00:01
All right, so the joint probability density function, given here, or x, y, and we're given the support here, and we're going to find the marginal p .s of x and y.
00:10
So these are given as the f of x, x, x is equal to the integral of our joint distribution, the integral with respect to y.
00:39
So let's go ahead and do this.
00:43
So we take our, this case, it's going to equal to the integral from 0 to 1, excuse me, because that's where our y's go from, 0 to 1.
00:57
4xy, d, y is equal to, let's see, it's 4xy squared over 2, evaluated for y equals 0, 2.
01:10
Let's see, that's going to be 2.
01:13
And it's going to divide up to be just two.
01:15
And then we can get one in for y.
01:19
So 2x.
01:21
And then the y -go -zero is just going to destroy that.
01:24
So it's 2 to x.
01:26
We'll do the same for y.
01:32
Integral 4xy with respect to x.
01:35
0 and 1.
01:36
So hopefully you're noticing things are looking the same, which is a good thing.
01:41
You're recognizing patterns.
01:43
But we're integrating with respect to x now.
01:54
So again, these four, the four over two will become two, and then we get one in for x, y, excuse me, one in for x, so it's two y.
02:04
And then zero is in for x, it just destroys it.
02:06
So there we go.
02:07
There's part eight.
02:11
Let's see.
02:12
Now the conditional pdf, conditional mean and conditional variance of x given y equals y.
02:18
All right.
02:19
So let's do the conditional pdf first.
02:28
So this is going to be half of, x given y and y.
02:59
It's going to be equal to, it's the joint probability, our joint distribution divided by the marginal of that given some y value.
03:27
We're going to conveniently use our marginal pdf for y that just found 2y, our joint is 4xy, and that divides out to be 2, the y is cancelled out, so we get 2x.
03:44
That's a conditional pdf...