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  1. Bayes Rule:
    The weatherman has predicted rain tomorrow. In recent years, it has rained only 73 days each year. When it actually rains, the weatherman correctly forecasts rain 70% of the time. When it doesn't rain, he incorrectly forecasts rain 30% of the time. What is the probability that it will rain tomorrow?
    Hint: Bayes Rule formula is:
    {$ P(X|E) = \frac{P(E|X)P(X)}{P(E)} $}
  2. We are given that the probability density function (pdf) of a continuous random variable X is
    {$ p(x) = \left\{ \begin{array}{ll} 4x & 0 \le x \le 1/2 \\ -4x+4 & 1/2 \le x \le 1\end{array} \right. $}
    What is the equation for the corresponding cumulative density function (cdf) P(x) and what does the graph look like?
  3. Calculate the expected value of {$X$}, {$E[X]$}, where {$X$} is a random variable representing the outcome of a roll of a trick die. Use the sample space {$x \in \{1,2,3,4,5,6\}$} and let
    {$ p(X=x) = \left\{ \begin{array}{ll} 1/2 & x=1 \\ 1/10 & x\neq 1 \end{array} \right. $}
  4. Use the properties of expectation to show that we can rewrite the variance of a random variable X (defined: {$ Var[X] = E[(X - \mu)^2]$}) as {$ Var[X] = E[X^2] - (E[X])^2 $}
  5. Consider the following system of equations:
    {$ \begin{array}{ccccccc} 2 x_{1} & + & x_{2} & + & x_{3} & = & 3, \\ 4x_{1} & & & + & 2x_{3} & = & 10, \\ 2 x_{1} & +& 2x_{2} & & & = & -2. \end{array} $}
    1. Write the system as a matrix equation of the form {$Ax = b . $}
    2. Write the solution of the system as a column {$S$} and verify by matrix multiplication that {$S $}satisfies the equation {$Ax = b . $}
    3. Write {$b $} as a linear combination of the columns in {$A. $}
  6. Consider the following matrix:
    {$\left[\begin{array}{ccc} 1 & 2 & 3 \\ 1 & 4 & 3\\ 1 & 3 & 4 \end{array}\right]$}
    1. Is the matrix invertible? (Hint: think about singularity and determinants...)
    2. What's the rank of the matrix?
  7. The eigenvalues of the matrix:
    {$ A= \left[\begin{array} {cc} 3 & 6 \\ 1 & 4 \end{array}\right]$} {$\quad$}
    are {$\lambda = 6$} and {$ \lambda\ = 1. $} Which of the following is an eigenvector for {$\lambda= 1? $}
    1. {$\left[\begin{array} {c} 0\\ 1 \end{array}\right]$}
    2. {$\left[\begin{array} {c} -3\\ 1\end{array}\right]$}
    3. {$\left[\begin{array} {c} 3\\ 1 \end{array}\right]$}
    4. {$\left[\begin{array} {c} 1\\ 0 \end{array}\right]$}
  8. Find the 0, 1, 2 and {$\infty$} norms of
    {$ x= \left(\begin{array} {c}2 \\ 1\\-4 \\ -2 \end{array}\right)$} {$\quad$}
    {$ \lim_{p \rightarrow 0} ||x||_{p}^{p} $} (The zero norm is the number of non-zero elements.)
    {$ ||x||_{1} = \sum_{i=1}^{n} |x_{i}|$} (The grid norm)
    {$ ||x||_{2} = \left(\sum_{i=1}^{n} |x_{i}|^{2}\right)^{1/2} $} (The euclidean norm)
    {$ ||x||_{\infty} = \lim_{p \rightarrow \infty} ||x||_{p} = \lim_{p \rightarrow \infty} \left(\sum_{i=1}^{n} |x_{i}|^{p}\right)^{1/p} = \max_{i}|x_{i}|$} (The max norm)

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