Homework 7

BSTA 550

Modified

November 6, 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.

Extra problems do not need to be turned in!

Questions

  1. Prove that for a r.v. \(X\) and constants \(a\) and \(b\), that \[\mathrm{Var}[aX+b]=a^2\mathrm{Var}[X].\]

  2. Let \(\overline{X}\) be the random variable for the sample mean, \(\overline{X}=\frac{\sum_{i=1}^nX_i}{n}\), where the \(X_i\) are i.i.d. random variables with common mean \(\mu\) and variance \(\sigma^2\).

    1. Find \(\mathbb{E}[\overline{X}]\).

    2. Find \(Var[\overline{X}]\).

  3. Let \(f_X(x)=\lambda e^{-\lambda x}\) for \(x>0\), where \(\lambda>0\).

    1. Show \(Var[X]=\frac{1}{\lambda^2}\). You may use the result from class for \(\mathbb{E}[X]\) without first proving it.

  4. A shipping company handles containers in three different sizes: (1) 27 \(ft^3\) (3 x 3 x 3), (2) 125 \(ft^3\), and (3) 512 \(ft^3\). Let \(X_i\) (\(i = 1, 2, 3\)) denote the number of type \(i\) containers shipped during a given week. Suppose that \(\mu_1 =200,\sigma_1=10,\mu_2 =250,\sigma_2=12,\mu_3 =100,\sigma_3=8\).

    1. Assuming that \(X_1,X_2,X_3\) are independent, calculate the expected value and variance of the total volume shipped.

    2. Would your calculations necessarily be correct if the \(X_i\)’s were not independent? Explain.

  5. Consider the joint density of \(X\) and \(Y\): \[f_{X,Y}(x,y)= \dfrac32 x y\] for \(0 \leq x\) and \(0 \leq y\) and \(x+y \leq 2\) and \(f_{X,Y}(x,y)=0\) otherwise. Find \(E(Y)\).

Extra problems

  1. There is a bowl containing 30 cashews, 20 pecans, 25 almonds, and 25 walnuts. I am going to randomly pick and eat 3 nuts (without replacement). Find the expected value of the number of cashews by defining the number of cashews as a sum of random variables. (This one takes a little while if we don’t rely on the

  2. Let \(\hat{p}\) be the random variable for the sample proportion, \(\hat{p}=\frac{X}{n}\), where \(X\) is the number of successes in a random sample of size \(n\). Assume the probability of success is \(p\).

    1. Find \(\mathbb{E}[\hat{p}]\).

    2. Find \(Var[\hat{p}]\).

  3. Suppose your waiting time for a bus in the morning is uniformly distributed on [0, 8] (minutes), whereas waiting time in the evening is uniformly distributed on [0, 10] (minutes) independent of morning waiting time. Make sure to FIRST set up an equation for calculating the total waiting time in each question before calculating the mean and variance of the total waiting time. You may use results from class for the expected value and variance of uniform r.v.’s without proving them.

    1. If you take the bus each morning and evening for a week (7 days), what is your total expected waiting time?

    2. What is the variance of your total waiting time?

    3. What are the expected value and variance of the difference between morning and evening waiting times on a given day?

    4. What are the expected value and variance of the difference between total morning waiting time and total evening waiting time for a particular week?