180C Hmwork

Follow the links to see homework assignments.

Homework #1: (Due Thursday, March 31, 2011.)

	Read the article: Shark attacks and the Poisson approximation, by Byron Schmuland at http://www.stat.ualberta.ca/people/schmu/preprints/poisson.pdf.
	Play around with the Excel spread sheet at Poisson.xlsx or in the older format at Poisson.xls. (You can generate new random data by hitting the F9 key.) See if you can understand how the data is being generated. Notice the Poisson clumping in the samples that are generated. (If you do not own Excell you can use gnumeric instead. Go to http://projects.gnome.org/gnumeric/

This is a reading and computer exploration assignment. There is nothing to turn in for this assignment.
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Problems are from S. Karlin and H. Taylor (An Introduction to Stochastic Modeling, 3rd edition). 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 #2: (Due Thursday, April 7, 2011.)

	Look at but do not hand in: V.2.E1, V.2.P8
	Hand in: ∙ V.1.E1, V.1.E4, V.1.E9, V.1.P2, V.1.P6, V.1.P7 ∙ V.2.P4, V.2.P5 ∙ V.3.E8, V.3.E9, V.3.P9 You only need find the distribution of W_{r} using the ideas in the last two lines of the problem. You did the first part of the problem last quarter.

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Homework #3: (Due Thursday, April 14, 2011.)

	Look at but do not hand in: V.6.E2
	Hand in: ∙ V.4.E2, V.4.E3, V.4.E5, V.4.P4, V.4.P8 ∙ V.5.E3, V.5.P4, V.5.P6* ∙ V.6.E3, V.6.E5, V.6.P9

* Suggestion: Let M>0 be a large number. First find the distribution of the number, K_{M}, of spheres with centers at most M -- units from the origin which cover the origin. Then take the limit as M--> infinity.

** This problem really wants us to fix a point in space, say 0, and then compute the distribution of R -- the distance to the star nearest 0.
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Test #1: Monday 04/18/2011

(Will cover Poisson Processes and related material.)
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Homework #4: (Due Thursday, April 21, 2011.)

	Read and hand in the 5 exercises in the Matrix Exponential work sheet: Matrix Exponential Work Sheet.
	Read through VI.1 of Karlin and Taylor, pages 333-340.
	Play around with the Excel sheet continuous time homogeneous Markov Chain with 4 sites, either use;

Continuous MC.xls or Continuous MC.xlsx.

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Homework #5: (Due Thursday, April 28, 2011.)

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

* Please show directly (without appeal to the general theory) 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 one of the ODE results in the lecture notes useful.
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Homework #6: (Due Thursday, May 5, 2011.)

Hand in:
(p. 365 --): VI.3.E3, VI.3P3, VI.3P4
(p. 377 --): VI.4.E2, VI.4E4, VI.4P1, VI.4.P3, VI.4P4, VI.4P6
(p. 405 --): VI.6.P2, VI.6.P4**

** Another badly worded problem. Moreover, the process in VI.6.P4 is not going to be a Markov process unless we choose the interpretation of the problem carefully. So please modify the problem so as to;

Assume the service center can fix both machines simultaneously when necessary, i.e. when both are broken. The y value now represents the number of severely broken machines.

For example the state (0,1) now means both machines are broken with one being severely broken. Both of these machines are being worked on simultaneously. The chain can now transition to (1,1) if the lightly broken machine is fixed first and to (1,0) if the severely broken machine is fixed first.
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Homework #7: (Due Thursday, May 12, 2011.)

Hand in:
(p. 392 --): VI.5.5.P2
(p. 424-426): VII.1.E2, VII.1.E3, VII.1.P1 (second part only, i.e. the part starting with; Carry out ...)
(p. 431-432): VII.2 .E1
(p. 435-437): VII.3.E1*, VII.3.E3**, VII.3.P2

*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 VII.3.E1.
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Test #2: Wednesday, May 18, 2011.

(Will cover continuous time Markov Chains and just a bit of Renewal theory.)
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Homework #8: (Due Thursday, May 19, 2011.)

Study for the test on Wednesday and start your final review. No other homework is due this week.

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Homework #9: (Due Thursday, May 26, 2011.)

Hand in:
(p. 424-426): VII.1.P3
(p. 435-437): VII.3.P4
(p. 445-447): VII.4.E2, VII.4.E3, VII.4.E5, VII.4.P1, VII.4.P5*
(p. 455-457): VII.5.E1, VII.5.P1

*See comments in the lecture notes.

Look at but do not hand in:
(p. 455-457): VII.5.P4

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Homework #10: (Due Thursday, June 2, 2011.)

The remaining problems are from Chapter 24 (p.233- 237) of the lecture notes: Final Version of Notes

Hand in from Lecture Notes (Final Version of Notes):
Exercises: 24.1 -- 24.4 and 24.8 -- 24.10. (The latest version of the notes now contains the correct formula in Exercise 24.2.)

Look at but do not hand in from Lecture Notes (Final Version of Notes):
Exercises: 24.5 -- 24.7

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Final Exam, Thursday 06/9/2011