Index of /faculty/kerns/video/STAT3743

STAT 3743 Course Outline

STAT 3743 Course Outline

Except where otherwise indicated, all text materials are released under the GNU Free Documentation License version 1.3.

Copyright (c) 2010 G. Jay Kerns.

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.3 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation License".

2010-08-23 Mon

Beginnings

What are STATISTICS?

Definition

Experiments
deterministic vs. random
Sample space
set of all possible outcomes

Examples:

• toss coin
• number of pets
• color of student's eyes

Random variables

Example

1. Do a random experiment
2. Make (relative) frequency table
3. Make (relative) frequency histogram

Intuitive definition of probability

Probability mass functions (PMFs)

Mean, standard deviation of a random variable

Definition

Remarks

• mean is measure of center
• mean can be anything
• mean is "first moment about origin"

Example: calculate PMF, mean, variance

2010-08-25 Wed

Mean as a measure of center

Variance as a measure of spread

Shortcut formula for variance

Empirical distribution

Mean of empirical distribution (sample mean)

Variance of the empirical distribution

Sample variance, sample standard deviation

Dealing with data

Types of data

Quantitative versus qualitative

Collect many examples of variables

2010-08-30 Mon

• Types of data
• quantitative
• qualitative
• logical
• missing
• other types

Quantitative data

Remarks

Examples:

• `precip`
• `rivers`
• `discoveries`

Displaying quantitative data

Strip charts (dot plots)

• `overplot`, `stack`, `jitter`

Histograms

• bin width issues

Stemplots

Index plots: good for data ordered in time

• methods: spikes and points

Qualitative data and factors

ID versus factors

Examples

Factors: nominal versus ordinal

Examples:

• `state.abb`
• `state.region`

Displaying qualitative data

Contingency tables, `table` and `prop.table`

Bar graphs

• Ex: `state.region`

Pareto diagrams: like an ordered bar graph with some extra stuff

• Ex: `state.division`

Dot charts (good stuff)

• Ex: `state.region`

2010-09-01 Wed

Other data types

• logical
• missing

Features of data distributions

Four fundamental features:

• Center
• Shape
• Anything unusual

Symmetry versus skewness

• right and left skewed

Kurtosis

• leptokurtic, mesokurtic, platykurtic

Unusual features: clusters or gaps

• Ex: `faithful\$eruptions`
• granularity
• extreme observations
• possible sources

Descriptive statistics

Sample mean

Sample median

Trimmed mean

Order statistics

`sort`

Sample quantile, order p

First, third quartiles

Meaning of s

• Chebychev's Rule
• Empirical Rule

Interquartile range (IQR)

Median Absolute Deviation

Measures of shape

Sample skewness

• Remarks
• Rule of thumb

Sample excess kurtosis

• Remarks
• Rule of thumb

Why is it "-3"?

How to do it with R

`library(e1071)`, `skewness`, `kurtosis`

2010-09-06 Mon

University closed (Labor Day).

2010-09-08 Wed

Ex: `stack.loss`

Sample skewness

• Ex: `stack.loss`
• Ex: `discoveries` (check rule of thumb)

Sample excess kurtosis

• Ex: `stack.loss`
• Ex: `UKDriverDeaths` (check rule of thumb)

Exploratory data analysis

More on stemplots

• trim outliers
• split stems
• depths

Ex: `faithful\$eruptions`

Hinges and the Five number summary

Upper, lower hinges

5NS

`fivenum`

Boxplots

Visual display of the 5NS

Can judge CUSS

Outliers

Potential versus suspected

Rules of thumb for both

Ex: `rivers`

Standardizing variables

Measure of relative standing: z-score

`scale`

Ex: `precip`

Bi/Multivariate data

Data frames: rectangular arrays of data

• same number of rows each column
• data in each column must be same type
• indexing: two dimensional

Access columns of data frames in multiple ways

2010-09-13 Mon

Qualitative versus qualitative

Descriptive statistics: two-way tables

`xtabs`

`addmargins`

`colPercents`, `rowPercents`

Plotting two categorical variables

• Stacked bar charts
• Side-by-side bar charts
• Spine plots

Ex: `Titanic` data

Quantitative versus quantitative

Plotting two quantitative variables

Univariate displays separately

Association:

• linear versus nonlinear
• strong versus weak

Ex: `Puromycin` data

Ex: `attenu` data

Ex: `faithful` data

Ex: `iris` data

Measuring linear association

Sample Pearson product-moment correlation coefficient (r)

Properties of r

Ex: `iris` data

Ex: `attenu` data

• measures strength and direction of linear association
• rules of thumb
• r approx 0 doesn't mean no association

One quantitative, one qualitative

Comparison of groups

Ex: `weight ~ feed, data = chickwts`

Ex: `~ age | education, data = infert`

Ex: `~ count | spray, data = InsectSprays`

Multiple variables

Multi-way contingency tables

Mosaic plots, dot charts

Sample variance-covariance matrices

Scatterplot matrices (SPLOM)

Comparing groups: coplots

Ex: `LifeCycleSavings` data

Ex: `Titanic` data

Probability

Fundamental notions

Random experiment

Sample space

Subsets, events, empty set

Set theory review

Operations:

• union
• intersection
• difference
• complement

Algebra of sets

• identities
• commutative, associative, distributive

Examples: events in plain English, write set notation

Mutually exclusive, disjoint

2010-09-15 Wed

Probability as limiting relative frequency

Axioms for Probability

• P(A) nonnegative
• P(S) = 1
• Probability of disjoint union is sum of probabilities

Properties of probability

1. complement
2. probability of null event
3. monotone
4. P(A) between zero and one
5. General addition rule (for many events)
6. Boole's inequality

Probability assignment: equally likely model

P(A) = #(A)/#(S)

Examples: tossing coins, etc.

How to count

Multiplication principle

Examples

2010-09-20 Mon

Selecting ordered samples

• examples

Selecting unordered samples

More about binomial coefficients, Pascal's triangle

Birthday problem

Poker hands

• royal flush
• four of a kind

Conditional probability

Example: draw 2 cards, without replacement

Definition

Example: toss a coin twice

Example: toss a die twice

• A = {outcomes match}
• B = {sum of outcomes >= 8}

Properties

• P(B|A) nonnegative
• P(S|A) = 1
• cond. prob of disjoint union is sum of cond. probs

More properties

Multiplication rule
P(A and B) = P(A)*P(B|A)

Mult. rule for several events

Examples

Good example: transferred ball

Independence

Example: toss two coins

Definition, intuition

Proposition: A, B independent then so are complements

Mutual Independence

Example: toss 100 coins

Pairwise doesn't imply mutual

Example: computers in series

Bayes' Rule

The theorem

Picture

What does it mean?

2010-09-22 Wed

What does [Bayes' Rule] mean?

Example: misfiling assistants

Random variables

Definition

Support set

Examples

• toss coin n times
• toss coin, wait until lands

Discrete random variables

Definition

Probability mass functions (PMFs)

Every PMF satisfies…

Example: toss coin four times, X = #Heads

• set up function, find PMF, draw graph

Mean, variance, standard deviation

• calculate mean for previous example

Remarks

• μ can be any number
• μ is measure of center
• interpretation

Cumulative distribution functions (CDFs)

Definition

Properties

• nondecreasing
• right-continuous
• limits at infinity

Example: toss coin 3 times

• find CDF

Discrete uniform distribution

Definition, PMF, notation

Examples

Find mean and variance of =disunif=(m)

Example: find mean/variance for rolling die

Bernoulli/Binomial model

Bernoulli model

Mean and variance

Binomial model

• n Bernoulli trials
• trials independent
• prob p doesn't change

Definition, PMF, notation

Check sum of f(x) is 1

Binomial series

Find mean, variance of =binom=(n,p)

Example: five child family

How to do it with R

Example: roll 15 dice

Find/plot binomial CDFs

Connection to `distr***` family of packages

2010-09-27 Mon

Expectation

Definition

Examples: mean/variance

Properties

1. E(c)=c
2. E(cX) = cE(X)
3. E[g(X)+h(X)] = Eg(X) + Eh(X)

Ex: discrete uniform

Ex: binomial mean (from definition)

Moment generating functions (MGF)

Definition

M(0) = 1

Ex: discrete uniform MGF

Ex: binomial MGF

• Why do they call it a MGF?

M'(0) = E(X) etc.

Ex: binomial mean

Ex: binomial variance

• Applications

Uniqueness theorem

Ex: binomial MGF

• Another connection

Taylor series of f about a

Definition

• mean
• variance
• CDF

Hypergeometric distribution

Dependent Bernoulli trials

Example

PMF, mean, variance

Example

2010-09-29 Wed

Example of R functions for discrete probabilities

• `dbinom`
• `pbinom`
• `pbinom`, `lower.tail = FALSE`
• `distrEx` package, `E(X)`, `var(X)`
• plots

Waiting time distributions

Geometric distribution

PMF, mean, variance, MGF

Ex: Jeff Reed

Negative Binomial distribution

PMF, mean, variance, MGF

Poisson distribution

PMF, mean, variance, MGF

Continuous distributions

Definition

Probability density functions (PDF)

Properties

• nonnegative
• integral is one
• probability is area under the curve
• remarks

2010-10-04 Mon

Continuous CDFs

• nondecreasing
• continuous function
• limits at infinity
• derivative of CDF is PDF

Expectation

• mean, variance, MGF

Examples

• finite support: `beta(4,1)`
• integral 1, probability, μ, σ
• infinite support: `exp(rate = 5)`
• integral 1, prob, μ, σ

Continuous uniform distribution

PDF, CDF, notation, R code, MGF

Find the mean, variance

Special case: `unif(min = 0, max = 1)`

Normal distribution

PDF, notation, R code

Standard normal: Z

φ, Φ

If X ~ `norm(m,s)` then Z = (X-m)/s ~ `norm(0,1)`

MGF of Z

MGF of X

MGF of Y = a + bX

68-95-99.7 (Empirical) Rule

Example: find P(a < X < b) for something, say, IQ

Quantiles and the quantile function

Want to go in reverse: given area, find the IQ

2010-10-06 Wed

Quantile functions

Definition of quantile function:

Q(p) = min{x : F(x) >= p}, 0 < p < 1

Properties:

1. defined, finite for 0 < p < 1
2. left continuous
3. reflect F about line y=x (continuous case)
4. boundary limits exist but may be infinite

Special case: normal distribution

Define zα

Examples

z0.025, z0.01, z0.005

Functions of random variables

The idea

Discrete case

Make a table, accumulate probabilities

Example: X ~ `binom`, Y = something

PDF method

Introduce the formula and conditions

Intuitive formula

Example: X ~ `norm`, Y = linear transformation

Fact: if X is any norm then lin. trans. is also normal

Example: X ~ `norm`, Y = exp(X) (lognormal)

CDF method

Introduce the method

Example: X ~ `unif(0,1)`, Y = -ln(X) (exponential)

Probability Integral Transform

Why the PIT is important

2010-10-11 Mon

Probability Integral Transform

How to do it with R

Other distributions

Exponential distribution

PDF, Graph, Notation, CDF, Graph

Find the MGF

Find the mean, variance

Memoryless property

Relationship with Poisson model

Gamma distribution

PDF, Notation, Parameters

The Gamma Function

Properties of the Gamma Function

MGF, mean and variance

Motivation: relationship with Poisson process

Example: car wash

How to do it with R

Special case: Chi square distribution

Even more continuous distributions

• Cauchy
• Beta
• Logistic
• Lognormal
• Weibull
• Student's t distribution
• F distribution

Multivariate Distributions

Comparison of univariate versus multivariate

Discrete case: joint PMFs, CDFs

Example: roll fair die twice

2010-10-13 Wed

Joint distributions

Example: roll fair die twice

Marginal distributions

Continuous case: joint PDFs, marginals

Definition

Independence

Analogy with independent events

Definition (pairwise)

Comment for multivariate mutual independence

Independent, Identically Distributed (IID)

Simple Random Sample of size n

Expectation of functions of independent random variables

Sampling Distributions

Simple Random Samples

BACKGROUND

Mean and variance of linear combination of independent sample

2010-10-18 Mon

Mean and variance of X-bar

MGF of linear combination of independent sample

MGF of X-bar

Example: SRS(11) from `gamma`, find dist'n of X-bar

Proposition: SRS(n) from `norm`, then X-bar is normal, too

Sum of independent `chisq` is `chisq`, and `df`'s add

Sum of squared standard normals is `chisq` with `df` = n

Theorem: SRS(n) from `norm`, then

1. X-bar and S2 are independent
2. S2 is scaled `chisq` with `df` = n - 1

Linear combinations of independent normals is normal, too

Central Limit Theorem

statement of the theorem

Remarks:

• if X's are normal, then X-bar is normal
• if X's are close to normal, then X-bar is close to normal, too
• Discrete/continuous doesn't matter
• how large is large?

2010-10-20 Wed

Example: SRS(40) have μ = 21 and σ = 7. Find probability for X-bar.

Student's t distribution and the F distribution

Student's t

where it comes from, notation

Remarks

• looks like `norm` but with heavy tails
• as `df` gets large approaches `norm(0,1)`

Why it's called "Student's t" and William Sealy Gosset

F distribution

where it comes from, notation

Sir Ronald Aylmer Fisher and Snedecor

Graphs of the PDF

If F ~ `f(n1,n2)` then 1/F ~ `f(n2,n1)`

Other types of sampling distributions

Two `norm` samples, difference in means

Simulation based

Estimation

Maximum likelihood

Example: go fishing, count the number of sharks

Likelihood function, L

Maximize L, for us, with derivative

X-bar pops out of nowhere!

2010-10-25 Mon

Check it's a max with FDT or SDT

General case for maximum likelihood

Remarks

• MLE: maximum likelihood estimator
• point estimators of a parameter
• sometimes take logs before differentiating

Example: MLE for `geom` distribution

More remarks

• can do it for more than one parameter
• sometimes need sophisticated numerical methods
• MLEs not unique, in general
• sometimes an MLE does not exist
• MLEs are just one of many types of point estimator

Definition of unbiased estimator

Example: X-bar is unbiased in the sharks problem

Example: in two-parameter `norm` case, MLE of variance is biased, but sample variance is unbiased

Confidence intervals for means

why CI's are important for estimation

intuition for the z-interval

Definition of z-interval, confidence interval, confidence coefficient

Example: 90% confidence interval for μ

Remarks

• for fixed confidence, as sample size increases the CI gets shorter
• for fixed sample size, as confidence increases the CI gets wider

Example: SRS(10) from pop'n, find 95% CI for μ

1. identify Parameter of interest, in context
2. check Assumptions
3. choose relevant Name of procedure based on above
4. actually calculate the Interval
5. state Conclusion, in context of problem

2010-10-27 Wed

PANIC

If σ is unknown, use s instead

Remarks

• if n is large then approx 100(1 - α )% confidence
• if n small

– if pop'n normal then use Student's t critical value, get t-interval – if pop'n non-normal — if approx normal then still not too bad — if highly skewed or outliers, ask statistician

• can have one-sided intervals, too

Example: SRS(7) from pop'n, assume `norm`, find CI for μ

Confidence intervals for difference of two means

intuition for the confidence interval

First suppose variances known: two-sample z-interval

If variances are unknown:

• samples both large: use sample variances, approximate 100(1-α )% confidence
• one (or both) samples small, then in trouble unless

– pop'ns both normal — if variances equal: pooled t-interval — variances unequal: Welch interval

Example: independent SRS(4) and SRS(7), `norm`

2010-11-01 Mon

Welch-Aspin interval explained

How to do it with R

Confidence intervals for proportions

Examples of proportions

The asymptotic z-interval for proportions

Example: SRS(4791) from `binom`, find 95% CI for p

Score interval: see book

Confidence intervals (asymptotic) for two proportions

One-sided CIs for p

How to do it with R

Sample size

margin of error, E

sample sizes for means, easy

Example: find n such that x-bar plus/minus E is 95% CI

Remarks

• always round up

For proportions, little bit harder

1. replace p with prior guess
2. replace p(1 - p) with 1/4, can't do worse

Example: monkeys who don't like bananas, find sample size for proportion, both methods

Sample size for small pop'ns, use hypergeometric distribution

Hypothesis Testing

For proportions

Example: shotgun machine, want to improve performance, install thing-a-majig

Terminology

• null hypothesis
• alternative hypothesis

2010-11-03 Wed

Basic hypothesis testing procedure

1. set up hypotheses, collect some data
2. assume null is true, construct 95% CI for parameter
3. if CI doesn't cover null value, reject H0, otherwise, don't reject

More terminology

• Type I error, Type II error
• significance level
• rejection region
• two-sided and one-sided tests

Tests of hypotheses for one proportion (asymptotic)

Draw pictures

Example: observe bunch of shotguns, do hypothesis test

Example: do another hypothesis test, but for different significance levels

p-value of a hypothesis test

Example: find p-value for previous test

Hypothesis testing: "reject H0 if p-value is small"

Tests of hypotheses for two proportions

For one mean and/or one variance

Given SRS(n) from `norm`, unknown mean

For means

• σ known: z-tests for one mean
• σ unknown: t-tests for one mean

If σ unknown but n is large then use z-test

Example: SRS(9) from `norm`, hypothesis test for the mean, find p-value

2010-11-15 Mon

Standard error of the sample mean

Standard errors in general

For variances

Tests of hypothesis for a variance

Example: `salary` from `RcmdrTestDrive`

How to do it with R

`sigma.test` from the `TeachingDemos` package

For two means and/or two variances

For two means

Independent two-sample t-test (pooled)

Example: mean `before` by `smoking` in `RcmdrTestDrive`

Remarks

• two-sample z-test
• large sample approximation

For two variances

Two-sample F-test

Example: variance of `before` by `smoking` in `RcmdrTestDrive`

How to do it with R

`var.test` from base R

2010-11-17 Wed

Remarks

• side-by-side displays
• watch assumptions

– normality – equal variances

For paired samples

Paired t-test

Example: compare `before` vs. `after` in `RcmdrTestDrive`

Simple Linear Regression

Motivation

Regression Assumptions

What does it mean?

Maximum likelihood derivation of least-squares

2010-11-24 Mon

Parameter estimates

Example: `salary` modeled by `order` in `RcmdrTestDrive`

How to do it with R: the `lm` function

Take a look at the fitted line

Residuals = ACTUAL - PREDICTED

MLE of residual standard error

Why is it called "Regression"?

Interval estimation

Sampling distributions

Confidence intervals for regression parameters

How to do it with R

2010-11-29 Mon

Residual analysis

Graphically/numerically

Assess normality

Assess constant variance

Assess independence

Other topics

Chi-square goodness of fit

Karl Pearson (1900)

Motivation/derivation

Remarks

• observed versus expected
• rule of thumb: expected counts at least 5

Example: Gregor Mendel's corn

How to do it with R

2010-12-01 Wed

Analysis of Variance (ANOVA)

Motivation

ANOVA Decomposition

Between vs. within

Sampling dist's of SSTO, SSE

Cochran's Theorem

Sampling dist'n of SST

ANOVA Table

Example: `breaks` by `tension` in `warpbreaks` data

How to do it with R

the `aov` function in base R

Remarks

• constant variance
• normality assumption
• multi-way ANOVA
• balanced sample sizes

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You may add a passage of up to five words as a Front-Cover Text, and a passage of up to 25 words as a Back-Cover Text, to the end of the list of Cover Texts in the Modified Version. Only one passage of Front-Cover Text and one of Back-Cover Text may be added by (or through arrangements made by) any one entity. If the Document already includes a cover text for the same cover, previously added by you or by arrangement made by the same entity you are acting on behalf of, you may not add another; but you may replace the old one, on explicit permission from the previous publisher that added the old one.

The author(s) and publisher(s) of the Document do not by this License give permission to use their names for publicity for or to assert or imply endorsement of any Modified Version.

1. COMBINING DOCUMENTS

You may combine the Document with other documents released under this License, under the terms defined in section 4 above for modified versions, provided that you include in the combination all of the Invariant Sections of all of the original documents, unmodified, and list them all as Invariant Sections of your combined work in its license notice, and that you preserve all their Warranty Disclaimers.

The combined work need only contain one copy of this License, and multiple identical Invariant Sections may be replaced with a single copy. If there are multiple Invariant Sections with the same name but different contents, make the title of each such section unique by adding at the end of it, in parentheses, the name of the original author or publisher of that section if known, or else a unique number. Make the same adjustment to the section titles in the list of Invariant Sections in the license notice of the combined work.

In the combination, you must combine any sections Entitled "History" in the various original documents, forming one section Entitled "History"; likewise combine any sections Entitled "Acknowledgements", and any sections Entitled "Dedications". You must delete all sections Entitled "Endorsements".

1. COLLECTIONS OF DOCUMENTS

You may make a collection consisting of the Document and other documents released under this License, and replace the individual copies of this License in the various documents with a single copy that is included in the collection, provided that you follow the rules of this License for verbatim copying of each of the documents in all other respects.

You may extract a single document from such a collection, and distribute it individually under this License, provided you insert a copy of this License into the extracted document, and follow this License in all other respects regarding verbatim copying of that document.

1. AGGREGATION WITH INDEPENDENT WORKS

A compilation of the Document or its derivatives with other separate and independent documents or works, in or on a volume of a storage or distribution medium, is called an "aggregate" if the copyright resulting from the compilation is not used to limit the legal rights of the compilation's users beyond what the individual works permit. When the Document is included in an aggregate, this License does not apply to the other works in the aggregate which are not themselves derivative works of the Document.

If the Cover Text requirement of section 3 is applicable to these copies of the Document, then if the Document is less than one half of the entire aggregate, the Document's Cover Texts may be placed on covers that bracket the Document within the aggregate, or the electronic equivalent of covers if the Document is in electronic form. Otherwise they must appear on printed covers that bracket the whole aggregate.

1. TRANSLATION

Translation is considered a kind of modification, so you may distribute translations of the Document under the terms of section 4. Replacing Invariant Sections with translations requires special permission from their copyright holders, but you may include translations of some or all Invariant Sections in addition to the original versions of these Invariant Sections. You may include a translation of this License, and all the license notices in the Document, and any Warranty Disclaimers, provided that you also include the original English version of this License and the original versions of those notices and disclaimers. In case of a disagreement between the translation and the original version of this License or a notice or disclaimer, the original version will prevail.

If a section in the Document is Entitled "Acknowledgements", "Dedications", or "History", the requirement (section 4) to Preserve its Title (section 1) will typically require changing the actual title.

1. TERMINATION

You may not copy, modify, sublicense, or distribute the Document except as expressly provided under this License. Any attempt otherwise to copy, modify, sublicense, or distribute it is void, and will automatically terminate your rights under this License.

However, if you cease all violation of this License, then your license from a particular copyright holder is reinstated (a) provisionally, unless and until the copyright holder explicitly and finally terminates your license, and (b) permanently, if the copyright holder fails to notify you of the violation by some reasonable means prior to 60 days after the cessation.

Moreover, your license from a particular copyright holder is reinstated permanently if the copyright holder notifies you of the violation by some reasonable means, this is the first time you have received notice of violation of this License (for any work) from that copyright holder, and you cure the violation prior to 30 days after your receipt of the notice.

Termination of your rights under this section does not terminate the licenses of parties who have received copies or rights from you under this License. If your rights have been terminated and not permanently reinstated, receipt of a copy of some or all of the same material does not give you any rights to use it.

1. FUTURE REVISIONS OF THIS LICENSE

The Free Software Foundation may publish new, revised versions of the GNU Free Documentation License from time to time. Such new versions will be similar in spirit to the present version, but may differ in detail to address new problems or concerns. See http://www.gnu.org/copyleft/.

Each version of the License is given a distinguishing version number. If the Document specifies that a particular numbered version of this License "or any later version" applies to it, you have the option of following the terms and conditions either of that specified version or of any later version that has been published (not as a draft) by the Free Software Foundation. If the Document does not specify a version number of this License, you may choose any version ever published (not as a draft) by the Free Software Foundation. If the Document specifies that a proxy can decide which future versions of this License can be used, that proxy's public statement of acceptance of a version permanently authorizes you to choose that version for the Document.

1. RELICENSING

"Massive Multiauthor Collaboration Site" (or "MMC Site") means any World Wide Web server that publishes copyrightable works and also provides prominent facilities for anybody to edit those works. A public wiki that anybody can edit is an example of such a server. A "Massive Multiauthor Collaboration" (or "MMC") contained in the site means any set of copyrightable works thus published on the MMC site.

"Incorporate" means to publish or republish a Document, in whole or in part, as part of another Document.

An MMC is "eligible for relicensing" if it is licensed under this License, and if all works that were first published under this License somewhere other than this MMC, and subsequently incorporated in whole or in part into the MMC, (1) had no cover texts or invariant sections, and (2) were thus incorporated prior to November 1, 2008.

The operator of an MMC Site may republish an MMC contained in the site under CC-BY-SA on the same site at any time before August 1, 2009, provided the MMC is eligible for relicensing.

To use this License in a document you have written, include a copy of the License in the document and put the following copyright and license notices just after the title page:

Copyright (c) YEAR YOUR NAME. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.3 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation License".

If you have Invariant Sections, Front-Cover Texts and Back-Cover Texts, replace the "with…Texts." line with this:

with the Invariant Sections being LIST THEIR TITLES, with the Front-Cover Texts being LIST, and with the Back-Cover Texts being LIST.

If you have Invariant Sections without Cover Texts, or some other combination of the three, merge those two alternatives to suit the situation.

If your document contains nontrivial examples of program code, we recommend releasing these examples in parallel under your choice of free software license, such as the GNU General Public License, to permit their use in free software.

Date: 2010-12-03 15:08:39 EST

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