Aufgaben:Exercise 1.3: Fictional University Somewhere - Revision history

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	[[Category:Theory of Stochastic Signals: Exercises\|^1.2 Set Theory Basics^]]		[[Category:Theory of Stochastic Signals: Exercises\|^1.2 Set Theory Basics^]]
	[[de:Aufgaben:1.~~3_Fiktive_Uni_Irgenwo~~]]		[[de:Aufgaben:Aufgabe 1.3: Fiktive Uni Irgendwo]]

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	[[Category:Theory of Stochastic Signals: Exercises\|^1.2 Set Theory Basics^]]		[[Category:Theory of Stochastic Signals: Exercises\|^1.2 Set Theory Basics^]]
			[[de:Aufgaben:1.3_Fiktive_Uni_Irgenwo]]

Guenter at 14:09, 25 November 2021

2021-11-25T14:09:52Z

← Older revision		Revision as of 16:09, 25 November 2021
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	{{ML-Kopf}}		{{ML-Kopf}}

	'''(1)'''  From simple geometric considerations, we arrive at the results:		'''(1)'''  From simple geometric considerations,  we arrive at the results:

	:$${\rm Pr}(B) = 3/4 \cdot 1 = 3/4\hspace{0.3cm}(\text{absolute:}\ 720),$$		:$${\rm Pr}(B) = 3/4 \cdot 1 = 3/4\hspace{0.3cm}(\text{absolute:}\ 720),$$
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			'''(2)'''  <u>Proposed solutions 2, 3, 5 and 6</u>  are correct   ⇒   proposed solutions 1, 4, 7 are consequently incorrect:
	'''(2)'''  <u>Proposed solutions 2, 3, 5 and 6</u> are correct   ⇒   proposed solutions 1, 4, 7 are consequently incorrect:		*There are also female IT students,  although very few.
	*There are also female IT students, although very few.		*The union of  $B$,  $I$  and  $T$  gives the universal set,  but not a complete system  (not all combinations of  $B$,  $I$  and  $T$  are disjoint).
	*The union of  $B$,  $I$  and  $T$  gives the universal set, but not a complete system (not all combinations of  $B$,  $I$  and  $T$  are disjoint).		*For the same reason,  the intersection of  $B$,  $I$  and  $T$  does not yield the empty set.
	*For the same reason, the intersection of  $B$,  $I$  and  $T$  does not yield the empty set.




	[[File:P_ID181__Sto_A_1_3_d_neu.png\|right\|frame\|Geometric solution of a probability problem]]		[[File:P_ID181__Sto_A_1_3_d_neu.png\|right\|frame\|Geometric solution of a probability problem]]
	'''(3)'''  In set theory, an IT student is the intersection of  $I$  and  $W$  <br>(shown as a shaded area in the upper left of the graph):		'''(3)'''  In set theory,  an IT student is the intersection of  $I$  and  $W$  <br>(shown as a shaded area in the upper left of the graph):

	:$$\text{Pr[female IT student] = Pr}(I \cap W) = {1}/{2}\cdot {1}/{4} \cdot {1}/{4} = {1}/{32} \hspace{0.15cm}\underline { \thickapprox 3.13 \%}.$$		:$$\text{Pr[female IT student] = Pr}(I \cap W) = {1}/{2}\cdot {1}/{4} \cdot {1}/{4} = {1}/{32} \hspace{0.15cm}\underline { \thickapprox 3.13 \%}.$$

	In words, there are  $30$  female IT students among the  $960$  students.		In words,  there are  $30$  female IT students among the  $960$  students.





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	:$$p_2 ={\rm Pr}( \overline{B} \cap I \cap \overline{T}) = {\rm Rectangle\hspace{0.1cm}(DEFG)}= \frac{1}{4}\hspace{0.02cm}\cdot \hspace{0.02cm} \frac{1}{4}\hspace{0.02cm}+ \hspace{0.02cm}\frac{1}{2}\hspace{0.02cm}\cdot \hspace{0.02cm} \frac{1}{4}\hspace{0.02cm}\cdot \hspace{0.02cm} \frac{1}{4} = \frac{3}{32}\hspace{0.1cm}\underline{\approx 0.0938},$$		:$$p_2 ={\rm Pr}( \overline{B} \cap I \cap \overline{T}) = {\rm Rectangle\hspace{0.1cm}(DEFG)}= \frac{1}{4}\hspace{0.02cm}\cdot \hspace{0.02cm} \frac{1}{4}\hspace{0.02cm}+ \hspace{0.02cm}\frac{1}{2}\hspace{0.02cm}\cdot \hspace{0.02cm} \frac{1}{4}\hspace{0.02cm}\cdot \hspace{0.02cm} \frac{1}{4} = \frac{3}{32}\hspace{0.1cm}\underline{\approx 0.0938},$$

	:$$p_3 ~~= {\rm Pr}( B \cap \overline{I} \cap \overline{T}) ={\rm Rectangle\hspace{0.1cm}(HIJK)}~~= {\rm Triangle\hspace{0.1cm}(~~HLK~~)}- {\rm Triangle\hspace{0.1cm}(~~ILJ~~)} = ~~\frac~~{1}{2}\hspace{0.~~02cm~~} \cdot \hspace{0.~~02cm} \frac{5}{4~~}\hspace{0.~~02cm~~} \cdot \hspace{0.~~02cm} \frac{5}{8~~}\hspace{0.~~02cm~~} - \hspace{0.~~02cm~~}~~\frac~~{1}{2}\hspace{0.~~02cm~~} \cdot \hspace{0.~~02cm~~} ~~\frac~~{1}{4} \cdot ~~\frac~~{1}{4} = ~~\frac~~{23}{64~~}\hspace{0.1cm}\underline{\approx 0.3594~~}.$$		:$$p_3 = {\rm Triangle\hspace{0.1cm}(HIC)}- {\rm Triangle\hspace{0.1cm}(KJC)} ={1}/{2}\hspace{0.1cm} \cdot \hspace{0.1cm} 1 \hspace{0.1cm} \cdot \hspace{0.1cm} 1 \hspace{0.1cm} - \hspace{0.1cm}{1}/{2}\hspace{0.05cm} \cdot \hspace{0.1cm} {3}/{4} \cdot {3}/{8} = {23}/{64}.$$

	$\text{Or:}\hspace{0.3cm}$		$\text{Or:}\hspace{0.3cm}$
	:$$p_3 = {\rm Triangle\hspace{0.1cm}(~~HIC~~)}- {\rm Triangle\hspace{0.1cm}(~~KJC~~)} ={1}/{2}\hspace{0.~~1cm~~} \cdot \hspace{0.~~1cm~~} 1 \hspace{0.~~1cm~~} \cdot \hspace{0.~~1cm~~} 1 \hspace{0.~~1cm~~} - \hspace{0.~~1cm~~}{1}/{2}\hspace{0.~~05cm~~} \cdot \hspace{0.~~1cm~~} {3}/{4} \cdot {3}/{8} = {23}/{64}.$$
			:$$p_3 = {\rm Pr}( B \cap \overline{I} \cap \overline{T}) ={\rm Rectangle\hspace{0.1cm}(HIJK)}= {\rm Triangle\hspace{0.1cm}(HLK)}- {\rm Triangle\hspace{0.1cm}(ILJ)} = \frac{1}{2}\hspace{0.02cm} \cdot \hspace{0.02cm} \frac{5}{4}\hspace{0.02cm} \cdot \hspace{0.02cm} \frac{5}{8}\hspace{0.02cm} - \hspace{0.02cm}\frac{1}{2}\hspace{0.02cm} \cdot \hspace{0.02cm} \frac{1}{4} \cdot \frac{1}{4} = \frac{23}{64}\hspace{0.1cm}\underline{\approx 0.3594}.$$

	*The sum of these three probabilities leads to the final result  $ \text{Pr[one field of study] } = 31/64 \;\underline {\approx 48.43 \%}$.		*The sum of these three probabilities leads to the final result  $ \text{Pr[one field of study] } = 31/64 \;\underline {\approx 48.43 \%}$.




	'''(5)'''  This probability is expressed by the triangle  $\text{Triangle(AGK)}$ .  This has the area		'''(5)'''  This probability is expressed by the triangle  $\text{Triangle(AGK)}$ .  This has the area
	:$$\rm Pr[three fields of study] = {1}/{2}\cdot {1}/{4}\cdot {1}/{8} = {1}/{64}\hspace{0.15cm}\underline{\approx 1.56 \%}.$$		:$$\text{Pr[three fields of study]} = {1}/{2}\cdot {1}/{4}\cdot {1}/{8} = {1}/{64}\hspace{0.15cm}\underline{\approx 1.56 \%}.$$


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	form a complete system.  Thus, using the results of the last subtasks, we obtain:		form a complete system.  Thus,  using the results of the last subtasks,  we obtain:
	:$$\rm Pr[two fields of study] = 1- \text{Pr[one field of study] } - \~~rm Pr~~[three fields of study]= 1- {31}/{64} - {1}/{64} \hspace{0.15cm}\underline{= 50\%}.$$		:$$\text{Pr[two fields of study]} = 1- \text{Pr[one field of study] } - \text{[three fields of study]}= 1- {31}/{64} - {1}/{64} \hspace{0.15cm}\underline{= 50\%}.$$
	One would arrive at exactly the same result – but with considerably more effort – in the direct way accordingly:		One would arrive at exactly the same result – but with considerably more effort – in the direct way accordingly:
	:$${~~\rm~~ Pr[two fields of study] = Pr}(B\cap I \cap\overline{T}) + {\rm Pr}(B\cap\overline{I}\cap{T}) + {\rm Pr}(\overline{B}\cap I \cap T).$$		:$$\text{Pr[two fields of study]} = {\rm Pr}(B\cap I \cap\overline{T}) + {\rm Pr}(B\cap\overline{I}\cap{T}) + {\rm Pr}(\overline{B}\cap I \cap T).$$
	{{ML-Fuß}}		{{ML-Fuß}}

Guenter at 13:37, 25 November 2021

2021-11-25T13:37:04Z

← Older revision		Revision as of 15:37, 25 November 2021
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	*The topic of this chapter is illustrated with examples in the  (German language)  learning video		*The topic of this chapter is illustrated with examples in the  (German language)  learning video
	:[[Mengentheoretische_Begriffe_und_Gesetzmäßigkeiten_(Lernvideo)\|Mengentheoretische Begriffe und Gesetzmäßigkeiten]]   $\Rightarrow$   "Set Theoretical Concepts and Laws".		:[[Mengentheoretische_Begriffe_und_Gesetzmäßigkeiten_(Lernvideo)\|Mengentheoretische Begriffe und Gesetzmäßigkeiten]]   $\Rightarrow$   "Set Theoretical Concepts and Laws".
	*The areas in the above diagram are to scale,&nbsp, so you can easily give the  (percentage)  occupancy figures using the numerical values given and simple geometric considerations.		*The areas in the above diagram are to scale,  so you can easily give the  (percentage)  occupancy figures using the numerical values given and simple geometric considerations.

Guenter at 13:36, 25 November 2021

2021-11-25T13:36:28Z

← Older revision		Revision as of 15:36, 25 November 2021
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	[[File:P_ID89__Sto_A_1_3.png\|right\|frame\|Fictional University Somewhere]]		[[File:P_ID89__Sto_A_1_3.png\|right\|frame\|Fictional University Somewhere]]
	From the adjacent graph you can read some information about  $\rm FUS$  (''Fictional University Somewhere'') ~~ablesen~~. The whole square represents the universal set  $G$  of  $960$  students. Of these		From the adjacent graph you can read some information about  $\rm FUS$  ("Fictional University Somewhere").  The whole square represents the universal set  $G$  of  $960$  students.  Of these
	*$25\%$  female  (set  $W$, purple rectangle),		*$25\%$  female  (German:  "weiblich")   (set  $W$,  purple rectangle),
	*$75\%$  male  (set  $M$, yellow rectangle).		*$75\%$  male  (German:  "männlich")   (set  $M$,  yellow rectangle).


	At the university there are the faculties of		At the university there are the faculties of
	*Theology  (set  $T$, black triangle),		*Theology  (set  $T$,nbsp; black triangle),
	*Information Technology  (set  $I$, blue triangle),		*Information Technology  (set  $I$,nbsp; blue triangle),
	*Business Administration  (set  $B$, green ~~triangle).~~		*Business Administration  (set  $B$,nbsp; green rectangle).


	~~Each student must be assigned to at least one of these faculties, but can belong to two or three faculties at the same time.~~

	~~(Bitte diesen Satz überprüfen)~~
	~~The areas in the above diagram are to scale, so you can easily indicate the (percentage) occupancy ("Vielleicht 'enrollment' besser?"~~) ~~figures using the numerical values given and simple geometric considerations~~.



			Each student must be assigned to at least one of these faculties,nbsp; but can belong to two or three faculties at the same time.


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	Hints:		Hints:
	*The exercise belongs to the chapter  [[Theory_of_Stochastic_Signals/Set_Theory_Basics\|Set Theory Basics]].		*The exercise belongs to the chapter  [[Theory_of_Stochastic_Signals/Set_Theory_Basics\|Set Theory Basics]].
			*The topic of this chapter is illustrated with examples in the  (German language)  learning video
	*The topic of this chapter is illustrated with examples in the (German language) learning video [[Mengentheoretische_Begriffe_und_Gesetzmäßigkeiten_(Lernvideo)\|Mengentheoretische Begriffe und Gesetzmäßigkeiten]] $\Rightarrow$ Set Theoretical Concepts and Laws.		:[[Mengentheoretische_Begriffe_und_Gesetzmäßigkeiten_(Lernvideo)\|Mengentheoretische Begriffe und Gesetzmäßigkeiten]]   $\Rightarrow$   "Set Theoretical Concepts and Laws".
			*The areas in the above diagram are to scale,&nbsp, so you can easily give the  (percentage)  occupancy figures using the numerical values given and simple geometric considerations.


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	<quiz display=simple>		<quiz display=simple>

	{Calculate the number of students enrolled in the faculties.  As a check, enter the number of students in the Faculty of Theology  $(N_{\rm T})$~~ ~~.		{Calculate the number of students enrolled in the faculties.  As a check,  enter the number of students in the Faculty of Theology  $(N_{\rm T})$.
	\|type="{}"}		\|type="{}"}
	$N_{\rm T} \ = \ $ { 270 3% }		$N_{\rm T} \ = \ $ { 270 3% }
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	- $I$  is a subset of  $M$.		- $I$  is a subset of  $M$.
	+ $W$  is a subset of  $B$.		+ $W$  is a subset of  $B$.
	+ $W$  and  $M$  together form a complete system.		+ $W$  and  $M$  together form a  "complete system".
	- $B$,  $I$  and  $T$  ~~etogether~~ form a complete system.		- $B$,  $I$  and  $T$  together form a  "complete system".
	+ $W$  and  $T$  are disjoint sets.		+ $W$  and  $T$  are disjoint sets.
	+ The union of  $B$,  $I$  and  $T$  gives the universal set  $G$.		+ The union of  $B$,  $I$  and  $T$  gives the universal set  $G$.
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	{What is the proportion of ~~female~~ students with only one field of study?		{What is the proportion of students with only one field of study?
	\|type="{}"}		\|type="{}"}
	$\text{Pr}\big[\text{one field of study}\big] \ = \ $ { 48.43 3% } $\ \%$		$\text{Pr}\big[\text{one field of study}\big] \ = \ $ { 48.43 3% } $\ \%$

Noah at 17:29, 23 November 2021

2021-11-23T17:29:44Z

← Older revision		Revision as of 19:29, 23 November 2021
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	'''(6)'''  The three events		'''(6)'''  The three events
	* "only one field of study",		* "only one field of study",
	*"two fields of study" ~~und~~		*"two fields of study" and
	*"three fields of study"		*"three fields of study"

Noah at 17:29, 23 November 2021

2021-11-23T17:29:01Z

Show changes

Noah at 15:25, 23 November 2021

2021-11-23T15:25:57Z

← Older revision		Revision as of 17:25, 23 November 2021
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	{{quiz-Header\|Buchseite=~~Stochastische Signaltheorie~~/~~Mengentheoretische Grundlagen~~		{{quiz-Header\|Buchseite=Theory_of_Stochastic_Signals/Set_Theory_Basics
	}}		}}

	[[File:P_ID89__Sto_A_1_3.png\|right\|frame\|~~Fiktive Universität Irgendwo~~ ]]		[[File:P_ID89__Sto_A_1_3.png\|right\|frame\|Fictional University Somewhere]]
	~~Aus nebenstehender Grafik können Sie einige Informationen über die~~  $\rm ~~FUI~~$  (''~~Fiktive Universität Irgendwo~~'') ablesen. ~~Das gesamte Quadrat steht für die Grundmenge~~  $G$  ~~der~~  $960$  ~~Studierenden~~. ~~Von diesen sind~~		From the adjacent graph you can read some information about  $\rm FUS$  (''Fictional University Somewhere'') ablesen. The whole square represents the universal set  $G$  of  $960$  students. Of these
	*$25\%$  ~~weiblich~~  (~~Menge~~  $W$, ~~violettes Rechteck~~),		*$25\%$  female  (set  $W$, purple rectangle),
	*$75\%$  ~~männlich~~  (~~Menge~~  $M$, ~~gelbes Rechteck~~).		*$75\%$  male  (set  $M$, yellow rectangle).


	~~An der Universität gibt es die Fakultäten für~~		At the university there are the faculties of
	*Theologie  (~~Menge~~  $T$, ~~schwarzes Dreieck~~),		*Theologie  (set  $T$, black triangle),
	*~~Informationstechnik~~  (~~Menge~~  $I$, ~~blaues Dreieck~~),		*Information Technology  (set  $I$, blue triangle),
	*~~Betriebswirtschaft~~  (~~Menge~~  $B$, ~~grünes Viereck~~).		*Business Administration  (set  $B$, green triangle).


	~~Jeder Studierende muss mindestens einer dieser Fakultäten zugeordnet sein~~, ~~kann jedoch auch gleichzeitig zwei oder drei Fakultäten angehören~~.		Each student must be assigned to at least one of these faculties, but can belong to two or three faculties at the same time.

	~~Die Flächen~~ in ~~der obigen Darstellung sind maßstäblich~~, so ~~dass Sie anhand der angegebenen Zahlenwerte und einfachen geometrischen Überlegungen die~~ (~~prozentualen~~) ~~Belegungszahlen leicht angeben können~~.		(Bitte diesen Satz überprüfen)
			The areas in the above diagram are to scale, so you can easily indicate the (percentage) occupancy figures using the numerical values given and simple geometric considerations.


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	~~''Hinweise~~:''		Hints:
	*~~Die Aufgabe gehört zum Kapitel~~  [[Theory_of_Stochastic_Signals/~~Mengentheoretische_Grundlagen~~\|~~Mengentheoretische Grundlagen~~]].		*The exercise belongs to the chapter  [[Theory_of_Stochastic_Signals/Set_Theory_Basics\|Set Theory Basics]].

	*~~Eine Zusammenfassung der theoretischen Grundlagen mit Beispielen bringt das Lernvideo ~~ [[Mengentheoretische_Begriffe_und_Gesetzmäßigkeiten_(Lernvideo)\|Mengentheoretische Begriffe und Gesetzmäßigkeiten]].		*The topic of this chapter is illustrated with examples in the (German language) learning video [[Mengentheoretische_Begriffe_und_Gesetzmäßigkeiten_(Lernvideo)\|Mengentheoretische Begriffe und Gesetzmäßigkeiten]] $\Rightarrow$ Set Theoretical Concepts and Laws.

Guenter at 15:47, 12 October 2021

2021-10-12T15:47:29Z

← Older revision		Revision as of 17:47, 12 October 2021
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	[[Category:Theory of Stochastic Signals: Exercises\|^1.2 ~~Mengentheoretische Grundlagen~~^]]		[[Category:Theory of Stochastic Signals: Exercises\|^1.2 Set Theory Basics^]]

Guenter: Guenter moved page Aufgaben:Aufgabe 1.3: Fiktive Uni Irgendwo to Aufgaben:Exercise 1.3: Fictional University Somewhere

2021-10-12T15:43:47Z

Guenter moved page Aufgaben:Aufgabe 1.3: Fiktive Uni Irgendwo to Aufgaben:Exercise 1.3: Fictional University Somewhere

← Older revision	Revision as of 17:43, 12 October 2021
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