Homework 4

BSTA 550

Due: 11/02 at 11pm
Modified

October 30, 2025

Directions

Please turn in this homework on Sakai. Please submit your homework in pdf or html format.

You can type your work on your computer or submit a photo of your written work or any other method that can be turned into a pdf. The Adobe Scan phone app is an easy way to scan photos and compile into a PDF. Please let me know if you greatly prefer to submit a physical copy. We can work out another way for you to turn in homework.

You must show all of your work to receive credit.

Questions

  1. Mystery constant. Suppose \(X\) is a discrete random variable with a probability mass function \(p_X(x) = c(4- x)\) for \(x\) in \(\{-1, 2, 3\}\) and \(p_X(x) = 0\) otherwise.

    1. What is the value of c so that \(p_X(x)\) is a mass?
    2. Make a plot of the probability mass function and write the piecewise pmf
    3. What is the CDF of \(X\)? Write the piecewise CDF.
    4. Make a plot of the CDF.

  2. Wastebasket basketball. Chris tries to throw a ball of paper in the wastebasket behind his back (without looking). He estimates that his chance of success each time, regardless of the outcome of the other attempts, is \(1/3\). Let \(X\) be the number of attempts required. If he is not successful within the first 5 attempts, then he quits, and he lets \(X = 6\) in such a case.

    1. Draw the mass of \(X\) and define the piecewise pmf.
    2. Draw the CDF of \(X\) and define the piecewise CDF.
    3. Simulate 10,000 trials of this experiment in R and plot the approximate probability distribution.

  3. Suppose a density \(f_X(x)\) increases linearly from \((16, 0)\) to \(\left(24, \dfrac{1}{4}\right)\).

    1. Find the CDF of \(X\).
    2. What is the value of \(a\) so that \(P(X > a) = 0.75\)?

  4. For the following pdf \[f_X(x) = \begin{cases} kx^9(1-x)^2 & \text{if } 0 \leq x \leq 1 \\ 0 & \text{otherwise} \end{cases}\]

    1. What is the constant \(k\) that makes the following function a valid density?
    2. What is the cumulative distribution function (cdf) \(F_X(x)\)?