MCS 341 Problem Assignments

3. Discrete Random Variables

Sections to read	Topics	Suggested problems	Assigned problems	Due Date
3.1 3.2	Basic definition The probability distribution for a discrete random variable	3.3	3.4, 3.6	M 10/4
3.3	Expected values		3.10, 3.23, (A)[see below]	M 10/4
3.4	The binomial probability distribution	3.25 (shows binom. approx. still good), 3.29, 3.45	3.32, 3.35, 3.38	F 10/8
3.4	The binomial probability distribution	Extra credit	See below.	F 10/8
3.4	MLE	Example 3.10	3.48, 3.49	M 10/11
3.5	The geometric probability distribution	Learn a + ar + ar² ... + ar^n-1 = ?	3.54, 3.55, 3.67	M 10/11
3.6	The negative binomial distribution	3.73, 3.75	3.74, 3.76, World Series problem (ps) (pdf)	W 10/13
3.6	The negative binomial distribution	Extra credit	Prove Theorem 3.9 by direct calculation.
3.7	The hypergeometric distribution	3.91, 3.92	3.85, 3.86
3.7	The hypergeometric distribution	Extra credit	Prove Theorem 3.10 by direct calculation.
Class notes	Variance, covariance of sums, independent r.v.s		Problems given in class	F 10/15
3.8	The Poisson distribution	Know the series for e^x.	3.98, 3.99, 3.106
3.9	Moments & moment generating functions	Review Maclaurin series.	3.117, 3.118
3.10	Probability generating functions	Extra credit: 3.128, 3.129, 3.130	None	M 10/18
3.11	Tchebysheff's inequality	Extra credit: 3.133	None	M 10/18

"Suggested problems" are not to be turned in.

(A) Assume that the range of the r.v. Y is {y₁, y₂, ..., y_n} and that these values are equally likely. Determine (show your work) the expected value, E(Y), and the variance, V(Y), and relate your answers to pp. 8-9 and my "Notes on the Text."

Extra credit: Let Y have a binomial distribution with parameters p and n. Derive a simple formula for the "r^th factorial moment" of Y, E(Y^r).

Homework rules

Students are encouraged to discuss course topics with one another. On assigned work, though, you are expected to do the work on your own. Your discussion with other students should be confined to matters related to understanding the problems and general approaches to doing problems, and you should acknowledge those who help you in this way. You should not collaborate on problem solutions unless directed otherwise.

Acknowledge your sources (people and texts).
In nontrivial problems, show how you get your answers.
Turn in neat, well-written solutions, not messy first drafts. Trim "fringes." Staple multiple-page submissions.
Do not copy collaborative solutions; write up solutions in your own words.
Turn in homework on time. Each class day late reduces the possible points by 33%.
Do extra credit problems entirely on your own. Turn in solutions separately from regular homework.

Honor pledge: "On my honor, I pledge that I have not given, received, or tolerated others' use of unauthorized aid in completing this work."