Problem 1

Let \(X_1,...,X_{10}\sim N(10, 3)\), all independent. Find

  1. \(P(X_1>13)\)
  2. \(P(X_1+X_2>26)\)
  3. \(P(\sum_{i=1}^{ 10}X_i>130)\)

Problem 2

Say \((X,Y)\) has a bivariate normal distribution with \(\mu_x=1,\mu_y=2,\sigma_x^2=5,\sigma_y^2=7\) and \(\rho=-0.3\). Find \(E[XY]\).

Problem 3

Let \((X,Y,Z)\) have a multivariate normal distribution with mean vector \(\pmb{\mu} = \begin{pmatrix} 1&0&1 \\ \end{pmatrix}^T\) and variance-covariance matrix

\[ \pmb{\Sigma} = \begin{pmatrix} 5 & 2& 0\\ 2 & 7 & 2 \\ 0 & 2 & 10\\ \end{pmatrix} \] Find

  1. \(P(X<3)\)
  2. \(P(X+Y<3)\)
  3. \(P(X+Y+Z<3)\)
  4. \(P(X<3|Y=1)\)
  5. \(P(X<Y)\)