HOMEWORK

Homework is due in class on Wednesdays. Please hand in the starred problems.

Homework 1, due Wed January 12 in class.
1*. Let {X(i), i=1, 2, ....} be i.i.d. where E[max(X(i),0)] is infinite and E[max(-X(i),0)] is finite. If S(n) = X(1) + ...+ X(n), prove that S(n)/n tends to infinity a.s. as n tends to infinity. Hint: First consider what happens if you replace X(i) by min(X(i), M) for M>0.
2*. Suppose that an elevator has periods when it is functioning followed by periods when it is being repaired. Suppose that the elevator starts out functioning and the length of the ith functioning period is X(i). This is followed by a repair period lasting Y(i) time units. Suppose that X(i), Y(i) , i=1, 2, ... are all mutually independent and positive and that the X(i) all have a common distribution with finite mean E(X(1)) and the Y(i) all have a common distribution with finite mean E(Y(1)). Let W(t) be the amount of time that the elevator is functioning in the time period [0, t]. Show that W(t)/t tends to E[X(1)]/(E[X(1)]+E[Y(1)]) almost surely as t tends to infinity. Hint: You may use the result from renewal theory on pages 222-223 of Resnick, which was derived last quarter.
Solutions to homework 1.

Homework 2, due Wed January 19 in class.
Chapter 8 of Resnick: 2*, 3(a), 4(e), 5, 9*, 10, 11, 18*.
Note: you can ignore the last sentence of problem 4(e); it seems to be non-sensical.
Solutions to homework 2.

Homework 3, due Wed February 2, 2011 in class.
Chapter 9 of Resnick: 9*, 14, 16(a)*, 26, 27, 28*, 32.
Solutions to homework 3.

Homework 4, due Wednesday, February 9, 2011, in class.
Chapter 9 of Resnick: 2*, 5, 15 (i),(ii), 44*.
Solutions to homework 4.

Homework 5, due Wednesday, February 16, 2011, in class.
Chapter 10 of Resnick, 2, 3*, 5*, 7(b), 9, 14*.
Solutions to homework 5.

Homework 6, due Wednesday, February 23, 2011, in class.
Chapter 10 of Resnick, 15*, 16, 17(a), 23, 46*. (Note: there is a typo in problem 46, the indicators should be of B_i not B_n)
Solutions to homework 6.

MIDTERM, Monday, February 28, 2011, in class.
There will be a midterm exam on Monday, February 28, 2011, in class. It will test the material covered in lectures and in related readings and from the homework assignments given up to that point. Please bring your student ID, a blue book or two, and pens/pencils to write with. No books or notes are allowed. You may not bring a calculator -- you will not need one. Make sure to justify your answers (credit will not be given for "inspired'' answers). Remember that part of each problem is to set it up and to arrive at the answer by a progression of logical steps. Please start each problem on a new page, write legibly, and put your name on your blue book.
Midterm and solutions.

Wednesday, March 2, 2011, class: there will be no formal lecture on this day. The TA will go over the solutions to the midterm that day.

Last homework: Chapter 10 of Resnick: 19, 23, 29*, 39, 51*, due by 5pm on Friday, March 11. Note that in problem 51b, the i.i.d. sequence {Y_n} should start from n=0 and there is a brace missing after X_n in the hint.
Solutions to homework 7.