00:01
So this question we want to show first of all that for some random variable x, you want to show the variance of x is the expected value of x squared minus the mean of x squared.
00:16
So the variance of x is the expected value of x minus the mean of x squared.
00:25
And we can expand inside this expected value, so let's do that.
00:29
It's the expected value of x squared minus 2x mean of x plus mean of x squared.
00:38
Now we can use the linear property of the expectation value to split this sum up.
00:42
So the expected value of x squared minus twice the expected value of x times the mean of x plus the mean of x squared.
00:51
Because when we take the expected value of constant, we just get that constant.
00:57
But we know that the expected value of x is the mean of x.
00:59
So this is the expected value of x minus twice the mean of x squared plus the mean of x squared.
01:08
And now we're done because this is just the expected value of x squared minus the mean of x squared.
01:18
Okay, so now let's move on.
01:21
We have the, so we want to show that the covariance of a plus bx plus cv with y is equal to b times the covariance of x and y plus c times the covariance of x and y plus c times the covariance.
01:35
Of v and y.
01:39
So what we can use is that the covariance of a plus bx plus cv and y is the expected value of a plus bx plus cv minus the expected value of all of that times y minus the expected value of y all inside the expected value...