View source for Theory of Stochastic Signals/Some Basic Definitions

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== # OVERVIEW OF THE FIRST MAIN CHAPTER # ==
<br>
This first chapter brings a brief summary of&nbsp; &raquo;'''probability calculation'''&laquo;,&nbsp; which surely many of you already know from your school days and which is an important prerequisite for understanding the chapters that follow.

This chapter includes
# some&nbsp; &raquo;definitions&laquo;&nbsp; such as&nbsp; &raquo;random experiment&laquo;&nbsp;,&nbsp; &raquo;outcome&laquo;&nbsp;,&nbsp; &raquo; event&laquo;&nbsp;, and&nbsp; &raquo;probability&laquo;&nbsp;,
# the&nbsp; &raquo;set-theoretical basics&laquo;&nbsp; relevant for probability theory,
# the clarification of&nbsp; &raquo;statistical dependence&laquo;&nbsp; and&nbsp; &raquo;statistical independence&laquo;,
# the mathematical treatment of statistical dependence by&nbsp; &raquo;Markov chains&laquo;.


==Experiment and outcome==
<br>
The starting point of any statistical investigation is a&nbsp; &raquo;'''random experiment'''&laquo;.&nbsp; By this,&nbsp; one understands
*an experiment that can be repeated as often as desired under the same conditions with an uncertain&nbsp; &raquo;'''outcome'''&laquo;&nbsp; &nbsp;$($German:&nbsp; "$\rm E\hspace{0.02cm}$rgebnis"$)$&nbsp; $E$,
 
*in which,&nbsp; however,&nbsp; the quantity&nbsp;  $ \{E_μ \}$&nbsp; of the possible outcomes is specifiable.


{{BlaueBox|TEXT=  
$\text{Definition:}$&nbsp; The number of possible outcomes is called the&nbsp; &raquo;'''outcome set size'''&laquo;&nbsp; $M$.&nbsp; Then holds:
:$$E_\mu \in G = \{E_\mu\}=  \{E_1, \hspace{0.1cm}\text{...} \hspace{0.1cm}, E_M \} .$$
#The variable&nbsp; $μ$&nbsp; can take all integer values between&nbsp; $1$&nbsp; and&nbsp; $M$.&nbsp;  
#$G = \{E_\mu\}$&nbsp; is called the event space or the&nbsp; &raquo;'''universal set'''&laquo;&nbsp; $($German:&nbsp; "Grundmenge" &nbsp; &rArr; &nbsp; letter:&nbsp; "G"$)$&nbsp; with&nbsp; $M$&nbsp; possible outcomes.}}


{{GraueBox|TEXT=  
$\text{Example 1:}$&nbsp; 
*In the experiment&nbsp; &raquo;coin toss&laquo;&nbsp; there are only two possible outcomes,&nbsp; namely&nbsp; &raquo;heads&laquo;&nbsp; and&nbsp; &raquo;tails&laquo;  &nbsp; ⇒  &nbsp; $M = 2$.&nbsp;  

*In contrast,&nbsp; in the random experiment&nbsp; &raquo;throwing a roulette ball&laquo;&nbsp; a total of&nbsp; $M = 37$&nbsp; different outcomes are possible,&nbsp; and it holds for the universal set in this case:
:$$G = \{E_\mu\} = \{0, 1, 2, \text{...} \hspace{0.1cm} , 36\}.$$}}

==Classical definition of probability==
<br>
We assume that each trial results in exactly one outcome from&nbsp; $G$&nbsp; and that each of these&nbsp; $M$&nbsp; outcomes is possible in the same way&nbsp; $($without preference or disadvantage$)$.

{{BlaueBox|TEXT=  
$\text{Definition:}$&nbsp; With this assumption,&nbsp; the&nbsp; &raquo;'''probability'''&laquo;&nbsp; of each outcome&nbsp; $E_μ$&nbsp; is equally:
:$$\Pr (E_\mu) = 1/{M}.$$}}


This is the&nbsp; &raquo;classical definition of probability&laquo;.&nbsp; ${\rm Pr}(\text{...} )$&nbsp; stands also for&nbsp; &raquo;probability&laquo;&nbsp; and is to be understood as a mathematical function.

{{GraueBox|TEXT=  
$\text{Example 2:}$&nbsp;
In the random experiment&nbsp; &raquo;coin toss&laquo;,&nbsp; the probabilities of the two possible outcomes are
:$$\rm Pr(heads)=Pr(tails)=1/2.$$

*This assumes that each attempt ends either with&nbsp; &raquo;heads&laquo;&nbsp; or with&nbsp; &raquo;tails&raquo;&nbsp; and that the coin cannot come to rest on its edge during an attempt.

*In the experiment&nbsp; &raquo;throwing a roulette ball&laquo;&nbsp; the probabilities&nbsp; ${\rm Pr}( E_μ) = 1/37$&nbsp; are equal for all numbers from&nbsp; $0$&nbsp; to&nbsp; $36$&nbsp; <br>only if the roulette table has not been manipulated.}}


Note: &nbsp; '''Probability theory – and the statistics based on it – can only provide well-founded statements if all implicitly agreed conditions are actually fulfilled'''.&nbsp; 
*Checking these conditions is not the task of statistics,&nbsp; but of those who use them.&nbsp;
 
*Since this basic rule is often violated,&nbsp; statistics has a much worse reputation in society than it actually deserves.

==Event and event probability==
<br>
{{BlaueBox|TEXT=  
$\text{Definitions:}$&nbsp;

'''(1)''' &nbsp; By an&nbsp; &raquo;'''event'''&laquo;&nbsp; we mean a set or summary of outcomes.&nbsp; We refer to the set of all events as the &nbsp; &raquo;'''event set'''&laquo;&nbsp; $\{A_i \}$. 

::Since the number&nbsp; $I$&nbsp; of possible events&nbsp; $\{A_i \}$&nbsp; is generally not the same as the number&nbsp; $M$&nbsp; of possible outcomes  &nbsp; &rArr; &nbsp;  the elements of&nbsp; $G = \{ E_μ \}$,&nbsp; <br>different indices are chosen here.

'''(2)''' &nbsp;If an event&nbsp; $A_i$&nbsp; is composed of&nbsp; $K$&nbsp; $($elementary$)$&nbsp; outcomes,&nbsp; the&nbsp; &raquo;'''event probability'''&laquo;&nbsp; is defined as follows:
:$${\rm Pr} (A_i) = \frac{K}{M} = \frac{\rm Number\hspace{0.1cm}of\hspace{0.1cm}favorable\hspace{0.1cm}outcomes}{\rm Number\hspace{0.1cm}of\hspace{0.1cm}possible\hspace{0.1cm}outcomes}.$$}}


This equation is called the&nbsp; [https://en.wikipedia.org/wiki/Pierre-Simon_Laplace &raquo;'''Laplace probability definition'''&laquo;].  
*Here,&nbsp; &raquo;favorable outcomes&laquo;&nbsp; are those outcomes that belong to the composite event&nbsp; $A_i$.

*From this definition it is already clear that a probability must always lie between&nbsp; $0$&nbsp; and&nbsp; $1$&nbsp;&nbsp; $($including these two limits$)$. 


{{GraueBox|TEXT=  
$\text{Example 3:}$&nbsp;
We now consider the experiment&nbsp; &raquo;throwing a die&laquo;.&nbsp; The possible outcomes&nbsp; $($number of points$)$&nbsp; are thus&nbsp;  $E_μ ∈ G = \{1, 2, 3, 4, 5, 6\}$. 

Let us now define two events&nbsp; $(I = 2)$, viz.
* $A_1 = \big[$the outcome is even$\big] = \{2, 4, 6\}$,&nbsp; and 
 
* $A_2 = \big[$the outcome is odd$\big] = \{1, 3, 5\}$, 


then the event set&nbsp;  $\{A_1, A_2\}$&nbsp; is equal to the universe&nbsp; $G$.&nbsp; For this example,&nbsp; the events&nbsp; $A_1$&nbsp; and&nbsp; $A_2$&nbsp; represent a so-called&nbsp; [[Theory_of_Stochastic_Signals/Set_Theory_Basics#Complete_system|&raquo;complete system&laquo;]].

On the other hand,&nbsp; the further event set&nbsp;  $\{A_3, A_4\}$&nbsp; is not equal to the universe&nbsp; $G$,&nbsp; if we define the single events as follows:
* $A_3 = \big[$the outcome is smaller than&nbsp; 3$\big] = \{1, 2\}$,
 
* $A_4 =\big[$the outcome is bigger than&nbsp; 3$\big] = \{4, 5, 6\}$. 


Here,&nbsp; the event set&nbsp; $\{A_3, A_4\}$&nbsp; does not include the element&nbsp; $3$.&nbsp; The probabilities of the events defined here are&nbsp; 
:$${\rm Pr}( A_3) = 1/3,\ {\rm Pr}( A_1) ={\rm Pr}(A_2) = {\rm Pr}(A_4) = 1/2.$$}}


&rArr; &nbsp; The topic of this chapter is illustrated with examples in the&nbsp; $($German language$)$&nbsp; learning video&nbsp;<br> &nbsp; &nbsp;  &nbsp; &nbsp; &nbsp; &nbsp; [[Klassische_Definition_der_Wahrscheinlickeit_(Lernvideo)|&raquo;Klassische Definition der Wahrscheinlickeit&laquo;]] &nbsp; &rArr; &nbsp; &raquo;Classical definition of probability&laquo;.


==Exercises for the chapter==
<br>
[[Aufgaben:Exercise_1.1:_A_Special_Dice_Game|Exercise 1.1: A Special Dice Game]]

[[Aufgaben:Exercise_1.1Z:_Sum_of_Two_Ternary_Signals|Exercise 1.1Z: Sum of Two Ternary Signals]]


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