Eleisha's Segment 18: The Correlation Matrix

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To Calculate:

1. Random points i are chosen uniformly on a circle of radius 1, and their $\displaystyle (x_i,y_i)$ coordinates in the plane are recorded. What is the 2x2 covariance matrix of the random variables $\displaystyle X$ and $\displaystyle Y$ ? (Hint: Transform probabilities from $\displaystyle \theta$ to $\displaystyle x$ . 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 $\displaystyle \lambda$ uniformly random in $\displaystyle (0,1)$ . Then a point's$\displaystyle (x,y,z)$ coordinates are $\displaystyle (\alpha\lambda,\beta\lambda,\gamma\lambda)$ . What is the covariance matrix of the random variables $\displaystyle (X,Y,Z)$ in terms of $\displaystyle \alpha,\beta,\text{ and }\gamma$ ? What is the linear correlation matrix of the same random variables?

1. Suppose you want to get a feel for what a linear correlation$\displaystyle r=0.3$ (say) looks like. How would you generate a bunch of points in the plane with this value of $\displaystyle r$ ? Try it. Then try for different values of $\displaystyle r$ . As $\displaystyle r$ 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 $\displaystyle (x,y)$ 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 $\displaystyle r$  ?