Aufgaben:Exercise 1.3: Fictional University Somewhere - Revision history

Guenter at 14:09, 25 November 2021

2021-11-25T14:09:52Z

Guenter at 13:37, 25 November 2021

2021-11-25T13:37:04Z

Guenter at 13:36, 25 November 2021

2021-11-25T13:36:28Z

Noah at 17:29, 23 November 2021

2021-11-23T17:29:44Z

Noah at 17:29, 23 November 2021

2021-11-23T17:29:01Z

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Noah at 15:25, 23 November 2021

2021-11-23T15:25:57Z

Guenter at 15:47, 12 October 2021

2021-10-12T15:47:29Z

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 Aufgabe 1.3: Fiktive Uni Irgendwo to Exercise 1.3: Fictional University Somewhere

Javier: Text replacement - "„" to """

2021-05-28T14:23:25Z

Text replacement - "„" to """

Javier: Text replacement - "”" to """

2021-05-28T14:21:36Z

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@@ Line 72: / Line 72: @@
 {{ML-Kopf}}
-'''(1)'''&nbsp; From simple geometric considerations, we arrive at the results:
+'''(1)'''&nbsp; From simple geometric considerations,&nbsp; we arrive at the results:
 :$${\rm Pr}(B) = 3/4 \cdot 1 = 3/4\hspace{0.3cm}(\text{absolute:}\ 720),$$
@@ Line 81: / Line 81: @@
+'''(2)'''&nbsp; <u>Proposed solutions 2, 3, 5 and 6</u>&nbsp; are correct &nbsp; ⇒  &nbsp; proposed solutions 1, 4, 7 are consequently incorrect:
-'''(2)'''&nbsp; <u>Proposed solutions 2, 3, 5 and 6</u> are correct &nbsp; ⇒  &nbsp; proposed solutions 1, 4, 7 are consequently incorrect:
+*There are also female IT students,&nbsp; although very few.
-*There are also female IT students, although very few.
+*The union of&nbsp; $B$,&nbsp; $I$&nbsp; and&nbsp; $T$&nbsp; gives the universal set,&nbsp; but not a complete system&nbsp; (not all combinations of&nbsp; $B$,&nbsp; $I$&nbsp; and&nbsp; $T$&nbsp; are disjoint).
-*The union of&nbsp; $B$,&nbsp; $I$&nbsp; and&nbsp; $T$&nbsp; gives the universal set, but not a complete system (not all combinations of&nbsp; $B$,&nbsp; $I$&nbsp; and&nbsp; $T$&nbsp; are disjoint).
+*For the same reason,&nbsp; the intersection of&nbsp; $B$,&nbsp; $I$&nbsp; and&nbsp; $T$&nbsp; does not yield the empty set.
 [[File:P_ID181__Sto_A_1_3_d_neu.png|right|frame|Geometric solution of a probability problem]]
-'''(3)'''&nbsp; In set theory, an IT student is the intersection of&nbsp; $I$&nbsp; and&nbsp; $W$&nbsp; <br>(shown as a shaded area in the upper left of the graph):
+'''(3)'''&nbsp; In set theory,&nbsp; an IT student is the intersection of&nbsp; $I$&nbsp; and&nbsp; $W$&nbsp; <br>(shown as a shaded area in the upper left of the graph):
 :$$\text{Pr[female IT student] = Pr}(I \cap W) = {1}/{2}\cdot {1}/{4} \cdot {1}/{4} = {1}/{32} \hspace{0.15cm}\underline { \thickapprox 3.13 \%}.$$
-In words, there are&nbsp; $30$&nbsp; female IT students among the&nbsp; $960$&nbsp; students.
+In words,&nbsp; there are&nbsp; $30$&nbsp; female IT students among the&nbsp; $960$&nbsp; students.
@@ Line 109: / Line 103: @@
 :$$p_2 ={\rm Pr}( \overline{B} \cap I \cap \overline{T}) =  {\rm Rectangle\hspace{0.1cm}(DEFG)}= \frac{1}{4}\hspace{0.02cm}\cdot \hspace{0.02cm}  \frac{1}{4}\hspace{0.02cm}+ \hspace{0.02cm}\frac{1}{2}\hspace{0.02cm}\cdot \hspace{0.02cm} \frac{1}{4}\hspace{0.02cm}\cdot \hspace{0.02cm} \frac{1}{4} = \frac{3}{32}\hspace{0.1cm}\underline{\approx 0.0938},$$
-:$$p_3 = {\rm Pr}( B \cap \overline{I} \cap \overline{T}) ={\rm Rectangle\hspace{0.1cm}(HIJK)}= {\rm Triangle\hspace{0.1cm}(HLK)}- {\rm Triangle\hspace{0.1cm}(ILJ)} = \frac{1}{2}\hspace{0.02cm} \cdot \hspace{0.02cm} \frac{5}{4}\hspace{0.02cm} \cdot \hspace{0.02cm} \frac{5}{8}\hspace{0.02cm} -  \hspace{0.02cm}\frac{1}{2}\hspace{0.02cm} \cdot \hspace{0.02cm} \frac{1}{4} \cdot  \frac{1}{4} =  \frac{23}{64}\hspace{0.1cm}\underline{\approx 0.3594}.$$
+:$$p_3 = {\rm Triangle\hspace{0.1cm}(HIC)}- {\rm Triangle\hspace{0.1cm}(KJC)} ={1}/{2}\hspace{0.1cm} \cdot \hspace{0.1cm} 1 \hspace{0.1cm} \cdot \hspace{0.1cm} 1 \hspace{0.1cm} -  \hspace{0.1cm}{1}/{2}\hspace{0.05cm} \cdot \hspace{0.1cm} {3}/{4} \cdot  {3}/{8} =  {23}/{64}.$$
 &nbsp;$\text{Or:}\hspace{0.3cm}$
-:$$p_3 = {\rm Triangle\hspace{0.1cm}(HIC)}- {\rm Triangle\hspace{0.1cm}(KJC)} ={1}/{2}\hspace{0.1cm} \cdot \hspace{0.1cm} 1 \hspace{0.1cm} \cdot \hspace{0.1cm} 1 \hspace{0.1cm} -  \hspace{0.1cm}{1}/{2}\hspace{0.05cm} \cdot \hspace{0.1cm} {3}/{4} \cdot  {3}/{8} =  {23}/{64}.$$
 *The sum of these three probabilities leads to the final result&nbsp; $ \text{Pr[one field of study] } = 31/64 \;\underline {\approx 48.43 \%}$.
 '''(5)'''&nbsp; This probability is expressed by the triangle&nbsp; $\text{Triangle(AGK)}$&nbsp;.&nbsp; This has the area
-:$$\rm Pr[three fields of study] = {1}/{2}\cdot {1}/{4}\cdot {1}/{8} = {1}/{64}\hspace{0.15cm}\underline{\approx 1.56 \%}.$$
+:$$\text{Pr[three fields of study]} = {1}/{2}\cdot {1}/{4}\cdot {1}/{8} = {1}/{64}\hspace{0.15cm}\underline{\approx 1.56 \%}.$$
@@ Line 130: / Line 124: @@
-form a complete system.&nbsp; Thus, using the results of the last subtasks, we obtain:
+form a complete system.&nbsp; Thus,&nbsp; using the results of the last subtasks,&nbsp; 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 &ndash; but with considerably more effort &ndash; 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}}

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

@@ Line 4: / Line 4: @@
 [[File:P_ID89__Sto_A_1_3.png|right|frame|Fictional University Somewhere]]
-From the adjacent graph you can read some information about&nbsp; $\rm FUS$&nbsp; (''Fictional University Somewhere'') ablesen. The whole square represents the universal set&nbsp; $G$&nbsp; of&nbsp; $960$&nbsp; students. Of these
+From the adjacent graph you can read some information about&nbsp; $\rm FUS$&nbsp; ("Fictional University Somewhere").&nbsp; The whole square represents the universal set&nbsp; $G$&nbsp; of&nbsp; $960$&nbsp; students.&nbsp; Of these
-*$25\%$&nbsp; female&nbsp; (set&nbsp; $W$, purple rectangle),
+*$25\%$&nbsp; female&nbsp; (German:&nbsp; "weiblich") &nbsp; (set&nbsp; $W$,&nbsp;  purple rectangle),
-*$75\%$&nbsp; male&nbsp; (set&nbsp; $M$, yellow rectangle).
+*$75\%$&nbsp; male&nbsp; (German:&nbsp; "männlich") &nbsp; (set&nbsp; $M$,&nbsp; yellow rectangle).
 At the university there are the faculties of
-*Theology&nbsp; (set&nbsp; $T$, black triangle),
+*Theology&nbsp; (set&nbsp; $T$,nbsp; black triangle),
-*Information Technology&nbsp; (set&nbsp; $I$, blue triangle),
+*Information Technology&nbsp; (set&nbsp; $I$,nbsp; blue triangle),
-*Business Administration&nbsp; (set&nbsp; $B$, green triangle).
+*Business Administration&nbsp; (set&nbsp; $B$,nbsp; green rectangle).
@@ Line 29: / Line 23: @@
 Hints:
 *The exercise belongs to the chapter&nbsp; [[Theory_of_Stochastic_Signals/Set_Theory_Basics|Set Theory Basics]].
+*The topic of this chapter is illustrated with examples in the&nbsp;  (German language)&nbsp;  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]] &nbsp; $\Rightarrow$ &nbsp; "Set Theoretical Concepts and Laws".
@@ Line 36: / Line 31: @@
 <quiz display=simple>
-{Calculate the number of students enrolled in the faculties.&nbsp; As a check, enter the number of students in the Faculty of Theology&nbsp; $(N_{\rm T})$&nbsp;.
+{Calculate the number of students enrolled in the faculties.&nbsp; As a check,&nbsp; enter the number of students in the Faculty of Theology&nbsp; $(N_{\rm T})$.
 |type="{}"}
 $N_{\rm T} \ = \ $  { 270 3% }
@@ Line 45: / Line 40: @@
 - $I$&nbsp; is a subset of&nbsp; $M$.
 + $W$&nbsp; is a subset of&nbsp; $B$.
-+ $W$&nbsp; and&nbsp; $M$&nbsp; together form a complete system.
++ $W$&nbsp; and&nbsp; $M$&nbsp; together form a&nbsp; "complete system".
-- $B$,&nbsp; $I$&nbsp; and&nbsp; $T$&nbsp; etogether form a complete system.
+- $B$,&nbsp; $I$&nbsp; and&nbsp; $T$&nbsp; together form a&nbsp; "complete system".
 + $W$&nbsp; and&nbsp; $T$&nbsp; are disjoint sets.
 + The union of&nbsp; $B$,&nbsp; $I$&nbsp; and&nbsp; $T$&nbsp; gives the universal set&nbsp; $G$.
@@ Line 57: / Line 52: @@
-{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="{}"}
 $\text{Pr}\big[\text{one field of study}\big] \ = \ $ { 48.43 3% } $\ \%$

@@ Line 131: / Line 131: @@
 '''(6)'''&nbsp; The three events
 * "only one field of study",
-*"two fields of study" und
+*"two fields of study" and
 *"three fields of study"

@@ Line 1: / Line 1: @@
-{{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&nbsp; $\rm FUI$&nbsp; (''Fiktive Universität Irgendwo'') ablesen. Das gesamte Quadrat steht für die Grundmenge&nbsp; $G$&nbsp; der&nbsp; $960$&nbsp; Studierenden. Von diesen sind
+From the adjacent graph you can read some information about&nbsp; $\rm FUS$&nbsp; (''Fictional University Somewhere'') ablesen. The whole square represents the universal set&nbsp; $G$&nbsp; of&nbsp; $960$&nbsp; students. Of these
-*$25\%$&nbsp; weiblich&nbsp; (Menge&nbsp; $W$, violettes Rechteck),
+*$25\%$&nbsp; female&nbsp; (set&nbsp; $W$, purple rectangle),
-*$75\%$&nbsp; männlich&nbsp; (Menge&nbsp; $M$, gelbes Rechteck).
+*$75\%$&nbsp; male&nbsp; (set&nbsp; $M$, yellow rectangle).
-An der Universität gibt es die Fakultäten für
+At the university there are the faculties of
-*Theologie&nbsp; (Menge&nbsp; $T$, schwarzes Dreieck),
+*Theologie&nbsp; (set&nbsp; $T$, black triangle),
-*Informationstechnik&nbsp; (Menge&nbsp; $I$, blaues Dreieck),
+*Information Technology&nbsp; (set&nbsp; $I$, blue triangle),
-*Betriebswirtschaft&nbsp; (Menge&nbsp; $B$, grünes Viereck).
+*Business Administration&nbsp; (set&nbsp; $B$, 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)
@@ Line 26: / Line 27: @@
-''Hinweise:''
+Hints:
-*Die Aufgabe gehört zum  Kapitel&nbsp; [[Theory_of_Stochastic_Signals/Mengentheoretische_Grundlagen|Mengentheoretische Grundlagen]].
+*The exercise belongs to the chapter&nbsp; [[Theory_of_Stochastic_Signals/Set_Theory_Basics|Set Theory Basics]].
-*Eine Zusammenfassung der theoretischen Grundlagen mit Beispielen bringt das Lernvideo&nbsp; [[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.

← Older revision		Revision as of 15:47, 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^]]

← Older revision	Revision as of 15:43, 12 October 2021
(No difference)

@@ Line 129: / Line 129: @@
 '''(6)'''&nbsp; Die drei Ereignisse
-* &bdquo;nur ein Studienfach",
+* "nur ein Studienfach",
-*&bdquo;zwei Studienf&auml;cher" und
+*"zwei Studienf&auml;cher" und
-*&bdquo;drei Studienf&auml;cher"
+*"drei Studienf&auml;cher"