Theory of Stochastic Signals/Some Basic Definitions - Revision history

Guenter at 16:47, 30 November 2023

2023-11-30T16:47:16Z

Guenter at 15:20, 13 December 2022

2022-12-13T15:20:45Z

Guenter at 14:10, 13 December 2022

2022-12-13T14:10:52Z

Hwang at 20:21, 11 December 2022

2022-12-11T20:21:25Z

Hwang at 18:35, 8 December 2022

2022-12-08T18:35:29Z

Guenter at 14:39, 14 November 2021

2021-11-14T14:39:01Z

Guenter at 12:46, 14 November 2021

2021-11-14T12:46:46Z

Guenter at 17:04, 13 November 2021

2021-11-13T17:04:23Z

Guenter at 16:24, 13 November 2021

2021-11-13T16:24:14Z

Guenter at 15:10, 13 November 2021

2021-11-13T15:10:34Z

@@ Line 19: / Line 19: @@
 <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; $E$,
+*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.
@@ Line 26: / Line 27: @@
 $\text{Definition:}$&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;
+#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.}}
+#$G = \{E_\mu\}$&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; "coin toss"&nbsp; there are only two possible outcomes,&nbsp; namely&nbsp; "heads"&nbsp; and&nbsp; "tails"  &nbsp; ⇒  &nbsp; $M = 2$.&nbsp;
+$\text{Example 1:}$&nbsp;
-In contrast, in the random experiment&nbsp; "throwing a roulette ball"&nbsp; a total of&nbsp; $M = 37$&nbsp; different outcomes are possible,&nbsp; and it holds for the universal set in this case:
+*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 here 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 (without preference or disadvantage).
+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=
@@ Line 45: / Line 47: @@
-This is the&nbsp; "classical definition of probability".&nbsp; ${\rm Pr}(\text{...} )$&nbsp; stands for&nbsp; "probability"&nbsp; and is to be understood as a mathematical function.
+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; "coin toss",&nbsp; the probabilities of the two possible outcomes are
+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; "heads"&nbsp; or with&nbsp; "tails"&nbsp; and that the coin cannot come to rest on its edge during an attempt.
+*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.
-*Also in the experiment&nbsp; "throwing a roulette ball"&nbsp; the probabilities&nbsp; ${\rm Pr}( E_μ) = 1/37$&nbsp; are equal for all numbers from&nbsp; $0$&nbsp; to&nbsp; $36$&nbsp; only if the roulette table has not been manipulated.}}
+*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.
@@ Line 68: / Line 71: @@
 '''(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.
+::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) outcomes, the&nbsp; &raquo;'''event probability'''&laquo;&nbsp; is defined as follows:
+'''(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 $\text{Laplace probability definition}$].
+This equation is called the&nbsp; [https://en.wikipedia.org/wiki/Pierre-Simon_Laplace &raquo;'''Laplace probability definition'''&laquo;].
-*Here,&nbsp; "favorable outcomes"&nbsp; are those outcomes that belong to the composite event&nbsp; $A_i$.
+*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; "throwing a die".&nbsp; The possible outcomes&nbsp; (number of points)&nbsp; are thus&nbsp;  $E_μ ∈ G = \{1, 2, 3, 4, 5, 6\}$.
+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|$\text{complete system}$]].
+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; ${\rm Pr}( A_3) = 1/3$&nbsp; and&nbsp; ${\rm Pr}( A_1) ={\rm Pr}(A_2) = {\rm Pr}(A_4) = 1/2$.}}
+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;
-&rArr; &nbsp; The topic of this chapter is illustrated with examples in the&nbsp; (German language)&nbsp; learning video&nbsp; [[Klassische_Definition_der_Wahrscheinlickeit_(Lernvideo)|"Klassische Definition der Wahrscheinlickeit"]] &nbsp; &rArr; &nbsp; "Classical definition of probability".
+&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;.

@@ Line 10: / Line 10: @@
 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;,
+# 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&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 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;.
+# the mathematical treatment of statistical dependence by&nbsp; &raquo;Markov chains&laquo;.

← Older revision		Revision as of 14:10, 13 December 2022
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−	The topic of this chapter is illustrated with examples in the  (German language)  learning video  [[Klassische_Definition_der_Wahrscheinlickeit_(Lernvideo)\|"Klassische Definition der Wahrscheinlickeit"]]   ⇒   "Classical definition of probability".	+	⇒   The topic of this chapter is illustrated with examples in the  (German language)  learning video  [[Klassische_Definition_der_Wahrscheinlickeit_(Lernvideo)\|"Klassische Definition der Wahrscheinlickeit"]]   ⇒   "Classical definition of probability".

← Older revision		Revision as of 20:21, 11 December 2022
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−	then the event set  $\{A_1, A_2\}$  is equal to the universe  $G$.  For this example,  the events  $A_1$  and  $A_2$  represent a so-called  [[Theory_of_Stochastic_Signals/Set_Theory_Basics#~~Vollst.C3.A4ndiges_System~~\|$\text{complete system}$]].	+	then the event set  $\{A_1, A_2\}$  is equal to the universe  $G$.  For this example,  the events  $A_1$  and  $A_2$  represent a so-called  [[Theory_of_Stochastic_Signals/Set_Theory_Basics#Complete_system\|$\text{complete system}$]].

	On the other hand,  the further event set  $\{A_3, A_4\}$  is not equal to the universe  $G$,  if we define the single events as follows:		On the other hand,  the further event set  $\{A_3, A_4\}$  is not equal to the universe  $G$,  if we define the single events as follows:

@@ Line 7: / Line 7: @@
 == # OVERVIEW OF THE FIRST MAIN CHAPTER # ==
 <br>
-This first chapter brings a brief summary of&nbsp; '''probability calculation''',&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 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
@@ Line 18: / Line 18: @@
 ==Experiment and outcome==
 <br>
-The starting point of any statistical investigation is a&nbsp; '''random experiment'''.&nbsp; By this,&nbsp; one understands
+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; '''outcome'''&nbsp; &nbsp;(German:&nbsp; "$\rm E\hspace{0.02cm}$rgebnis")&nbsp; $E$,
+*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; '''outcome set size'''&nbsp; $M$.&nbsp; Then holds:
+$\text{Definition:}$&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; '''universal set'''&nbsp; $($German:&nbsp; "Grundmenge" &nbsp; &rArr; &nbsp; letter:&nbsp; "G"$)$&nbsp; with&nbsp; $M$&nbsp; possible outcomes.}}
+*$G = \{E_\mu\}$&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.}}
@@ Line 41: / Line 41: @@
 {{BlaueBox|TEXT=
-$\text{Definition:}$&nbsp; With this assumption,&nbsp; the&nbsp; '''probability'''&nbsp; of each outcome&nbsp; $E_μ$&nbsp; is equally:
+$\text{Definition:}$&nbsp; With this assumption,&nbsp; the&nbsp; &raquo;'''probability'''&laquo;&nbsp; of each outcome&nbsp; $E_μ$&nbsp; is equally:
 :$$\Pr (E_\mu) = 1/{M}.$$}}
@@ Line 66: / Line 66: @@
 $\text{Definitions:}$&nbsp;
-'''(1)''' &nbsp; By an&nbsp; '''event'''&nbsp; we mean a set or summary of outcomes.&nbsp; We refer to the set of all events as the &nbsp; '''event set'''&nbsp; $\{A_i \}$.
+'''(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) outcomes, the&nbsp; '''event probability'''&nbsp; is defined as follows:
+'''(2)''' &nbsp;If an event&nbsp; $A_i$&nbsp; is composed of&nbsp; $K$&nbsp; (elementary) outcomes, 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 Laplace probability definition].
+This equation is called the&nbsp; [https://en.wikipedia.org/wiki/Pierre-Simon_Laplace $\text{Laplace probability definition}$].
 *Here,&nbsp; "favorable outcomes"&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).
@@ Line 88: / Line 88: @@
-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#Vollst.C3.A4ndiges_System|"complete system"]].
+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#Vollst.C3.A4ndiges_System|$\text{complete system}$]].
 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:
@@ Line 98: / Line 98: @@
-The topic of this chapter is illustrated with examples in the&nbsp; (German language)&nbsp; learning video&nbsp; [[Klassische_Definition_der_Wahrscheinlickeit_(Lernvideo)|Klassische Definition der Wahrscheinlickeit]] &nbsp; &rArr; &nbsp; "Classical definition of probability".
+The topic of this chapter is illustrated with examples in the&nbsp; (German language)&nbsp; learning video&nbsp; [[Klassische_Definition_der_Wahrscheinlickeit_(Lernvideo)|"Klassische Definition der Wahrscheinlickeit"]] &nbsp; &rArr; &nbsp; "Classical definition of probability".

← Older revision		Revision as of 12:46, 14 November 2021
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−	The topic of this chapter is illustrated with examples in the  (German language)  learning video  [[Klassische_Definition_der_Wahrscheinlickeit_(Lernvideo)\|~~Classical~~ Definition ~~of Probability~~]]   ⇒   "Classical definition of probability".	+	The topic of this chapter is illustrated with examples in the  (German language)  learning video  [[Klassische_Definition_der_Wahrscheinlickeit_(Lernvideo)\|Klassische Definition der Wahrscheinlickeit]]   ⇒   "Classical definition of probability".

@@ Line 81: / Line 81: @@
 {{GraueBox|TEXT=
 $\text{Example 3:}$&nbsp;
-We now consider the experiment&nbsp; "throwing a die".&nbsp; The possible outcomes are thus&nbsp;  $E_μ ∈ G = \{1, 2, 3, 4, 5, 6\}$.
+We now consider the experiment&nbsp; "throwing a die".&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 number of points is even$\big] = \{2, 4, 6\}$,&nbsp; and
+* $A_1 = \big[$the outcome is even$\big] = \{2, 4, 6\}$,&nbsp; and
-* $A_2 = \big[$the number of eyes is odd$\big] = \{1, 3, 5\}$,
+* $A_2 = \big[$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, the events&nbsp; $A_1$&nbsp; and&nbsp; $A_2$&nbsp; represent a so-called&nbsp; [[Theory_of_Stochastic_Signals/Set_Theory_Basics#Vollst.C3.A4ndiges_System|complete system]]&nbsp;.
+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#Vollst.C3.A4ndiges_System|"complete system"]].
-On the other hand, the further event set&nbsp;  $\{A_3, A_4\}$&nbsp; is not equal to the universe&nbsp; $G$, if we define the single events as follows:
+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 number is smaller 3$\big] = \{1, 2\}$,
+* $A_3 = \big[$the outcome is smaller than&nbsp; 3$\big] = \{1, 2\}$,
-* $A_4 =\big[$the number is bigger than 3$\big] = \{4, 5, 6\}$.
+* $A_4 =\big[$the outcome is bigger than&nbsp; 3$\big] = \{4, 5, 6\}$.
-Here, the event set&nbsp; $\{A_3, A_4\}$&nbsp; does not include the element „3”.&nbsp; The probabilities of the events defined here are&nbsp; ${\rm Pr}( A_3) = 1/3$&nbsp; and&nbsp; ${\rm Pr}( A_1) ={\rm Pr}(A_2) = {\rm Pr}(A_4) = 1/2$.}}
+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; ${\rm Pr}( A_3) = 1/3$&nbsp; and&nbsp; ${\rm Pr}( A_1) ={\rm Pr}(A_2) = {\rm Pr}(A_4) = 1/2$.}}
-The topic of this chapter is illustrated with examples in the learning video&nbsp; [[Klassische_Definition_der_Wahrscheinlickeit_(Lernvideo)|Classical Definition of Probability]]&nbsp;.
+The topic of this chapter is illustrated with examples in the&nbsp; (German language)&nbsp; learning video&nbsp; [[Klassische_Definition_der_Wahrscheinlickeit_(Lernvideo)|Classical Definition of Probability]] &nbsp; &rArr; &nbsp; "Classical definition of probability".
 ==Exercises for the chapter==
 <br>
-[[Aufgaben:1.1 Würfelspiel_Mäxchen|Aufgabe 1.1: Würfelspiel Mäxchen]]
+[[Aufgaben:Exercise_1.1:_A_Special_Dice_Game|Exercise 1.1: A Special Dice Game]]
-[[Aufgaben:1.1Z_Summe_zweier_Ternärsignale|Aufgabe 1.1Z: Summe zweier Ternärsignale]]
+[[Aufgaben:Exercise_1.1Z:_Sum_of_Two_Ternary_Signals|Exercise 1.1Z: Sum of Two Ternary Signals]]
 {{Display}}