00:01
For this problem, to begin, it's going to be very helpful to just sketch out the region where we have our variables defined.
00:08
We know that x is going to be defined between 0 and 1, and y is defined between 0 and x.
00:17
So, if we say, for instance, that x is equal to 0 .75, oh, that's not quite far along enough, if we have x is at about 0 .75, then y must be between 0 and 0 .75.
00:35
If x is equal to, or is approaching 1, then y can approach 1.
00:42
We would have a midpoint at 0 .5, and then, for instance, at 0 .25, then y must be between 0 and 0 .25, and so on.
00:51
So we can see that we have a perfect y equals x diagonal, and we have that everything valid is in the region underneath that line.
01:01
So, to find the marginal densities of x and y, f of x, starting out with finding the marginal density of x, what we do is we integrate f of x, y over the region where y is defined as a function of x.
01:28
So, if we have that y must be between 0 and x, then the lower bound of our integration must be 0, and the upper bound of our integration must be x, and we are integrating over y.
01:42
So, we know that our joint density is f of x, y is equal to 2, so we have, oh, that should be, let me fix that there.
01:53
We have that the upper bound should be still just x there.
01:57
I'm just going to change one thing there...