# Difference between revisions of "Seg18. The Correlation Matrix"

## Skilled problem

### problem 1

Random points i are chosen uniformly on a circle of radius 1, and their $(x_i,y_i)$ coordinates in the plane are recorded. What is the 2x2 covariance matrix of the random variables $X$ and $Y$? (Hint: Transform probabilities from $\theta$ to $x$. Second hint: Is there a symmetry argument that some components must be zero, or must be equal?)

The matrix would be $\begin{bmatrix}  Cov(X,X) & Cov(X,Y) \\ Cov(Y,X) & Cov(Y,Y) \\ \end{bmatrix}$


Since the sample space is symmetric and the sampling is uniform, it's easy to see the mean for x,y are both 0. And the covarience of X and X is it's variance. So the matrix is actually:

$\begin{bmatrix}  <x^2>-<x>^2 & <xy> \\ <yx> & <y^2>-<y>^2 \\ \end{bmatrix}$