Table of Contents
"Introduction to Statistics"
INTRODUCTION
COMBINATORY CALCULATION
ELEMENTARY STATISTICS
ADVANCED STATISTICS
STATISTICAL INFERENCE
STOCHASTIC PROCESSES
"Introduction to Statistics"
SIMONE MALACRIDA
All mathematical topics related to statistics are presented in this book:
combinatorial calculus
probability and elementary statistics
random variables
continuous and discrete probability distributions
estimation theory and hypothesis testing
regression and Bayesian inference
stochastic processes
Simone Malacrida (1977)
Engineer and writer, has worked on research, finance, energy policy and industrial plants.
ANALYTICAL INDEX
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INTRODUCTION
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I – COMBINATORY CALCULATION
Definitions
Operations
Applications
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II – ELEMENTARY STATISTICS
Chance
Conditional probability and Bayes' theorem
Elementary statistics
Applications
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III - ADVANCED STATISTICS
Introduction
Random variables, distributions and properties
Notable inequalities
Convergence
Discrete distributions
Continuous distributions
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IV - STATISTICAL INFERENCE
Introduction
Estimation theory
Hypothesis testing
Regression
Bayesian inference
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V - STOCHASTIC PROCESSES
Definitions
Markov chains and other processes
INTRODUCTION
Statistics is now characterized as a discipline in its own right with respect to mathematics, even though its foundations are in the latter.
The expansion of statistical applications is now evident to everyone: statistical concepts are present in every scientific and technological field, in economics and politics, in sociology and in the human sciences.
The first two chapters of this book summarize the basic concepts, already understandable at the high school level.
Combinatorial calculus and the concept of probability are the first fundamentals of statistics.
Subsequently, the great leap forward given by the definitions of random variables and their distributions or laws of probability is presented.
Through the study of these distributions it is possible to study the characteristics of many relevant aspects at the application level.
Estimation theory, statistical inference and hypothesis testing are the main fields in which statistics play a predominant role.
Finally, we must not forget the statistical processes, i.e. those models that can be taken as a reference to compare practical cases.
It goes without saying that such knowledge requires in-depth mathematical preparation at university level.
I
COMBINATORY CALCULATION
Definitions
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Combinatorial calculus is the branch of mathematics that studies the possible configurations for grouping the elements of a finite set.
To do this it is necessary to introduce some operations that we are now going to expose.
We define the factorial operation of any positive integer as the multiplication of the first n positive integers less than or equal to that number.
The factorial symbol is given by an exclamation point following the number itself.
In formulas we have:
By definition 0!=1 and therefore the factorial operation is also computed recursively:
The binomial coefficient between two integers (positive and negative) is given by :
The properties of the binomial coefficient are given by:
The penultimate property generalizes the construction of the binomial coefficients according to the Tartaglia triangle.
The last property is used instead to define the binomial theorem, also called Newton's formula or Newton's binomial or binomial expansion which expresses the expansion in the n-th power of any binomial:
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Operations
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A simple permutation (or without repetitions) is an ordered sequence of the elements of a set in which each element is present once and only once.
The number of simple permutations of a set composed of n elements is given by:
If, on the other hand, the sequence contains repeating elements, we speak of permutations with repetitions .
Said k the number of times that a single element repeats itself and generalizing up to r elements, the number of permutations with repetitions is:
This formula generalizes the previous one, in fact by setting all the coefficients k equal to one, we see that we fall back into the case of permutations without repetitions.
A permutation is a bijective function within the same starting set.
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Verlag: BookRix GmbH & Co. KG
Tag der Veröffentlichung: 19.04.2023
ISBN: 978-3-7554-3944-8
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