00:01
Okay, so we're asked to find the correlation between the variables x, y, and z.
00:04
However, z isn't defined in the picture that you've uploaded.
00:09
So i'm just going to compute the correlation between x and y, which we call rxy.
00:16
And that is given by the covariance of x and y, divided by the standard deviation of x times the standard deviation of y.
00:25
So first we need the covariance of x and y.
00:28
Y.
00:29
This is defined as the expectation of x minus the expectation of x times y minus the expectation of y.
00:40
And so to start off with, we need the expectation of x and y.
00:44
Now we can see that the distribution of x, they've sort of written it down for us, is 0, 1, 2, those are the values x can take, and they take those values with probability a quarter, a half, and a quarter.
00:56
So the expectation of x is just the sum of the x values times the probabilities, so that's 0 times a quarter plus 1 times a half plus 2 times a quarter, and that gives you 1.
01:11
Similarly, y can take value 0 or 1 and does so with probability a half of each, and so the expectation of y, calculating it in the exact same way, is a half.
01:21
So the covariance of x and y is given by the expectation of x minus 1 times y minus a half, which if you expand it out is xy minus a half x minus y plus a half...