Instructor: Eric Slud, Math. Dept. Rm. 2314, X 5-5469, evs@math.umd.edu
Office hours: Monday 12-2 (initially) or by appointment.
Prerequisite: Math 140-141. Genuine proficiency in
the very basic operations of
single-variable calculus is needed, i.e.,summation of simple series,
integration (by
substitution and by parts), differentiation including chain rule. The
course is also
geared to the solution of word problems.
Text: Probability & Statistics for Engineering
and the Sciences, 6th ed. (2004),
by J. L. Devore, Duxbury Press.
Coverage:In the Devore text: Chapter 2; Chapter 3 (omitting
negative binomial
distribution, section 3.5, and Poisson process, section 3.6; Chapter
4 (omitting
Sections 4.5, 4.6); Extra topic based on handout(s): Distributions
of Functions of
Random Variables, applied to Computer Simulation of Random Data; Section
5.1
(subsections on: discrete joint probability mass functions, independent
random
variables); Section 5.2 (initial subsection only); Sections 5.3-5.5;
Chapter 1,
especially section 1.2; Chapters 6, 7, 8; Chapter 9, sections 9.1-9.3
as time
permits.
Grading: The grade in the course will be based 35% on homeworks
(about 9, graded),
40% on 3 in-class tests (approximately equally spaced across the term,
the first on
October 3), and 25% on a comprehensive final (Saturday, Dec. 20, 8-10a.m.).
Note on using the text. With this text, you should not need another
text for backup,
but it would still be a good idea to use Schaum's Outline
in Probability & Statistics
or some other introductory book as a source of auxiliary (worked) problems.
The exam and at least some of the tests
will be open-book, so throughout the
course you should maintain your own list of the most important definitions
and formulas
for easy reference.
Note on HW: Homework assignments are to be handed in, on time,
and will be graded.
Initially, about half of the assigned problems (chosen at random by
me, after you hand
your homework papers in) will be graded. You should show complete reasoning
on the
problems you hand in to get full credit since odd-numbered solutions,
at least, can be
found at the back of the book.
First Assignment: read Ch. 2, sections 2.1, 2.2 and 2.3 by next
Monday (9/8), and
the rest of Chapter 2 by the following Friday (9/12).
Problem Set 1 due Wednesday, 9/10: Sec 2.1 (p. 57) # 4, 6; Sec.
2.2 (pp. 65-66)
# 18, 20, 26; Sec. 2.3 (pp. 74-75), # 34, 40, 41; Sec. 2.4, # 48.
Note: I will grade only problems which are
in sections we have already covered
by the due-date of the assignment (and for
now, not all of those either). This means
that the last problem will not be graded this
time around.
Problem Set 2 due Friday, 9/19:
Sec. 2.3 # 39, 42. Sec. 2.4 # 50, 51, 60. Sec. 2.5 # 72, 78.
Sec. 3.1 #4 (Also give the probabilities of your
selected outcomes for X, if all zip codes
00000 to 99999 were equiprobable); Sec. 3.2 #
18, 20.
Solutions to past (selected) HW
problems [the trickiest ones] are
available at this link.
Problem Set 3 due Wednesday, 10/1:
Sec. 2.4 #62; Sec 2.5 #82, 83; Sec. 3.2 #14, 22;
Sec. 3.3 #35, 38. Sec. 3.4 #46, 62; Sec. 3.5
#64, 66; Sec. 3.6 #76, 84. (6 will be graded)
A sample test for in-class test 1, can be found here .
Problem Set 4 due Friday, 10/17:
Sec. 3.5 #65, 71; Sec. 3.6 #79; Sec. 3.3 #31,
Sec. 4.1 #2, 3, 6; Sec. 4.2 # 11, 20; Sec. 4.3
# 32, 50; Sec. 4.4 #59. (6 will be graded)
NOTE change of due-date for HW4 !!
Problem Set 5 due Wednesday, 10/29:
Sec. 4.2 #24; Sec. 4.3 # 35, 42, 51;
Sec. 4.4 # 64; Sec. 4.5 # 66, 72; Sec. 5.1 #3,
7; Sec. 5.2 # 26, and for this problem also
find the correlation of X and Y; Sec.
5.3 #38, 42.
In addition, read
the Handouts Transformation of
Random Variables and
Random Number Generation and Simulation
below.We
will cover this material in
the next couple of classes., and do Problems
TRAN.1
and TRAN.2 and Sim.1 from
these handouts, to be be handed in with HW5.
Problem Set 6 due Wednesday, 11/5:
Sec. 5.2 #22; and for this problem also
find the correlation of X and Y; Sec.5.3
#37; Sec. 5.4 #46, 48, 56; Sec. 5.5 #58, 60;
plus Sim.3 from the Random
Number Generation and Simulation handout below.
A sample test for in-class test 2, can be found here .
Problem Set 7 due Monday, 11/24:
Sec. 5.5 #58, 60; Sec. 6.1 #3, 6, 9;
Sec. 1.3 #34(a,b), 39. Sec. 7.2 #14, 20.
Also hand in:
(1) Using the data of #20 in Section 1.2, (a)
graph the empirical distribution function,
(b) use (a) to find
the sample median and 0.75 quantile, and (c) construct and
graph a scaled
relative frequency histogram using the intervals boundaries 0, 1000,
2000, 3000, 4000, 5000,
and 6000. Note that for it to be scaled, the vertical units
must be chosen so that
the total area in the histogram bars is 1.
(2) Suppose that random variables X_1,
X_2, ...., X_60 are independent and Expon(2)
distributed. Find approximately
the numerical probability that X_1 + ... + X_60 falls
between 26 and
36.
Problem Set 8 due Wednesday, 12/3:
Sec. 7.2 #21, 25. Sec. 7.3 # 30, 33, 34, 38.
Supplementary Exercises
Ch.7: #48, 52. Ch. 8 Sec. 1: # 9, 10, Sec 8.2: #16, 22.
A sample test for in-class test
3, can be found here
.
NOTE: although I did not
explicitly
list the types of problems:
sample
quantiles and SCALED relative frequency histograms;
sample-size
calculations (to achieve specified precision);
and prediction intervals
in my summary of questions
which MAY appear on the
in-class exam (Test 3)
on Friday, December 5, they ARE
fair game and you should
review these topics and prepare
your cheat-sheets accordingly.
A Review Session for the Final
Exam will be held on Wednesday,
December 17, 2003, from 3 to
5pm, in Room Mth 0103
.
A sample of Final Exam problems can be found here .
See also a quick summary
and outline of topics for you to study
in reviewing for the exam.
To see the Tests given earlier this term, along with
the Solutions
handed out for Tests 2 and 3, click here
.
Handouts:
(1)
9/29/03 This handout concerns numerical calculations for the Binomial
approximation
to Hypergeometric random variables, and the Poisson
approximation to the Binomial.
In addition, some simulated-data results are given
to show that the expectations and
probability mass functions behave as they should
according to the relative-frequency
interpretation of probabilities.
(2) 10/20/03 There
are two handouts here, respectively on Transformation
of Random
Variables
and on Random
Number Generation and Simulation . These
topics are very
important for the rest of the course, as they
allow us to generate and interpret `artificial data'
to illustrate the meaning of our Probability
Limit Theorems (Law of Large Numbers, Central
Limit Theorem) and later statistical results
(Consistent Statistical estimators, Confidence
Intervals). In addition, Simulation gives us
an `experimental' avenue to calculate via artificial
data probabilities which may be too difficult
to figure analytically.
(3) 10/22/03 The
handout on Normal
Approximation to Binomial Distribution contains
a
word-problem worked example, as well as some
numerical examples of the quality of the
normal approximation to the Binomial. This
example is continued below, in a statistical
setting (confidence interval for estimate of
a population proportion in a political opinion poll)
in handout (7) below, dated 11/19/03.
A graph comparing the
distribution function values of Binom(100,.3) with its
approximating normal distribution N(30,21)
can
be found here.
(4) 10/27/03 A
preliminary version of this handout was distributed and discussed in class:
this is our first Example
of Simulation for Calculating Probability and Expectation.
(5) 11/3/03 This
picture handed out in class shows the behavior
of sample averages Sn/n
as a function of n from 1,...,2000
on each of four sets of simulated data, from different types
ofrandom variables. Within each picture, the
sample averages Sn/n are based on progres-
sively larger segments of the same 2000 data-values,
and the point is to see that these
averages settledown to the place where the Law
of Large numbers guarantee they should
for large enough n, namely the theoretical
expectation ofthe individual r.v.'s.
(6) 11/12/03 Pictures
handed out in class to show the
behavior of scaled relative frequency
histograms
by comparison with densities . Pictures
in this document show the overlaid plots of
histograms in large simulated samples with the
theoretical densities they are supposed to
represent. Pictures overlaying empirical distribution
functions with the theoretical cdf's the
data in large simulated datsets are supposed
to represent, are also available, in two settings:
(i) The overlaid empirical
and theoretical cdf's for 1000 simulated values of Z1+Z2 (sum of
two independent standard normal deviates) can
be found here
.
(ii) The overlaid empirical
and theoretical cdf's for 1000 simulated values of U_1+...+U_100
(sum of 100 independent Uniform[0,1] independent
deviates) can be found here
.
(7) 11/19/03 The
word-problem on political opinion polling begun in handout (3) above,
dated 10/22/03, is continued here from the vantage
point of statistics, particularly
confidence
intervals for estimates of a population proportion in a political opinion
poll.
Class # Sections
Topics
& Date
(1) 9/3
1.1, 2.1 Overview &
Sample Spaces: Probability as limiting relative
frequency under replicated experiments.
(2) 9/5
2.2
Events, Prob. axioms, Equiprobable outcomes.
(3) 9/8,9/10 2.3
Counting techniques, combinatorial examples.
(4) 9/12
2.4
Conditional prob. via urn problems, Bayes rule.
(5) 9/15
2.5
Independence & dependence from sampling with &
without replacement. Examples like craps (prob that A
occurs before B in repeated trials) or other combinatorics.
(6) 9/17
3.1-3.2 Definition of random
variable, prob. mass fcn.,
combinatorial examples of calculation.
(7) 9/22
3.2
Distribution function, general & combinatorial examples.
(8) 9/24
3.4
Binomial distribution, tables, word-problems.
(9) 9/26
3.5
Hypergeometric distribution: def'n through finite sampling
(or through conditional dist of binomial X given X+Y).
Binomial as large-population limit of hypergeom.
(10) 9/29
3.6
Poisson as limit of binomial. Examples, overview of discrete
distributions (uniform, binom, hypergeom, geometric, Poisson)
determined by qualitative properties ("word problem types").
(11) 10/1
3.3
Expectation, def'n as large-sample average via relative freq's.
Calculation for binomial and made-up gambling examples.
(12) 10/8-10 3.3
Expectation of function of r.v., with variance (via formula
EX^2-(EX)^2) as illustration. EITHER additional moment-
calculations OR word-problem examples. In addition, mean
and variance for binomial, Poisson, Hypergeometric, geometric.
(13) 10/13 4.1-4.2
Definition of continuous r.v.'s, density and distribution fcns.
(14) 10/15 4.3-4.4
Uniform, exponential & normal densities: probability calculations.
(15) 10/17 4.2
Calculations of expectations, variances: interpretations of param-
eters in Uniform, exponential, normal. [Extra topic omitted:
gamma integrals for exponential moments & gamma density.]
Quantiles of continuous random variables.
(16) 10/20 [Extra topic] Change of variable:
distribution and density of function of r.v.
(18)-(19) [Extra topic] Application of univariate
change of variable either to develop other
10/22
distributional examples (Weibull, lognormal) as in Section 4.5, OR
to simulate from density F' as F^{-1}(U), where U ~ Unif[0,1].
(20) 10/24 5.1
Only subsections on discrete joint prob. mass functions, indep. rv's.
(21) 10/27 5.2
Initial subsection only. [Optional extra topic: covariance and
correlation as measure of dependence.]
(22)-(23) 11/3 5.3-5.4 The
Central Limit Theorem, Law of Large Numbers.
(24) -(25) 11/5 5.4-5.5 Applications of CLT. Normal
approximation to binomial dist'n.
Mean & variance formulas for weighted sums of iid variables.
REVIEW 11/5 FOR IN-CLASS TEST 2, to be held 11/7/03
(26) 11/10
More on CLT, exactness of CLT for normal r.v.'s.
(27) 11/12 1.2
Scaled relative frequency histograms & emipirical distribution
functions. Computer-simulated examples. Connection of sample
quantiles (ie, quantiles in empirical distribution) with true
population quantiles.
(28) 11/14 6.1
Notion of a parameter (e.g. unknown mean or variance) of a
distribution. (More discussion of empirical df's & histograms.)
(29) 11/17 6.1, 6.2
Statistic minus parameter as r.v. Verification that sample
variance is unbiased and consistent for true variance.
Method of moments estimators (skip MLE).
(30) 11/19 7.1, 7.2
Confidence intervals, def'ns and terminology. Large-sample
(31) 11/21 "
intervals for mean or proportion (variance known or estimated).
(32) 11/24 7.3
Confidence interval, (finite-sample, unknown-variance,
normal data). t distribution and tables.
(33) 11/26 7.2-7.4
One-sided intervals, additional word-problem examples.
(34) 12/1 8.1
Prediction intervals. Beginning hypothesis test terminology.
REVIEW 12/3 FOR IN-CLASS TEST 3, to be held 12/5/03
(35) 12/8 8.1-8.2
Hypothesis test terminology. Normal-data example, known variance.
Hypothesis test about population mean, more generally.
(36) 12/10 8.3-8.4
Examples of calculating power. Sample size formulas.
Hypothesis test for population proportion.
(37) 12/12 8.5
P-values. Duality between hypothesis tests & confidence intervals.
REVIEW SESSION FOR FINAL EXAM will be scheduled 12/15 or 12/16
COURSE OUTLINE
I. Descriptive Statistics and Data Presentation
Sample space, events as subsets
(Scaled) relative-frequency histograms, sample
quantiles and moments.
II. Probability Fundamentals
Probabilities as limiting relative frequencies.
Probability axioms.
Counting techniques, equally likely outcomes.
Conditional probability, (mutually) independent
events.
Bayes' rule.
*Subjective probabilities as betting-odds.
III. Discrete Random Variables
Probability mass function, distribution function,
expected values, moments.
Binomial, hypergeometric, Poisson distributions.
Binomial as limit of hypergeometric, and Poisson
as limit of binomial.
IV. Continuous Random Variables
Probability density function, distribution
function, expected values, moments.
*Theoretical quantiles for continuous random
variables.
Uniform, exponential, Normal distributions.
*Gamma function and gamma distribution.
*Transformation of random variables (by smoothly
invertible functions):
distribution function and density.
*Simulation of pseudo-random variates of specified
distribution (as inverse
d.f. of Uniform).
V. Joint distributions, Random Sampling.
Bivariate random variables, joint (discrete)
probability mass functions.
*Expectation of function of jointly
distributed random variables.
*Covariance and correlation.
Mutually independent random variables.
Sums of independent random variables,
and their means and variances.
Law of Large Numbers, Central Limit
Theorem.
*Connection between scaled histograms
of random samples and probability density.
VI. Point Estimation
Populations, statistics, parameters,
and sampling.
Properties of estimators: consistency
(*and unbiasedness)
Estimation of mean, variance, proportion.
Method of moments estimation.
*Estimators as population characteristics
of the Empirical Distribution.
VII. Confidence Intervals
Large sample confidence intervals for
means and proportions using Central Limit Thm.
Small sample methods for normal populations,
Student t distribution.
*Small sample confidence interval for
variance in normal population, chi-square distribution.
Hypothesis testing about means using
Confidence Interval.
Hypothesis testing definitions (type
I and II errors, significance level, power), examples
using binomial and normal data.