00:01
So we're told we have x and z, which are standard normal variables, and they're independently distributed.
00:07
So the mean is zero, variance is one.
00:10
And we're told that y is equal to x squared plus z.
00:15
And the first part, we want to show that the expected value of y, even x, is equal to x squared.
00:27
So let's go ahead and do that.
00:29
So we'll replace y with our random variables here.
00:31
So it's the expected value of x squared plus z given x.
00:38
We can write this as the expected value of x squared given x plus the expected value of z given x.
00:56
So because these are independently distributed and they're both normally distributed, the mean of z is zero.
01:04
So x provides no information on z.
01:07
So we're just basically looking at the expected value of z.
01:12
The expected value of z is the mean, which is zero.
01:14
So now we're dealing with this.
01:15
So this is taking the expected value of x squared given some x.
01:19
So this is just going to be whatever that x value is.
01:24
Because we would think about this as expected value of x squared, given x equals some specified x value, whatever would be.
01:31
Therefore, it's just x squared.
01:33
So there we go.
01:34
Now this is only the case because z, and x are independently distributed and they're both standard normal.
01:41
All right.
01:44
Now moving on, part b asks us to show that the mean of y is equal to one.
01:54
Well, let's think about the mean of y is the expectation.
01:57
So the expectation of y, that's going to be equal to the expected value of x squared plus z.
02:04
So this gives us the expected value of x squared plus the expected value of z we are going to have the expected value of z is zero the expected value of x squared this is where we're going to take the definition of the second moment here so the expected value of x squared now i mean the definition in terms of the variance so it's equal to the variance and x squared, so let me get over that, variance, plus the expected value of x, quantity squared.
03:05
The expected value of x quantity squared, well the expected value of x is zero, squared that's still zero.
03:10
So there's that, and now we have the expected value of x squared is equal to the variance.
03:14
Well, what's the variance? it's one.
03:17
So therefore we've shown that the, right, because then the expected of x squared, is so therefore the expected value of y is one that's the mean one now we have c which is about the covariance and for that we want to show the covariance of x and y is equal to zero thus the correlation of x and y is also zero so basically if we show this then we're going to get the zero correlation so the covariance, let's rewrite y in terms of our variables.
03:59
So the covariance of x and x squared plus z.
04:05
So we can break this up into the covariance of x and x squared plus the covariance of x and z.
04:20
And the covariance of x and z, well they're independently distributed, they're independent, so there's no variance, so there's no covariance, so this is zero.
04:31
Now we have to do with this, the covariance of x and x squared.
04:33
For this, we're going to use the definition of expectation, is the expectation definition of covariance.
04:42
So it's the expected value of this variable, x, multiplied by the expected variable of the other variable, in this case x squared, minus the expected value of x, of that variable, multiplied by the expected value of this variable.
05:06
In this case, is x squared...