00:01
So we're given a joint distribution of x, y, x, y is equal to 1 eighth.
00:09
And this is on the support 0 to 4 for the x and 0 to 2 for the y.
00:17
And so it means, and it's always good to draw out the support here.
00:20
So x goes up to 4, y goes up to 2.
00:24
So we have this rectangular region in here.
00:29
And rest to, and of course it's 0 otherwise if you're outside the support here.
00:33
So the first thing we're going to do is find the marginal distribution of f of x.
00:38
And the way we do this is we take the integral of our joint over the other variable.
00:52
So x, we're going to integrate over y on the full range of y.
00:55
But just to make our lives easier, this is just the integral 0 to 2 of 1 eighth.
01:08
So using a little calculus, we get 1 to the 8th times y, that'll be 0 to 2, and we end up with 2 over 8, which is 1 quarter.
01:18
And this is on the support, 0 to 4.
01:22
And just because we're going to use it in a little bit, we're also going to find the marginal of y.
01:30
So same kind of idea, we integrate over the x, which is going to be from 0 to 4 of our joint, x.
01:36
Over x.
01:37
So we get 1 eighth x, 0 to 4.
01:40
So it's going to be 4 over 1 eighth, or 4 over 8, i mean, which is 1 half.
01:46
And this is on the full support there for 0 and 2.
01:49
Great.
01:49
So that's the first question.
01:50
The next thing we're going to do is find the probability that y is larger than x.
01:57
Y is greater than x.
02:00
So this is where our image will help us.
02:04
So here's 1, and here's 2, and years one and three there we go and so where where's the y bigger than x the only there's only one one little spot where the y is bigger than x and it's this line right here where y is bigger than x it's going to be in this region over this so what we're going to do think about i mean you could think about it as as part of a rectangle which is um one quarter of this rectangle so there's that but we're going to use calculus to do this because the answer is a quarter i'll tell theta.
02:40
It's a quarter.
02:40
I'm going to show you how we get it to quarter using our properties of distributions.
02:44
So what we're going to do is we're going to integrate.
02:46
Well, let's think about what we have here.
02:48
It's a double integral.
02:49
And it's of our joint distribution, so 1 eighth.
02:52
And we're going to integrate.
02:54
Let's see.
02:54
We want y to be bigger than x.
02:58
And so if we think about that, we're going to have this support on this is where, let me make some room here.
03:24
I'll put those back because we're going to use those a little bit.
03:37
Y has got to be bigger than x, but x is bound by zero.
03:40
So x is going to go from zero to y.
03:42
So what that means is we're going to integrate x from zero to y, d, x, d, y, and then we'll integrate y from zero up to...
03:55
So notice there's some...
03:57
We're going to have y here and y here.
03:59
So one thing that my old calculus or my old statistics professor taught us was to introduce dummy variables for these.
04:06
So instead of like x and y, think of them as like t and whatever variables you want.
04:18
T and v.
04:20
But no, remember, t is equal to x.
04:24
So this way, when we integrate, we don't get confused with the variables.
04:27
So let's go ahead and do the first one.
04:29
0 to 2.
04:30
We're going to integrate this one with respect to t.
04:32
But remember, t is x.
04:33
So we get 1 eighth t, 0 to y.
04:39
And we still have our dv out here, which remember, v is y.
04:42
Y.
04:42
So then we get 1 eighth times y minus 1 eighth times 0.
04:47
So that's just an eighth y.
04:51
We've got 1 eighth y dv.
04:55
And then we integrate here with respect to v.
05:03
Actually, i'm going to leave this as y.
05:05
This is going to go easier.
05:06
Just with x can be a little weird.
05:18
So now when we integrate we get 1 8th y squared over 2 2 and so let's see 1 8th times a half is going to be a 16th 2 goes in for y so it's 2 squared over 16 which is 4 over 16 which is a quarter that's how we know it's a quarter right so there's that and again um you can introduce variables here i'm i guess you don't really need to but sometimes you can if you want to keep your variables straight anyway it's it's a quarter.
05:53
Probably y is greater than x...