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STUDY GUIDE FOR MODERN MATHEMATICAL STATISTICS
MAT 446 WITH DAMIEN PITMAN
Introduction & Instructions
Content Goals. We will work towards an understanding of how to quantify uncertainty and how to make decisions in the face of uncertainty. First, we will lean how
to make statements about how likely an event is based on certain assumptions. This
forms the basis for the theory of probability. We will then learn to summarize and
visualize a collection of such statements that are all based on the same assumptions
or else based on observations made in an experiment. This forms the basis for the
theory of descriptive statistics. With the basic theories of probability and descriptive
statistics we will learn to make inferences and conclusions about unknown parameters of a population. This forms the basis for the theory of statistical inference.
General Goals. This course should reinforce much of the mathematics that you have
learned thus far in your life. This means that you are expected to use what you have
previously learned. This course should also develop your problem solving skills.
This means that I will go out of my way to challenge you to consider nontrivial
questions. This course should also be applicable to the world beyond the classroom.
This means that I will try to relate what we learn form the textbook to ideas that
you may not be familiar with from coursework in the math department. This could
mean that you have to do some independent learning, which brings me to the most
important goal of all. This course should develop your skills as an independent
learner. In my opinion, a good college course should make you feel like you took the
bull by the horns and wrestled your way to a greater understanding of life.
Personal Goals. Well, what are they?
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STUDY GUIDE FOR MODERN MATHEMATICAL STATISTICS
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Course Structure
: Reading the textbook is part of your homework and this document is your
guide for that reading as well as the homework problems that you are to turn
in.
: A typical class day will consist of two or three distinct segments. For the first
several minutes of class, we will have a sort of study hall, during which time
you should ask questions individually and work on homework from sections
that have already been discussed in class. After this, we will have a discussion
and work examples from the section that was assigned reading for the day.
Lastly, there could be a five minute quiz that will test your knowledge or understanding of a vocabulary word. The word could be any vocabulary word
from the assigned reading for that class day or any previous class day. The
discussion and examples will rely on a basic understanding of the assigned
reading, which is why I reserve the option to quiz you over the vocabulary at
any time.
: I recommend making a two-column reference sheet with every vocabulary
words in one column and the definition in the other.
: Do your best to complete all homework. Keep in mind that the goals for
homework are 1) to develop a practical understanding of the concepts from
the reading, and 2) to learn and practice the technical skills that will enable
you to apply your understanding of these concepts.
: While reading or working problems, write out any questions you have for me.
: Homework content is given in this document, but the timeline will be given in
class.
STUDY GUIDE FOR MODERN MATHEMATICAL STATISTICS
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Chapter 1: Overview and Descriptive Statistics
Homework Problems. 2(c), 16, 26, 34, 38, 46, 50, 52, 62
Additional Exercises. 5, 9, 11, 33, 39, 45, 71
1. Populations and Samples
Vocabulary. data, population, census, sample, variable, univariate, bivariate, multivariate, descriptive statistics, inferential statistics, concrete population, conceptual
population, simple random sample, stratified random sample, sampling frame
2. Pictorial and Tabular Methods in Descriptive Statistics
Vocabulary. sample size, stem-and-leaf display, dotplot, frequency, relative frequency,
histogram for counting data, classes, class boundary rule, histogram for measurement data: equal class widths, histogram for measurement data: unequal class
widths, total area of each rectangle in an histogram with unequal class widths, total
area of all rectangles in a histogram with unequal class widths, unimodal or bimodal
histograms, symmetric, positively or negatively skewed, qualitative data, multivariate data
3. Measures of Location
Vocabulary. sample mean x, population mean µ, sample median x̃, population median µ̃, quartiles (first, second, and third), trimmed mean, sample proportion
4. Measures of Variability
Vocabulary. range, deviations from the mean (signed, absolute, and squared), sample variance s2 , sample standard deviation s, population variance σ2 , population
standard deviation σ, degrees of freedom, Sxx , lower fourth, upper fourth, fourth
spread f s , boxplot, outlier (mild and extreme), boxplot showing outliers, comparative boxplot
Results. computational formula for s2 , outliers formulas
Chapter 2: Probability
Homework Problems. 2, 4, 10, 12, 22, 24, 26, 28, 30, 32, 34, 44, 48, 50, 58, 62, 66, 70,
78, 80
Additional Exercises. 11, 33, 84, 88-90, 104, 105
1. Sample Spaces and Events
Vocabulary. experiment, sample space, event (simple and compound), union, intersection, complement, disjoint or mutually exclusive events
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2. Axioms, Interpretations, and Properties of Probability
Vocabulary. axioms of probability, null set, equally likely events
Results. probability of the null set and implication for Axiom 3, sum of a geometric
series, complementary probability, sum rule for probabilities with or statements (A ∪
B)
3. Counting Techniques
Vocabulary. product rule for k-tuples, permutation Pk,n , factorial m!, combination
(nk) (or Ck,n )
Results. the computational formulas for each of the vocabulary words from this
section
4. Conditional Probability
Vocabulary. conditional probability of A given that B has occurred
Results. multiplication rule for probabilities with and statements (A ∩ B), law of total
probability, Bayes’ theorem
5. Independence
Vocabulary. independent events, dependent events, mutually independent events
Chapter 3: Discrete Random Variables and Probability Distributions
Homework Problems. 4, 7(a-d), 12, 14, 16, 22, 28, 30, 34, 60, 62, 70, 116, 119
Additional Exercises. 1, 11, 65, 113, 115
1. Random Variables
Vocabulary. random variable (rv), Bernoulli rv, discrete rv, continuous rv,
2. Probability Distributions for Discrete Random Variables
Vocabulary. probability distribution or probability mass function (pmf), probability
histogram, parameter, family of probability distributions, cumulative distribution
function (cdf), step function, F ( a−)
Reading Exercises.
(1) Is it true that the cdf of any discrete rv is a step function? Why?
3. Expected Values of Discrete Random Variables
Vocabulary. expected value or mean value of a random variable, expected value of
a function, variance of X σX2 = σ2 , standard deviation (SD) of X σx = σ
Results. computational formula for σ2
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5. The Binomial Probability Distribution
Vocabulary. trials, dichotomous, binomial experiment, without replacement, binomial random variable
Results. b( x; n, p) =?, Bin( x; n, p) =?
Chapter 4: Continuous Random Variables and Probability Distributions
Homework Problems. 2, 4, 12, 18, 20, 26, 40, 42, 44, 48, 54, 62
Additional Exercises. 1, 3, 5, 11, 27, 39, 41, 43
1. Probability Density Functions and Cumulative Distribution Functions
Vocabulary. probability density function (pdf) f ( x ), uniform distribution, cumulative distribution function (cdf) F ( x ), (100p)th percentile η ( p)
Results. P( a ≤ X ≤ b) = F (b) − F ( a), F 0 ( x ) = f ( x )
2. Expected Values and Moment Generating Functions
Vocabulary. expected or mean value, E[h( X )], variance of X, σX2 = V ( X )
Results. computational formula V ( X ) = E( X 2 ) − [ E( X )]2
3. The Normal Distribution
Vocabulary. normal distribution, standard normal distribution, normal random variable, standard normal random variable Z, standard normal cdf Φ(z), z critical values
zα , standardized variable
Results. standardized normal probabilities and percentiles, binomial approximation
by the normal
Chapter 5: Joint Probability Distributions
Homework Problems. 2, 12, 18, 21, 26
Additional Exercises. 1, 3, 17, 19
1. Jointly Distributed random Variables
Vocabulary. joint probability mass function, joint probability table, marginal probability mass function, joint probability density function, marginal probability density
function, independent random variables
2. Expected Values, Covariance, and Correlation
Vocabulary. covariance, correlation coefficient ρ
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Results. E(h( X, Y )) = . . ., Cov( X, Y ) = E( XY ) − µ X · µY , relationship between independence and ρ = 0
Chapter 6: Statistics and Sampling Distributions
Homework Problems. 2, 6, 12, 18, 24, 32, 42, 46, 47, 50(a,b)
Additional Exercises. 1, 11, 15, 27, 53
1. Statistics and Their Distributions
Vocabulary. statistic, sample mean X, sample standard deviation S, sample total
T0 , sampling distribution, random sample, independent and identically distributed
(i.i.d.), simulation experiment
Reading Questions. What is the difference between X and x? Does it make sense to
say that statistics with capital letters are random variables and those with lower case
letters are values of the relevant random variable? Does it make sense to say that
capital letters correspond to unobserved values and lower case letters correspond to
observed values?
2. The Distribution of the Sample Mean
Results. E( X ) = µ X , V ( X ) = σ2 /n, E( T0 ) = nµ, V ( T0 ) = nσ2 , sampling distribution
of a normal rv, Central Limit Theorem (CLT), Law of Large Numbers (LLN)
3. The Mean, Variance, (and MGF) for Several Variables
Vocabulary. linear combination (of rv’s)
Results. mean and variance of linear combinations (of rv’s), mean and variance of
the difference between two rv’s
4. Distributions Based on a Normal Random Sample
Vocabulary. chi-squared distribution χ2ν , degrees of freedom ν, t distribution, F distribution
√
Results. T = ( X − µ)/(S/ n)
Chapter 7: Point Estimation
Homework Problems. 4(a-c), 14
Additional Exercises. 1, 11
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1. General Concepts and Criteria
Vocabulary. point estimate, parameter, statistic, point estimator, bias, unbiased estimator, principle of unbiased estimation, principle of minimum variance unbiased
estimation, MVUE, standard error, estimated standard error,
Chapter 8: Statistical Intervals Based on a Single Sample
Homework Problems. 2, 6, 12, 20, 30, 34, 44, 46
Additional Exercises. 1, 5, 13, 21, 23, 29, 31, 35, 45, 47
1. Basic Properties of Confidence Intervals
Vocabulary. 100(1 − α)% confidence interval for the mean µ, bound on the error of
estimation (a.k.a. the error bound for the mean EBM or error bound for the proportion EBP), confidence level, α as error probability
2. Large-Sample Confidence Intervals for a Population Mean and
Proportion
Vocabulary. meaning of large sample confidence interval and necessary assumptions to use it
3. Intervals Based on a Normal Population Distribution
Vocabulary. t distribution, normality assumption for t distribution, properties of t
distribution, prediction interval
Chapter 9: Tests of Hypotheses Based on a Single Sample
Homework. 2, 6, 10, 22, 26, 38, 46, 48
Additional Exercises. 5, 7, 15, 19, 21, 35, 41, 45, 46, 51
1. Hypotheses and Test Procedures
Vocabulary. statistical hypothesis, null hypothesis, alternative hypothesis, null value,
test statistic, rejection region, type I error, type II error, significance α
2. Tests about a Population Mean
Vocabulary. assumption of normality, known σ, large sample test, unknown σ
3. Tests Concerning a Population Proportion
Vocabulary. np ≥ 10 assumption, small sample test for a proportion
4. P-Values
Vocabulary. P-value, decision rule based on P-value