# Segment 18. The Correlation Matrix

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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**
?