00:01
In this question we're given a density, f of x and y, which is equal to one -third x plus one -third y when x is between 0 and 1 and y is between 0 and 2, and otherwise it's 0.
00:23
So we want to get the marginal distributions of x and y.
00:26
So first of all, f of x, we take f of x and y, and we integrate out y between 0 and 2.
00:34
So that gives us a third x plus a third y integrated d y between zero and two, which gives us a third xy plus a sixth y squared between zero and two for y.
00:54
So when y is zero, we don't get anything.
00:56
When y is two, we get two thirds x plus four over six, which is another two thirds.
01:03
So that means that f of x is two thirds x plus two thirds for x between zero and one.
01:13
Now let's get f of y in the same way.
01:16
F of y is the integral from zero to one of one third x plus one third y the x.
01:23
So that's going to be one sixth x squared plus one third x y between x equals zero and x equals one.
01:31
In x equals zero we don't get anything and in x equals one we get one.
01:34
We get one sixth plus one third y and this is valid for y between 0 and 2...