00:01
Let's assume that x and y are continuous random variables.
00:13
Let f of xy be the joined pdf and let f of x be the marginal of x and f of y be the marginal of y.
00:24
Now let's start from the first question which is expectation of ax plus dy.
00:30
We have to prove what it says.
00:33
Then by definition, this is equal to integral over x, integral over y, a x plus b y into the joined period of both, which is f of x, then d y and d x.
00:49
Now let's split up this end, two integrals.
00:54
Before that we can just open the brackets like this, which is a x, f of x, d x, dx.
01:00
D .y, dx, and the second integral will be integral over x, integral or y, then by, f of x, y, d, d, x.
01:14
Now, we can rearrange that, here we have x, then a is a constant, so we can take that outside of the constant, then integral, a range of x, i state here.
01:27
Now again, x -i take it outside from the range of y.
01:32
Then we can see that integral over y, f -4, x -y, d -y, let's group this.
01:40
Then the remaining here, it is d -x.
01:43
And in second integral also, we can just make another arrangement, which is like this.
01:48
This b is a constant.
01:50
Then take integral of y outside.
01:53
And let's take y also outside.
01:57
The remaining things are integral over x, f of xy, d x, and i put d y outside...