Problems are from, An Introduction to Stochastic Modeling by S. Karlin and H. Taylor, or in a PDF file found by following the links.  (The convention here is that II.1: E1 refers to Exercise 1 of section 1 of Chapter II. While II.3: P4 refers to Problem 4 of section 3 of Chapter II.)

Homework 1 (due Monday, April 7)

Please click here for the first assignment.

Homework 2 (due Monday, April 14)

IV.1 (p. 208 --):  E5, E8, P1, P5
IV.3 (p. 243 --):  E1, E2, E3,
IV.4 (p.254 -- ):  E2

Homework 3 (due Monday, April 21):

Please see the beginning of the Lecture notes: 180Notes20080414 for the third homework assignment.

Homework 4 (due Monday, April 28):

VI.1 (p. 342 --):  E1, E2, E5, P3, P5*, P8**
IV.2 (p. 353 --):  E1, P2***

* Please show that W₁ and W₂ - W₁ are independent exponentially distributed random variables by computing P(W₁ > t and W₂ - W₁ > s) for all s,t>0.

**Hint: you can save some work using what we already have seen about two state Markov chains, see the notes or sections VI.3 or VI.6 of the book.

*** Depending on how you choose to do this problem you may find Lemma 2.7 in the lecture notes useful.

Homework 5 (due Monday, May 5):

VI.2 (p. 353 --): P2.3*
VI.3 (p. 365 --): E3.1, E3.3, P3.3, P3.4
VI.4 (p. 377 --): E4.2, P4.1

*Hint: look at the picture on page 345 to find an expression for the area in terms of the S_k.

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Test #1 is on Friday May 9
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Homework 6 (due Monday, May 12):

VI.4 (p. 377 --): P4.3
VI.5 (p. 392 --): P5.2
VI.6 (p. 405-):   P6.2

Homework 7 (due Monday, May 19):

VII.1 (p. 424-426): Ex. 1.2, 1.3; Pr. 1.1, 1.3
VII.2 (p. 431-432): Ex. 2.1
VII.3 (p. 435-437): Ex. 3.1*, 3.3**; Pr. 3.2

*Write the event {N(t)=n and W_{N(t)+1}>t+s} purely in terms of the Poisson process, N. Then use your knowledge of N in order to do the computations.

**Use facts you know about Poisson processes and make use of Ex. 3.1.

No Homework  (due Monday, May 26 Holiday):

Homework 8 (due Monday, June 2):

VII.3 (p. 435-437): Pr. 3.4
VII.4 (p. 445-447): Ex. 4.2, 4.3, 4.5; Pr. 4.1, 4.5
VII.5 (p. 455-457): Ex. 5.1; Pr. 5.1, 5.4
VIII.1 (p. 487-491): Ex. 1.1, 1.4, 1.5; Pr. 1.1  (I am going to change these!)

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The Final exam is on Wednesday, June 11 at 8:00 -- 11:00 AM in YORK 4080A.
    Please bring a blue book or paper!
    You may bring one 8 1/2 X 11 sheet (front and back) of notes if you wish.