1.49, 1.51, 1.55, 1.60, 1.66, 1.69, 2.3, 2.10, 2.23
This homework will not be collected. Exam #1 will be given on September 22.
From Text: 2.17, 2.22, 2.25, 2.28, 2.33, 2.37
The following link defines four joint density functions for random variables X and Y. For each of the four joint densities, answer the following questions.
a. Find and plot the marginal density for X.
b. Find and plot the marginal density for Y.
c. Find and plot the conditional density for Y given X.
d. Are the random variables independent? Explain.
Make up a new version of Exam #1. Use the same four basic concepts covered by the problems, but change the various density functions, etc. Give yourself 80 minutes to solve the exam that you created. Don't spend more than 20 minutes on any one problem, until you have worked on all problems. Practice showing all of your work and demonstrating as much as you can of what you know within 20 minutes. Use only your one page of notes as a reference.
3.2, 3.3, 3.5, 3.9, 3.12, 3.13
Find and sketch the density function for Z = X-Y for each of the density functions in the following link.
Z = X-Y, X and Y indep. and uniform in [0,1]. Find and sketch the density function for Z.
Z = X/Y, X and Y indep. and uniform in [0,1]. Find and sketch the density function for Z.
Z = max(X,Y), X and Y indep. and uniform in [0,1]. Find and sketch the density function for Z.
Verify 4.-6. using Monte Carlo Simulations (use MATLAB). Be prepared to turn in a printout of the MATLAB commands that you used to perform the Monte Carlo simulations, plots of the various histograms, theoretical calculations of the density functions for Z, and discussions of the comparisons between the theoretical density functions, and the corresponding histograms, which are estimates of the density functions (after normalization).
This homework will not be collected. Exam #2 will be given on October 20.
The exam will cover material from Homework #4 and Homework #5.
4.2, 4.3, 4.5, 4.6, 4.8, 4.9, 4.18, 4.19, 4.20, 4.25, 4.27
For each of the joint density functions given in Homework #3.2, find the means of X and Y, the correlation coefficient between X and Y, and the E[Y|X=-0.5].
Make up a new version of Exam #2. Use the same three basic problems, but change the various density functions and change the functions that map X and Y to Z or X to Y. Give yourself 80 minutes to solve the exam that you created. Don't spend more than 25 minutes on any one problem, until you have worked on all problems. Practice showing all of your work and demonstrating as much as you can of what you know within 25 minutes. Use only your one page of notes as a reference.
For problem 6.31, assume that there are two independent measurements X1 and X2. Assume that a is known to be 0. Consider two estimates for b: 1) X1 + X2, and 2) the maximum likelihood estimate. Which estimate is better? Explain your answer in detail, and show all work.
This homework will not be collected. Exam #3 will be given on November 25.
Make up a version of Exam #3. Make up three different problems based on the ideas from Homework #6 and Homework #7. Give yourself 75 minutes to solve the exam that you created. Don't spend more than 25 minutes on any one problem, until you have worked on all problems. Practice showing all of your work and demonstrating as much as you can of what you know within 25 minutes. Use only your one page of notes as a reference.
Handout. Review problems 6 and 7 from Homework #7 before starting this homework. Include the listing of all matlab commands that you use with this homework. This means a printout of each matlab script or function that you write, and a printout of the sequence of matlab commands that you issued in the command line.