MML - Practice for Exam 3

We will have our third exam this Friday! Here’s our in class practice sheet.

The problems

  1. Suppose that \(X\) has the continuous distribution \[ f(x) = \frac{1}{2} x \] over the interval \([0,2]\)

    1. Write down the computation that shows that \(f\) is a good probability distribution.
    2. Use an integral to compute the mean of \(X\).
    3. Using your computed mean from part (b), write down the integral that expresses the variance of \(X\).

  2. Use \(u\)-substitution to translate the normal integral \[\frac{1}{\sqrt{50\pi}}\int_2^6 e^{-(x-3)^2/50}\,dx\] to a standard normal integral.

  3. I’ve got a coin that might very well be unfair. Suppose I flip that coin 200 times and I get 60 heads.

    1. Based on that evidence, what’s your best guess of the probability \(p\) that the coin comes up heads?
    2. Given a value of \(p\), use the binomial distribution to write down a function \(f(p)\) that expresses the probability that the coin comes up heads 60 times in 200 flips.
    3. Use calculus to find the value of \(p\) that maximizes \(f\).

  4. Find the eigenvalues and corresponding eigenvectors of \[A = \left[\begin{array}{rr}3 & 1 \\ -2 & 0\end{array}\right].\]

  5. Let’s suppose that excessive basketball watching causes tardiness. To study this problem, I collected data on 100 people. Below we see this data plotted and in a partial table.

    Hours in March Late at least once
    55 1
    46 0
    52 1
    24 0
    \(\vdots\) \(\vdots\)
    1. Suppose we model this data using logistic regression. What is the primary objective?
    2. Logistic regression produces an estimator function that you use to achieve your objective. When we have one input variable (as in this case), the estimator function depends upon two parameters - \(a\) and \(b\). Write down the general formula for the estimator in terms of the parameters \(a\) and \(b\).
    3. Suppose I have the three candidate pairs of values of \(a\) and \(b\) shown in Table 1 together with their associated log-loss. Which candidate pair \((a,b)\) should I use for my estimator?
    4. What is the resulting probability estimate that an individual who watched 55 hours of basketball in March was late to work or school at least once during that time?
    5. Sketch a rough graph of your probability estimator function right on top of the plot.
    Table 1: LR parameter candidates and their log-loss
    \(a\) \(b\) Log-loss
    0.152 7.34 0.959
    0.23 966 0.828
    0.108 5.94 1.401