Homework 4

Problem 1

In a certain region, times (minutes) between occurrences of earthquakes (of any magnitude) have a distribution with percentiles displayed in the table below.

  1. Construct a spinner corresponding to this distribution.
  2. What percent of times are between 26.8 and 110.0 minutes?
  3. Let \(F\) be the cdf. Evaluate and interpret \(F(42.8)\).
  4. Let \(Q\) be the quantile function. Evaluate and interpret \(Q(0.9)\).
  5. Sketch (by hand) a histogram of this distribution.
Percentile Value (minutes)
10th 12.6
20th 26.8
30th 42.8
40th 61.3
50th 83.2
60th 110.0
70th 144.5
80th 193.1
90th 276.3

Problem 2

(Continued from previous HW.) In the Regina/Cady problem, let \(W=|R-Y|\) be the amount of time the first person to arrive has to wait for the second person. Recall that \(W\) is a continuous random variable with pdf \[ f_W(w) = 2(1-w), \quad 0 < w < 1. \] a. Let \(F\) be the cdf. Evaluate and interpret \(F(0.25)\). a. Find an expression for the cdf of \(W\). Set up an integral, but sketch a picture and use geometry to compute. b. Find and interpret the 25th percentile of \(W\). (You can do the next part first if you want, but it might help to start with a specific number like in this part.) c. Find the quantile function of \(W\). d. Sketch a spinner corresponding to the distribution of \(W\). Label the 25th, 50th, and 75th percentiles. e. Coding required. Use your simulation results from before to approximate the 25th, 50th, and 75th percentiles.

Problem 3

Solve Example 11.3 from the Normal distribution handout (the problem on daily high temperatures in SLO.)

Problem 4

Solve Example 11.4 from the Normal distribution handout (the problem on “effect size”.)

Problem 5

The latest series of collectible Lego Minifigures contains 3 different Minifigure prizes (labeled 1, 2, 3). Each package contains a single unknown prize. Suppose we only buy 3 packages and we consider as our sample space outcome the results of just these 3 packages (prize in package 1, prize in package 2, prize in package 3). For example, 323 (or (3, 2, 3)) represents prize 3 in the first package, prize 2 in the second package, prize 3 in the third package. Let \(X\) be the number of distinct prizes obtained in these 3 packages. Let \(Y\) be the number of these 3 packages that contain prize 1. Suppose that each package is equally likely to contain any of the 3 prizes, regardless of the contents of other packages. There are 27 possible, equally likely outcomes

box1 box2 box3 X Y
1 1 1
2 1 1
3 1 1
1 2 1
2 2 1
3 2 1
1 3 1
2 3 1
3 3 1
1 1 2
2 1 2
3 1 2
1 2 2
2 2 2
3 2 2
1 3 2
2 3 2
3 3 2
1 1 3
2 1 3
3 1 3
1 2 3
2 2 3
3 2 3
1 3 3
2 3 3
3 3 3
  1. Evaluate \(X\) and \(Y\) for each of the outcomes.
  2. Construct a two-way table representing the joint distribution of \(X\) and \(Y\).
  3. Sketch a plot representing the joint distribution of \(X\) and \(Y\).
  4. Construct a spinner corresponding to the joint distribution of \(X\) and \(Y\).
  5. Describe two ways to simulate an \((X, Y)\) pair.
  6. Identify the marginal distribution of \(X\), and construct a corresponding spinner.
  7. Identify the marginal distribution of \(Y\), and construct a corresponding spinner.
  8. Describe in detail in words how you could conduct a simulation and use the results to approximate. (This part is asking you to describe the process in words; not to write code.)
    1. \(\text{P}(X = 3)\)
    2. \(\text{P}(X = 2, Y = 1)\)
    3. \(\text{E}(X)\)
    4. \(\text{E}(Y)\)
    5. \(\text{E}(XY)\)
    6. \(\text{SD}(X)\)
    7. \(\text{SD}(Y)\)
    1. \(\text{Cov}(X, Y)\)
    1. \(\text{Corr}(X, Y)\)
  9. Coding required. Code and run the simulation from the previous part and use the simulation results to approximate each of
    1. \(\text{P}(X = 3)\)
    2. \(\text{P}(X = 2, Y = 1)\)
    3. \(\text{E}(X)\)
    4. \(\text{E}(Y)\)
    5. \(\text{E}(XY)\)
    6. \(\text{SD}(X)\)
    7. \(\text{SD}(Y)\)
    1. \(\text{Cov}(X, Y)\)
    1. \(\text{Corr}(X, Y)\)
  10. Compute
    1. \(\text{P}(X = 3)\)
    2. \(\text{P}(X = 2, Y = 1)\)
    3. \(\text{E}(X)\) iii.\(\text{E}(Y)\)
    4. \(\text{E}(XY)\)
    5. \(\text{SD}(X)\) iii.\(\text{SD}(Y)\)
    1. \(\text{Cov}(X, Y)\)
    1. \(\text{Corr}(X, Y)\)