# Homework for STA 2111F (Fall, 1996)

All homework questions are taken from the textbook, "Probability and Measure", 3rd ed., by Patrick Billingsley. (The questions are found at the end of the corresponding section; for example, question 2.4 is the fourth question at the end of section 2.) Note that some questions have hints provided, on pages 552-580 of the text.

These questions are tentative, and modifications may be announced during lectures. Students may discuss solutions to the homework questions, especially when they are stuck, but they should understand them and write them up entirely on their own. Directly copying is strictly forbidden!

Each homework is due by 4:00 p.m. on the day indicated, and should be placed in my mailbox (in Sid Smith 6018) or slid under my office door (Sid Smith 6011) by that time. Extensions will be considered, but only when discussed well in advance.

#### Homework #1, due Friday October 11, 4:00 p.m.

Include your name, department and year, e-mail address and/or phone number, a "brief physical description", and any comments you have about the course.

Text questions: 1.1, 2.3, 2.4, 2.13, 3.5, 4.2(a), 4.4, 4.6

(For question 3.5, recall the definition of inner measure from page 37.)

#### Homework #2, due Friday November 8, 4:00 p.m.

Text questions: 5.3, 5.4, 5.5, 5.13, 6.11, 6.13, 22.1, 22.5(a), 22.6

Bonus: 22.11 (do only if you've done all the rest!)

Notes:

• For questions 5.3 and 5.5, do not assume the random variables are simple.
• For question 5.5, recall that an inequality is sharp if it sometimes holds with equality.
• Question 5.13 is not clearly worded; you may take the question to be, "Prove the inclusion-exclusion formula".
• For question 22.1, you may assume that the X_i are indicator random variables, i.e. they only take the values 0 and 1. But Y is an arbitrary random variable.

#### Homework #3, due Thursday December 5, 4:00 p.m.

Text questions: 15.2, 17.7, 7.1, 7.7, 8.3, 8.13, 8.20, 22.8

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