinity of small f of the random vector x1, x2 at the point small x1, x2. And this integral is with respect to x2 because you are seeing that we have written there the x2 which is written there, focus on that so that you will know which variable of the integral we are taking. So you can see that in order to obtain the marginal of X1 we are having the integral with respect to x2. Exactly the same way we have for the other one and that is in front of you on the screen. Now let us take a simple example. Suppose that the random variables continuous random variables X1 and X2 have the joint PDF given by small f of X1, X2 is equal to X1 plus X2 and the limits of X1 and X2 are 0 to 1. X1 also goes from 0 to 1 and X2 also goes from 0 to 1. So we have this particular joint PDF and of course you can check for yourself that if you take the double integral of this PDF both with respect to X1 and X2 from 0 to 1 because we generally write minus infinity or infinity. But when you have that from 0 to 1 obviously you will write 0 to 1 for both of them. You can check it yourself that this double integral will come out to be equal to 1. So that is done. Now suppose that we are wanting to find the marginal PDF of X1. So just do exactly what I said a short while ago that simply integrate this joint PDF with respect to X2. So let us do it right now. Integral from 0 to 1 of X1 plus X2 with respect to X2 is equal to the integral of X1 with respect to X2 plus the integral of X2 with respect to X2. Now let us look at this separately. The first one is written in X1 integral that will come out because when we are integrating with respect to X2 students X1 then acts as a constant. Or if it acts like a constant then we can take it out. So when we take it out what are we left with inside? We are left with 1. Now if we take the integral of 1 with respect to X2 what do we get? Of course we get X2. After that apply limits 0 to 1. So upper limit minus the lower limit put on X2 1 minus 0 so that is equal to 1. And the one that was taken out X1 multiply it with X1. What do you get? X1 into 1 that is X1. This was the first part. This is how it is. Now let us focus on the second part the integral of X2 with respect to X2. So what will come? Obviously X2 square over 2. Now apply limits on this 0 to 1. Upper limit minus the lower limit. So X2 square over 2 instead of that 1 square over 2 minus 0 square over 2. Solve it. Obviously it is equal to half. So therefore what is the final result? The final result is that the marginal pdf of X1 is small f1 of X1 is equal to X1. The first integral was solved. Plus half to the second integral was solved. But students you cannot complete it. Just write this and think that it is complete. You must write the domain of this function. Or in other words the support of this function. Since this is the marginal of X1, it is obvious that you have to write the range of X1. X1 and X2 you know you remember. Both of them were going from 0 to 1. So it is obvious that you will go ahead with this. And then we will see. 0 less than X1 less than 1. And the more formal or rigorous or mathematically proper way of writing it. Under that you will also write 0 elsewhere. So this is how you can find a marginal of X1. And I leave it to you to find in a very similar manner the marginal pdf of X2.