00:01
All right, so we have a distribution here, a discrete distribution with these x and y values and these probabilities.
00:14
And we want to do a few things with this distribution.
00:16
The first thing i'm going to do is show that the covariance of x and y is equal to zero, and then we're also going to show that x and y are independent.
00:26
That's what i'm going to show.
00:27
All right, so what i've done is i've taken these values and put them into a table.
00:33
So you can have, these are just xy values, x, y, x, y, y, y, all the way down, right? x, y, x, y, y, y, y, y, and this just helps my brain organize it a little better.
00:43
And so, negative one, zero, x is negative one, y is zero, that probability is zero.
00:50
X is negative one, y is one, it's a quarter, right there, and zero, zero is six, zero, zero is a six, zero one, one, zero is a twelfth, which is right here, one zero is a twelfth, and then one one is a half, which is right there.
01:03
And you sum those all together, you get one.
01:06
And what i've done also summed up the marginal probabilities here, right? these are the f of x values.
01:11
You sum across the rows for negative 1, you get the probability of x equaling 1 is 0 .5, or 0 .25, and this is what it looks like if you kind of look at it as a column.
01:27
All right, so we're going to do, i do the same thing for y.
01:34
And so let's talk about how we find the co -bariance.
01:37
The covariance of x and y is found by taking the expected value of x, y, minus the expected value of x times the expected value of y.
01:50
So what we're going to do for these we need is expected values.
01:55
So the expected value of x is equal to the sum of all the x values multiplied by the probabilities of those x values.
02:02
So right f of x.
02:04
Likewise for the expected value of y is the sum of each y value multiplied by the probabilities of y.
02:12
And then the expected value of x, y, is where we take the sum of the product of x and y, multiplied by the joint probability of x and y.
02:22
So let's tackle these expected values first.
02:24
So i've got the x here, so i've got the distributions.
02:27
This column is filled with each x, negative 1 times a probability, negative 1 times a quarter, negative a quarter, 1.
02:33
Times the 6 to 0, or excuse me, 0 times the 6 to 0, 1 times 5 .83.
02:43
And we add those up, and there's our expected value, which is a third.
02:48
So the expected value of x is 1 third.
02:52
Now we're going to do the same thing for y.
02:55
This is what we get.
02:58
So i just, negative y is negative 1, it's the probability 0.
03:02
When y is 0 is probably a quarter, probably 1 is 0 .75.
03:05
To the whole product times the probability, the value times the probability.
03:11
We only get .75, so expected value y is 0 .75, which is 3 quarters.
03:18
Great.
03:18
So we can have our, we don't know the expected value of x, y, but we do know the expected value of x is a third.
03:24
Expectative value of y is three quarters.
03:29
Excellent.
03:29
So now we seem to find the expected value of x, y.
03:31
So what we're going to do is, how i broke this down, is i have this table which is filled with each x and y products.
03:38
So negative 1 times negative 1 is 1, which you don't really need these, but it's fine.
03:44
I did it just to show you.
03:46
Negative 1 times 0, right? negative 1 times 1 is negative 1 .0 times negative 1 .0 times negative 0...