# Segment 18. The Correlation Matrix

## Contents

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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 $and$ ? What is the linear correlation matrix of the same random variables?