Segment 18

1. Random points i are chosen uniformly on a circle of radius 1 and their $(x_i,y_i)$ coordinates in the plane are recorded. What is the 2x2 covariance matrix of the random variable $X$ and $Y$?
\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}
2. Points are generated in 3 dimensions: Choose $\lambda$ uniformly random in $(0,1)$. The a point's coordinates are $(\alpha \lambda , \beta \lambda, \gamma \lambda).$ What is the covariance matrix of the random variable in terms of $\alpha,\beta,\text{ and }\gamma$? What is the linear correlation matrix of the same random variables?