Segment 18. The Correlation Matrix

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The direct YouTube link is http://youtu.be/aW5q_P0it9E

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Problems

To Calculate

1. Random points i are chosen uniformly on a circle of radius 1, and their Failed to parse (unknown error): (x_i,y_i) coordinates in the plane are recorded. What is the 2x2 covariance matrix of the random variables Failed to parse (unknown error): X and Failed to parse (unknown error): Y ? (Hint: Transform probabilities from Failed to parse (unknown error): \theta to Failed to parse (unknown error): 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 Failed to parse (unknown error): \lambda uniformly random in Failed to parse (unknown error): (0,1) . Then a point's Failed to parse (unknown error): (x,y,z) coordinates are Failed to parse (unknown error): (\alpha\lambda,\beta\lambda,\gamma\lambda) . What is the covariance matrix of the random variables Failed to parse (unknown error): (X,Y,Z) in terms of Failed to parse (unknown error): \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 Failed to parse (unknown error): r=0.3 (say) looks like. How would you generate a bunch of points in the plane with this value of Failed to parse (unknown error): r ? Try it. Then try for different values of Failed to parse (unknown error): r . As Failed to parse (unknown error): 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 Failed to parse (unknown error): (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 Failed to parse (unknown error): r ?

Class Activity

Class Activity 3/5/14