No R packages are permitted for use in this assignment.

  1. Construct two random variables that have zero correlation, but are not independent.

  2. Recall the definition of the multivariate normal distribution from class:

    Definition 1. The \(p\)-dimensional random vector \(Y\) is said to follow the multivariate normal distribution with mean \(\mu\) and covariance matrix \(\Sigma\) if for every \(p\)-dimensional vector \(v\) such that \(v'\Sigma v \neq 0\),

    \[ v'Y \sim N(v'\mu,\, v'\Sigma v). \]

    Denote \(Y \sim N_p(\mu, \Sigma)\).

    \(\blacksquare\)

    Prove that if \(\Sigma\) is nonsingular, then \(Y \sim N_p(\mu, \Sigma)\) if and only if \(Y\) has density,

    \[ f(y) = \det(2\pi\Sigma)^{-\frac{1}{2}} e^{-\frac{1}{2}(y-\mu)'\Sigma^{-1}(y-\mu)}. \]

  3. Let \(U\) and \(V\) be independent \(N(0,1)\) random variables, and define \(Y := V\) and

    \[ X := \begin{cases} U & \text{if } UV \geq 0 \\ -U & \text{if } UV < 0 \end{cases} \]
    1. Show that \(X\) and \(Y\) each follow the standard normal distribution, but that \((X, Y)\) is not bivariate normal.
    2. Show that \(X^2\) and \(Y^2\) are independent.

  4. Let \(X \sim N_p(\mu, \Sigma)\). Show that for any partition of components, i.e.,

    \[ X = \begin{pmatrix} X_1 \\ \vdots \\ X_m \end{pmatrix}, \quad \mu = \begin{pmatrix} \mu_1 \\ \vdots \\ \mu_m \end{pmatrix}, \quad \Sigma = \begin{pmatrix} \Sigma_{11} & \cdots & \Sigma_{1m} \\ \vdots & \ddots & \vdots \\ \Sigma_{m1} & \cdots & \Sigma_{mm} \end{pmatrix}, \]

    \(X_1, \ldots, X_m\) are mutually independent if and only if \(\Sigma_{ij} = 0\) for every \(i \neq j\).

  5. Suppose that \((X, Y)\) has a bivariate distribution (not necessarily Gaussian) with mean \((\mu_X, \mu_Y)'\) and covariance matrix

    \[ \begin{pmatrix} \sigma_X^2 & \sigma_{X,Y} \\ \sigma_{Y,X} & \sigma_Y^2 \end{pmatrix}. \]
    1. Show that if \(E(Y \mid X) = \beta_0 + \beta_1 X\), then \(\beta_1 = \sigma_{Y,X}/\sigma_X^2\) and \(\beta_0 = \mu_Y - \beta_1\mu_X\).
    2. Show that if \(E(Y \mid X) = \beta_0 + \beta_1 X\) and \(\text{Var}(Y \mid X) = \tau^2\), then \(\tau^2 = \sigma_Y^2 - \sigma_{Y,X}^2/\sigma_X^2\).

  6. Let

    \[ Y \sim N_2\!\left(\begin{pmatrix} 1 \\ 1 \end{pmatrix}, \begin{pmatrix} 4 & 0 \\ 0 & 1 \end{pmatrix}\right), \quad A = \frac{1}{8}\begin{pmatrix} 1 & 2 \\ 2 & 4 \end{pmatrix}, \]

    and \(B = (1, -2)'\). Find the joint distribution of \(Y'AY\) and \(B'Y\).