Eleisha's Segment 18: The Correlation Matrix

To Calculate

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?)

2. Points are generated in 3 dimensions by this prescription: Choose $\lambda$ uniformly random in $(0,1)$. Then a point's$(x,y,z)$ coordinates are $(\alpha \lambda ,\beta \lambda ,\gamma \lambda )$ . What is the covariance matrix of the random variables $(X,Y,Z)$ in terms of $\alpha ,\beta ,{\text{ and }}\gamma$? What is the linear correlation matrix of the same random variables?

To Think About:

1. Suppose you want to get a feel for what a linear correlation$r=0.3$ (say) looks like. How would you generate a bunch of points in the plane with this value of $r$? Try it. Then try for different values of $r$. As $r$ increases from zero, what is the smallest value where you would subjectively say "if I know one of the variables, I pretty much know the value of the other"?

2. Suppose that points in the $(x,y)$ plane fall roughly on a 45-degree line between the points (0,0) and (10,10), but in a band of about width w (in these same units). What, roughly, is the linear correlation coefficient $r$ ?

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