# 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:

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


And since x,y is on the circle, thus $x = cos\theta, y = sin\theta$

Thus, the matrix would be $\begin{bmatrix}  \pi & 0 \\ 0 & \pi \\ \end{bmatrix}$


### problem 2

Points are generated in 3 dimensions by this prescription: Choose λ uniformly random in (0,1). Then a point's (x,y,z) coordinates are (αλ,βλ,γλ). What is the covariance matrix of the random variables (X,Y,Z) in terms of α,β, and γ? What is the linear correlation matrix of the same random variables?