Eleisha's Segment 18: The Correlation Matrix
1. Random points i are chosen uniformly on a circle of radius 1, and their coordinates in the plane are recorded. What is the 2x2 covariance matrix of the random variables and ? (Hint: Transform probabilities from to . 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 uniformly random in . Then a point's coordinates are . What is the covariance matrix of the random variables in terms of ? 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 (say) looks like. How would you generate a bunch of points in the plane with this value of ? Try it. Then try for different values of . As 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 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 ?
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