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Kaltenbach - A Concise Guide to Statistics

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Book:
A Concise Guide to Statistics
Author:
Kaltenbach / HansMichael
Publisher:
Springer Berlin Heidelberg : Imprint: Springer
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Books / Home and family
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2012
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Berlin;Heidelberg
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The text gives a concise introduction into fundamental concepts in statistics. Chapter 1: Short exposition of probability theory, using generic examples. Chapter 2: Estimation in theory and practice, using biologically motivated examples. Maximum-likelihood estimation in covered, including Fisher information and power computations. Methods for calculating confidence intervals and robust alternatives to standard estimators are given. Chapter 3: Hypothesis testing with emphasis on concepts, particularly type-I , type-II errors, and interpreting test results. Several examples are provided. T-tests are used throughout, followed important other tests and robust/nonparametric alternatives. Multiple testing is discussed in more depth, and combination of independent tests is explained. Chapter 4: Linear regression, with computations solely based on R. Multiple group comparisons with ANOVA are covered together with linear contrasts, again using R for computations.;Basics of Probability Theory -- Estimation -- Hypothesis Testing -- Regression -- References -- Index.

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Hans-Michael Kaltenbach SpringerBriefs in Statistics A Concise Guide to Statistics 10.1007/978-3-642-23502-3_1

Hans-Michael Kaltenbach 2012

1. Basics of Probability Theory

Hans-Michael Kaltenbach 1

(1)

ETH Zurich, Schwarzwaldallee 215, 4002 Basel, Switzerland

Hans-Michael Kaltenbach

Email:

Abstract

Statistics deals with the collection and interpretation of data. This chapter lays a foundation that allows to rigorously describe non-deterministic processes and to reason about non-deterministic quantities. The mathematical framework is given by probability theory, whose objects of interest are random quantities, their description and properties.

Keywords

Probability Distribution Moment Quantile

The laws of probability. So true in general. So fallacious in particular

Edward Gibbon

1.1 Probability and Events

In statistics, we are concerned with the collection, analysis, and interpretation of data, typically given as a random sample from a large set. We therefore need to lay a foundation in probability theory that allows us to formally represent non-deterministic processes and study their properties.

A first example . Let us consider the following situation: a dice is rolled leading to any of the numbers as a possible outcome With two dice the possible outcomes are described by - photo 1

as a possible outcome With two dice the possible outcomes are described by - photo 1

as a possible outcome . With two dice, the possible outcomes are described by the set

of size

The set of outcomes that lead to a sum of at least 10 is then

a set of size 6 A first definition of the probability that we will roll a sum - photo 4

a set of size 6. A first definition of the probability that we will roll a sum of at least 10 is given by counting the number of outcomes that lead to a sum larger or equal 10 and divide it by the number of all possible outcomes:

with the intuitive interpretation that 6 out of 36 possible outcomes are of the desired type. This definition implicitly assumes that each of the 36 possible outcomes has the same chance of occurring.

Any collection of possible outcomes A Concise Guide to Statistics - image 6

is called an event ; the previous definition assigns a probability of A Concise Guide to Statistics - image 7

to such an event. Events are sets and we can apply the usual operations on them: Let A be as above the event of having a sum of at least 10. Let us further denote by B the event that both dice show an even number; thus, B = {(2, 2), (2, 4), (2, 6), (4, 2), (4, 4), (4, 6), (6, 2), (6, 4), (6, 6)} and

The event C of rolling a sum of at least 10 and both dice even is then - photo 8

The event C of rolling a sum of at least 10 and both dice even is then described by the intersection of the two events:

and has probability

Similarly we can ask for the event of rolling a total of at least ten or both - photo 10

Similarly, we can ask for the event of rolling a total of at least ten or both dice even. This event corresponds to the union of A and B , since any of the elements of A or B will do:

The complement of an event corresponds to all possible outcomes that are not - photo 11

The complement of an event corresponds to all possible outcomes that are not covered by the event itself. For example, the event of not rolling an even number simultaneously on both dice is given by the complement of B , which is

with probability

The general case . Let

be the set of all possible outcomes of a particular experiment and denote by

any pair of events. Then, any function

with the properties

11 12 13 defines a probability measure or simpl - photo 17

(1.1)

12 13 defines a probability measure or simply a probability that - photo 18

(1.2)

13 defines a probability measure or simply a probability that allows to - photo 19

(1.3)

defines a probability measure or simply a probability that allows to compute the probability of events. The first requirement () gives us the algebraic rule how the probability of combined events is computed; importantly, this rule only applies for disjoint sets. Using the algebra of sets as above, we can immediately derive some additional facts:

Importantly there are multiple ways to define a valid probability measure for - photo 20

Importantly, there are multiple ways to define a valid probability measure for any given set

so these three requirements do not specify a unique such measure. For assigning a probability to discrete events like the ones discussed so far, it is sufficient to specify the probability A Concise Guide to Statistics - image 22

A Concise Guide to Statistics - image 22

for each possible outcome A Concise Guide to Statistics - image 23

of the experiment. For example, a die is described by its outcomes A Concise Guide to Statistics - image 24

One possible probability measure is A Concise Guide to Statistics - image 25

for each of the six possible outcomes it describes a fair die Another probability is in which case the - photo 26

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