Say \(Y\) is bivariate normal with mean vector \(\pmb{\mu}=\begin{pmatrix} 1 \\ 2\end{pmatrix}\) and covariance matrix \(\pmb{\Sigma}=\begin{pmatrix} 2 & 2 \\ 2 & 4\end{pmatrix}\). Find \(P(X<Y)\) and \(P\left(X<1\vert Y=0.5\right)\).
Say \(\pmb{X}\sim N(\begin{pmatrix}1\\1\end{pmatrix}, \begin{pmatrix}2&-1\\-1& 2\end{pmatrix}\). Find \(var(X_1^2+X_2^2)\).
\(X=\begin{pmatrix}X_1&...&X_n\end{pmatrix}\sim N(\pmb{0}, \pmb{\Sigma})\) with \(var(X_i)=2\) and \(cov(X_i,X_j)=1\) if \(\vert i-j\vert=1\), 0 otherwise. Find \(var(\sum_{i=1}^n X_i^2)\).
Say \(\pmb{X}\sim N(\begin{pmatrix}1\\1\end{pmatrix}, \begin{pmatrix}2&-1\\-1& 2\end{pmatrix}\). Find the \(cov(X_1+X_2, X_1^2+X_2^2)\).
Say we have a \(\pmb{y}=(y_1\text{ .. }y_n)'\) sample from \(N(\mu,\sigma^2)\). We want to test
\[H_0:\mu=\mu_0\text{ vs }H_a:\mu\ne\mu_0\] The usual test uses the test statistic \(T=\sqrt{n}\frac{\bar{y}-\mu_0}{s}\) and rejects \(H_0\) if \(|T|>t_{\alpha/2, n-1}\). Find the power of this test if in fact \(\mu=\mu_1\). As a numeric example use \(\mu_0=1,\mu_1=1.5,\sigma=2,n=100,\alpha=0.05\).
directly
using the noncentral t distribution