Segment 18: Daniel Shepard
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?)
This problem can be equivalently stated as follows: Consider a random variable . Define and . Determine the covariance matrix of .
To determine the covariance, the means must first be computed as
Now the elements of the covariance matrix can be computed as
The reason that and are not correlated is the symmetry of the problem. Given , is determined up to a sign ambiguity, which cancels when summed. The covariance matrix is thus
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?
Due to this prescription, is uniformly distributed on , is uniformly distributed on , and is uniformly distributed on . The variance of each of these random variables is thus , , and respectively. Since the random variables have a correlation of 1 (knowledge of any one of them provides full knowledge of the other two, the cross-terms in the covariance matrix are simply the multiplication of the corresponding standard deviations. This results in the covariance matrix
Therefore, the correlation matrix is simply
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 ?