Segment 18

From Computational Statistics (CSE383M and CS395T)
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Calculation Problems

1. Random points i are chosen uniformly on a circle of radius 1 and their <math>(x_i,y_i)</math> coordinates in the plane are recorded. What is the 2x2 covariance matrix of the random variable <math>X </math> and <math>Y</math>?

<math>\begin{align} Cov(x,y) &= \langle \left( x- \bar{x}\right)\left(y-\bar y\right) \rangle \\ &= \langle xy \rangle \\ &= \int_0^{2\pi} sin\theta cos\theta d\theta\\ &= 0\\ Cov(y,x) &= Cov(x,y) = 0\\ Cov(x,x) &= \langle \left( x- \bar{x}\right)\left( x-\bar x\right) \rangle \\ &= \langle x^2 \rangle \\ &= \int_0^{2\pi} cos^2\theta d\theta \\ &= \pi\\ Cov(x,x) &= \langle \left( y- \bar{y}\right)\left(y-\bar{y}\right) \rangle \\ &= \langle y^2 \rangle \\ &= \int_0^{2\pi} sin^2\theta d\theta \\ &= \pi \end{align} </math>

2. Points are generated in 3 dimensions: Choose <math> \lambda </math> uniformly random in <math>(0,1)</math>. The a point's coordinates are <math>(\alpha \lambda , \beta \lambda, \gamma \lambda).</math> What is the covariance matrix of the random variable in terms of <math> \alpha,\beta,\text{ and }\gamma</math>? What is the linear correlation matrix of the same random variables?

Food for Thought Problems

Class Activity