Introduction to Statistics

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