Question

Problem \(1 \quad\) Bivariate normal distribution

Assume that \(\boldsymbol{X}\) is a bivariate normal random variable with

$$ \boldsymbol{\mu}=E \boldsymbol{X}=\left(\begin{array}{l} 0 \\ 2 \end{array}\right) \quad \text { and } \quad \Sigma=\operatorname{Cov} \boldsymbol{X}=\left(\begin{array}{ll} 3 & 1 \\ 1 & 3 \end{array}\right) $$

Let

$$ \boldsymbol{Y}=\left(\begin{array}{l} Y_{1} \\ Y_{2} \end{array}\right)=\left(\begin{array}{lr} 1 / \sqrt{2} & -1 / \sqrt{2} \\ 1 / \sqrt{2} & 1 / \sqrt{2} \end{array}\right) \boldsymbol{X} $$

a) Find the mean vector and covariance matrix of \(Y\). What is the distribution of \(Y ?\) Are \(Y_{1}\) and \(Y_{2}\) independent random variables?

Let \(f\) be the pdf of \(\boldsymbol{X}\). Contours of \(f\) are the \(\boldsymbol{x}\) satisfying \(f(\boldsymbol{x})=a\) for some constant \(a>0,\) or equivalently \((x-\mu)^{\mathrm{T}} \Sigma^{-1}(x-\mu)=b\) for a corresponding constant \(b>0 .\) In the figure below, the contour for \(b=4.6\) is shown.

b) Explain the connections between the covariance matrix \(\Sigma,\) the chosen value of \(b\) and features of the ellipse (e.g. principal axes and their half-lengths). Mark these features on the figure (make a drawing or use the printed figure). What is the probability that \(\boldsymbol{X}\) falls within the given ellipse?

The following information might be useful:

Note-26-Feb-2021-2.pdf