Due 12/7

Complete the following problems.

1.

Suppose that p > 0 and that the the distribution function F is given by

$$F(x) = \left\{\begin{array} {cc} 0 & {\rm if}\;x\leq 1\ 1-\frac{1}{x^p} & {\rm if}\;x\geq 1\end{array}\right.$$

Suppose that for each positive integer k $V^{(k)} = (V_1^{(k)},\dots,V_k^{(k)})$ is a random sample from F, and that $M_k = \max V^{(k)}$. Find a sequence n_k so that M_k/n_k converges in distribution to a continuous distribution.

2.

Suppose that the random $2\times 1$ vector $\vec{X}$ has a bivariate normal distribution. Suppose that E[X₁] = E[X₂] = 0, ${\rm Var}(X_1) = 2$,${\rm Var}(X_2) = 1$, and $\rho(X_1,X_2) = 1/3$. Write down ${\rm Cov}(\vec{X})$, the density of $\vec{X}$ and the variance of X₁ + X₂.

3.

Suppose that the random $2\times 1$ vector $\vec{X}$ has a bivariate normal distribution with mean $\vec{0}$ and covariance $\Sigma$. Show that X₁ has a normal distribution with E[X₁] = 0 and ${\rm Var}(X_1) = \Sigma_{1,1}$.

4.

Suppose that the random $2\times 1$ vector $\vec{X}$ has a bivariate a bivariate normal distribution with mean $\vec{0}$ and covariance $\Sigma$.Suppose that ${\rm Var}(X_1) = 2$, ${\rm Var}(X_2) = 3$, and ${\rm Cov}(X_1,X_2) = 1$. Find a constant c and a normally distributed random variable Y so that X₁ and Y are independent and X₂ = cX₁ + Y.

5.

Suppose that A is a $k\times k$ invertible matrix and $\vec{X}$ is $k\times 1$ multivariate normal random vector with mean $\vec{\mu}$ and ${\rm Cov}(\vec{X}) = \Sigma$. Show that $\vec{Y}\equiv A^t\vec{X}$ is multivariate normal with mean $A^t\vec{\mu}$ and ${\rm Cov}(\vec{Y}) = A^t\Sigma A$. What happens if the columns of A are the eigenvectors of $\Sigma$? Work out what happens if k = 2, ${\rm E}(X_1) = {\rm E}(X_2)= 0$,${\rm Var}(X_1) = {\rm Var}(X_2) = 3$, and ${\rm Cov}(X_1,X_2) = 0$.