Aufgaben:Exercise 4.1Z: Appointment to Breakfast - Revision history

Guenter at 15:56, 22 February 2022

2022-02-22T15:56:09Z

Guenter at 12:17, 7 February 2022

2022-02-07T12:17:11Z

Guenter at 12:10, 7 February 2022

2022-02-07T12:10:15Z

Noah at 17:57, 17 January 2022

2022-01-17T17:57:24Z

Noah at 17:56, 17 January 2022

2022-01-17T17:56:59Z

Noah at 19:15, 13 January 2022

2022-01-13T19:15:11Z

Guenter at 10:50, 12 January 2022

2022-01-12T10:50:41Z

Guenter: Guenter moved page Aufgaben:Aufgabe 4.1Z: Verabredung zum Frühstück to Aufgaben:Exercise 4.1Z: Appointment to Breakfast

2022-01-12T10:44:59Z

Guenter moved page Aufgabe 4.1Z: Verabredung zum Frühstück to Exercise 4.1Z: Appointment to Breakfast

Javier: Text replacement - "„" to """

2021-05-28T14:35:11Z

Text replacement - "„" to """

Javier: Text replacement - "”" to """

2021-05-28T14:32:42Z

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@@ Line 52: / Line 52: @@
-[[File:EN_Sto_Z_4_1_d.png|right|frame|"Favorable area"&nbsp; for meeting]]
+[[File:EN_Sto_Z_4_1_d_neu.png|right|frame|"Favorable area"&nbsp; for meeting]]
 '''(2)'''&nbsp; If Ms. M. arrives at 8 a.m.,&nbsp; she meets Mr. S. only if he arrives before 8:15.
 *If Ms. M. arrives at 9 a.m.,&nbsp; Mr. S. must arrive after 8:45 a.m. so that they can meet.

@@ Line 48: / Line 48: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; If Mr. S. arrives at 8:30, he will meet Ms. M. if she arrives between 8:15 and 8:45. Thus the probability
+'''(1)'''&nbsp; If Mr. S. arrives at 8:30,&nbsp; he will meet Ms. M. if she arrives between 8:15 and 8:45.&nbsp; Thus the probability:
 :$$p_1 = \text{Pr(Mr. S. meets Ms. M.)}\hspace{0.15cm}\underline{=50\%}.$$
-[[File:EN_Sto_Z_4_1_d.png|right|frame|"Favorable area" for meeting]]
+[[File:EN_Sto_Z_4_1_d.png|right|frame|"Favorable area"&nbsp; for meeting]]
-'''(2)'''&nbsp; If Ms. M. arrives at 8 a.m., she meets Mr. S. only if he arrives before 8:15.
+'''(2)'''&nbsp; If Ms. M. arrives at 8 a.m.,&nbsp; she meets Mr. S. only if he arrives before 8:15.
-*If Ms. M. arrives at 9 a.m., Mr. S. must arrive after 8:45 a.m. so that they can meet.
+*If Ms. M. arrives at 9 a.m.,&nbsp; Mr. S. must arrive after 8:45 a.m. so that they can meet.
 *The probability of meeting is the same in both cases:
 :$$p_2 = \big[\text{Min Pr(Mr. S. meets Ms. M.)}\big]\hspace{0.15cm}\underline{=25\%}.$$
-'''(3)'''&nbsp; Of the two arrival times calculated in&nbsp;'''(2)''', 9 o'clock&nbsp; $(\underline{\text{Minute = 60}})$&nbsp; is more favorable, <br>&nbsp; &nbsp; &nbsp; since she &ndash; if Mr. S. is not there &ndash; can leave immediately.
+'''(3)'''&nbsp; Of the two arrival times calculated in&nbsp; '''(2)''',&nbsp; 9 o'clock&nbsp; $(\underline{\text{Minute = 60}})$&nbsp; is more favorable, <br>&nbsp; &nbsp; &nbsp; since she &ndash; if Mr. S. is not there &ndash; can leave immediately.
 '''(4)'''&nbsp; The probability&nbsp; $p_4$&nbsp; is given as the ratio of the red area in the graph to the total area&nbsp; $1$.
-*Using the triangular areas, one obtains:
+*Using the triangular areas,&nbsp; one obtains:
 :$$p_4=\rm 1-2\cdot\frac{1}{2}\cdot\frac{3}{4}\cdot\frac{3}{4}=\frac{7}{16}\hspace{0.15cm}\underline{=\rm 43.75\%}.$$

@@ Line 3: / Line 3: @@
 }}
-[[File:P_ID245__Sto_Z_4_1.jpg|right|frame|Candidates for Chancellor having breakfast in 2002]]
+[[File:P_ID245__Sto_Z_4_1.jpg|right|frame|Chancellor candidates &ndash; breakfast in 2002]]
 Ms. M. and Mr. S. are known to meet often for a joint breakfast:
 *Both promise to come to such a meeting on a certain day between 8 am and 9 am.
-*Further, they agree that each of them will arrive in this period (and only in this period) on "good luck" and wait up to fifteen minutes for the other.
+*Further,&nbsp; they agree that each of them will arrive in this period&nbsp; (and only in this period)&nbsp; on&nbsp; "good luck"&nbsp; <br>and wait up to fifteen minutes for the other.
@@ Line 17: / Line 13: @@
 Hints:
-*The task belongs to the chapter&nbsp; [[Theory_of_Stochastic_Signals/Two-Dimensional_Random_Variables|Two-Dimensional Random Variables]].
+*The exercise belongs to the chapter&nbsp; [[Theory_of_Stochastic_Signals/Two-Dimensional_Random_Variables|Two-Dimensional Random Variables]].
+*Use the minute of arrival time as the time in the following questions:&nbsp; <br>"Minute = 0"&nbsp; stands for 8 o'clock, "Minute = 60"&nbsp; for 9 o'clock.
-*Use the minute of arrival time as the time in the following questions:&nbsp; <br>"Minute = 0" stands for 8 o'clock, "Minute = 60" for 9 o'clock.
+*The exercise arose before the 2002 German Bundestag elections,&nbsp; when both Dr. Angela Merkel and Dr. Edmund Stoiber wanted to become the CDU/CSU's candidate for chancellor.
-*The task arose before the 2002 Bundestag elections, when both Dr. Angela Merkel and Dr. Edmund Stoiber wanted to become the CDU/CSU's candidate for chancellor.
+*At a joint breakfast in Wolfratshausen,&nbsp; Ms. Merkel renounced.&nbsp; The later election was won by Gerhard Schröder&nbsp; (SPD).
@@ Line 28: / Line 23: @@
 <quiz display=simple>
-{What is the probability $p_1$ that the two will meet when Mr. S. arrives at 8:30? Give reasons for your answer.
+{What is the probability&nbsp; $p_1$&nbsp; that the two will meet when Mr. S. arrives at 8:30?&nbsp; Give reasons for your answer.
 |type="{}"}
 $p_1 \ = \ $ { 50 1% } $\ \%$
-{Which arrival time should Ms. M. choose if she does not actually want to meet Mr. S., but still wants to keep to the agreement made? <br>What is the probability $p_2$ that Ms. M. and Mr. S. will meet?
+{Which arrival time should Ms. M. choose if she does not actually want to meet Mr. S.,&nbsp; but still wants to keep to the agreement made? <br>What is the probability&nbsp; $p_2$&nbsp; that Ms. M. and Mr. S. will meet?
 |type="{}"}
 $p_2 \ = \ $ { 25 1% } $\ \%$
-{Which arrival time should Ms. M. choose if she not only wants to avoid a meeting as much as possible, but also wants to minimize the waiting time?
+{Which arrival time should Ms. M. choose if she not only wants to avoid a meeting as much as possible,&nbsp; but also wants to minimize the waiting time?
 |type="{}"}
 $\rm minute \ = \ ${ 60 }
-{What is the probability&nbsp; $p_4$&nbsp; for a meeting in general, that is, if both actually appear on "Gut Gl&uuml;ck"?
+{What is the probability&nbsp; $p_4$&nbsp; for a meeting in general,&nbsp; that is,&nbsp; if both actually appear on "good luck"?
 |type="{}"}
 $p_4 \ = \ $ { 43.75 1% } $\ \%$

@@ Line 21: / Line 21: @@
 *Use the minute of arrival time as the time in the following questions:&nbsp; <br>"Minute = 0" stands for 8 o'clock, "Minute = 60" for 9 o'clock.
 *The task arose before the 2002 Bundestag elections, when both Dr. Angela Merkel and Dr. Edmund Stoiber wanted to become the CDU/CSU's candidate for chancellor.
-*At a joint breakfast in Wolfratshausen, Mrs. Merkel renounced.&nbsp; The later election was won by Gerhard Schröder (SPD).
+*At a joint breakfast in Wolfratshausen, Ms. Merkel renounced.&nbsp; The later election was won by Gerhard Schröder (SPD).
@@ Line 53: / Line 53: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; If Mr. S. arrives at 8:30, he will meet Mrs. M. if she arrives between 8:15 and 8:45. Thus the probability
+'''(1)'''&nbsp; If Mr. S. arrives at 8:30, he will meet Ms. M. if she arrives between 8:15 and 8:45. Thus the probability
 :$$p_1 = \text{Pr(Mr. S. meets Ms. M.)}\hspace{0.15cm}\underline{=50\%}.$$
@@ Line 59: / Line 59: @@
 [[File:EN_Sto_Z_4_1_d.png|right|frame|"Favorable area" for meeting]]
 '''(2)'''&nbsp; If Ms. M. arrives at 8 a.m., she meets Mr. S. only if he arrives before 8:15.
-*If Mrs. M. arrives at 9 a.m., Mr. S. must arrive after 8:45 a.m. so that they can meet.
+*If Ms. M. arrives at 9 a.m., Mr. S. must arrive after 8:45 a.m. so that they can meet.
 *The probability of meeting is the same in both cases:
 :$$p_2 = \big[\text{Min Pr(Mr. S. meets Ms. M.)}\big]\hspace{0.15cm}\underline{=25\%}.$$

@@ Line 3: / Line 3: @@
 }}
-[[File:P_ID245__Sto_Z_4_1.jpg|right|frame|Kanzlerkandidat(inn)en&ndash;Frühstück im Jahr 2002]]
+[[File:P_ID245__Sto_Z_4_1.jpg|right|frame|Candidates for Chancellor having breakfast in 2002]]
-Frau M. und Herr S. treffen sich ja bekanntlich &ouml;fter einmal zu einem gemeinsamen Fr&uuml;hst&uuml;ck:
+Ms. M. and Mr. S. are known to meet often for a joint breakfast:
-*Beide versprechen, an einem bestimmten Tag zwischen 8 Uhr und 9 Uhr zu einem solchen Treffen zu kommen.
+*Both promise to come to such a meeting on a certain day between 8 am and 9 am.
-*Weiter vereinbaren sie, dass jeder von ihnen in diesem Zeitraum (und nur in diesem) auf "Gut Gl&uuml;ck" eintrifft und bis zu einer Viertelstunde auf den Anderen wartet.
+*Further, they agree that each of them will arrive in this period (and only in this period) on "good luck" and wait up to fifteen minutes for the other.
@@ Line 15: / Line 15: @@
-''Hinweise:''
-*Die Aufgabe gehört zum  Kapitel&nbsp; [[Theory_of_Stochastic_Signals/Zweidimensionale_Zufallsgrößen|Zweidimensionale Zufallsgrößen]].
+Hints:
-*Verwenden Sie bei den folgenden Fragen als Zeitangabe die Minute der Ankunftszeit:&nbsp; <br>"Minute = 0" steht f&uuml;r 8 Uhr,&nbsp; "Minute = 60" f&uuml;r 9 Uhr.
+*Use the minute of arrival time as the time in the following questions:&nbsp; <br>"Minute = 0" stands for 8 o'clock, "Minute = 60" for 9 o'clock.
-*Die Aufgabe entstand vor der Bundestagswahl 2002, als sowohl Dr. Angela Merkel als auch Dr. Edmund Stoiber Kanzlerkandidat(in) der CDU/CSU werden wollten.
+*The task arose before the 2002 Bundestag elections, when both Dr. Angela Merkel and Dr. Edmund Stoiber wanted to become the CDU/CSU's candidate for chancellor.
-*Bei einem gemeinsamen Frühstück in Wolfratshausen verzichtete Frau Merkel.&nbsp; Die spätere Wahl gewann Gerhard Schröder (SPD).
+*At a joint breakfast in Wolfratshausen, Mrs. Merkel renounced.&nbsp; The later election was won by Gerhard Schröder (SPD).
-===Fragebogen===
+===Questions===
 <quiz display=simple>
-{Wie gro&szlig; ist die Wahrscheinlichkeit&nbsp; $p_1$, dass sich die beiden treffen, wenn Herr S. um 8 Uhr 30 ankommt? Begr&uuml;nden Sie Ihre Antwort.
+{What is the probability $p_1$ that the two will meet when Mr. S. arrives at 8:30? Give reasons for your answer.
 |type="{}"}
 $p_1 \ = \ $ { 50 1% } $\ \%$
-{Welche Ankunftszeit sollte Frau M. w&auml;hlen, wenn sie Herrn S. eigentlich nicht treffen m&ouml;chte, sich aber trotzdem an die getroffene Vereinbarung halten will? <br>Wie gro&szlig; ist dann die Wahrscheinlichkeit&nbsp; $p_2$, dass sich Frau M. und Herr S. treffen werden?
+{Which arrival time should Ms. M. choose if she does not actually want to meet Mr. S., but still wants to keep to the agreement made? <br>What is the probability $p_2$ that Ms. M. and Mr. S. will meet?
 |type="{}"}
-$p_2 \ = \  $ { 25 1% } $\ \%$
+$p_2 \ = \ $ { 25 1% } $\ \%$
-{Welche Ankunftszeit sollte Frau M. w&auml;hlen, wenn sie nicht nur ein Treffen m&ouml;glichst vermeiden, sondern die Wartezeit minimieren m&ouml;chte?
+{Which arrival time should Ms. M. choose if she not only wants to avoid a meeting as much as possible, but also wants to minimize the waiting time?
 |type="{}"}
-$\rm Minute \ =  \ ${ 60  }
+$\rm minute \ = \ ${ 60 }
-{Wie gro&szlig; ist die Wahrscheinlichkeit&nbsp; $p_4$&nbsp; f&uuml;r ein Zusammentreffen generell, das heißt, wenn beide tats&auml;chlich auf "Gut Gl&uuml;ck" erscheinen?
+{What is the probability&nbsp; $p_4$&nbsp; for a meeting in general, that is, if both actually appear on "Gut Gl&uuml;ck"?
 |type="{}"}
-$p_4 \ = \  $ { 43.75 1% } $\ \%$
+$p_4 \ = \ $ { 43.75 1% } $\ \%$
@@ Line 51: / Line 51: @@
 </quiz>
-===Musterlösung===
+===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; Kommt Herr S. um 8 Uhr 30, so trifft er Frau M., wenn diese zwischen 8 Uhr 15 und 8 Uhr 45 ankommt. Damit ist die Wahrscheinlichkeit
+'''(1)'''&nbsp; If Mr. S. arrives at 8:30, he will meet Mrs. M. if she arrives between 8:15 and 8:45. Thus the probability
-:$$p_1 = \text{Pr(Herr S. trifft Frau M.)}\hspace{0.15cm}\underline{=50\%}.$$
+:$$p_1 = \text{Pr(Mr. S. meets Ms. M.)}\hspace{0.15cm}\underline{=50\%}.$$
-[[File:EN_Sto_Z_4_1_d.png|right|frame|"Günstiger Bereich" für Zusammentreffen]]
+[[File:EN_Sto_Z_4_1_d.png|right|frame|"Favorable area" for meeting]]
-'''(2)'''&nbsp; Kommt Frau M. um 8 Uhr, so trifft sie Herrn S. nur dann, wenn dieser vor 8 Uhr 15 kommt.
+'''(2)'''&nbsp; If Ms. M. arrives at 8 a.m., she meets Mr. S. only if he arrives before 8:15.
-*Erscheint Frau M. um 9 Uhr, dann muss Herr S. nach 8 Uhr 45 angekommen sein, damit sich beide treffen k&ouml;nnen.
+*If Mrs. M. arrives at 9 a.m., Mr. S. must arrive after 8:45 a.m. so that they can meet.
-*Die Wahrscheinlichkeit f&uuml;r ein Zusammentreffen ist in beiden F&auml;llen:
+*The probability of meeting is the same in both cases:
-:$$p_2 = \big[\text{Min Pr(Herr S. trifft Frau M.)}\big]\hspace{0.15cm}\underline{=25\%}.$$
+:$$p_2 = \big[\text{Min Pr(Mr. S. meets Ms. M.)}\big]\hspace{0.15cm}\underline{=25\%}.$$
-'''(3)'''&nbsp; Von den beiden unter&nbsp;&nbsp; '''(2)''' berechneten Ankunftszeiten ist 9 Uhr&nbsp; $(\underline{\text{Minute = 60}})$&nbsp; g&uuml;nstiger, <br>&nbsp; &nbsp; &nbsp; &nbsp; da sie &ndash; falls Herr S. nicht da ist &ndash; sofort wieder gehen kann.
+'''(3)'''&nbsp; Of the two arrival times calculated in&nbsp;'''(2)''', 9 o'clock&nbsp; $(\underline{\text{Minute = 60}})$&nbsp; is more favorable, <br>&nbsp; &nbsp; &nbsp; since she &ndash; if Mr. S. is not there &ndash; can leave immediately.
-'''(4)'''&nbsp; Die Wahrscheinlichkeit&nbsp; $p_4$&nbsp; ergibt sich als das Verh&auml;ltnis der roten Fl&auml;che in der Grafik zur Gesamtfl&auml;che&nbsp; $1$.
+'''(4)'''&nbsp; The probability&nbsp; $p_4$&nbsp; is given as the ratio of the red area in the graph to the total area&nbsp; $1$.
-*Mit den Dreiecksfl&auml;chen erh&auml;lt man:
+*Using the triangular areas, one obtains:
 :$$p_4=\rm 1-2\cdot\frac{1}{2}\cdot\frac{3}{4}\cdot\frac{3}{4}=\frac{7}{16}\hspace{0.15cm}\underline{=\rm 43.75\%}.$$

← Older revision		Revision as of 19:15, 13 January 2022
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−	{{quiz-Header\|Buchseite=~~Stochastische Signaltheorie~~/~~Zweidimensionale Zufallsgrößen~~	+	{{quiz-Header\|Buchseite=Theory_of_Stochastic_Signals/Two-Dimensional_Random_Variables
	}}		}}

← Older revision		Revision as of 10:50, 12 January 2022
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−	[[Category:Theory of Stochastic Signals: Exercises\|^4.1 ~~Zweidimensionale Zufallsgrößen~~^]]	+	[[Category:Theory of Stochastic Signals: Exercises\|^4.1 Two-Dimensional Random Variables^]]

@@ Line 6: / Line 6: @@
 Frau M. und Herr S. treffen sich ja bekanntlich &ouml;fter einmal zu einem gemeinsamen Fr&uuml;hst&uuml;ck:
 *Beide versprechen, an einem bestimmten Tag zwischen 8 Uhr und 9 Uhr zu einem solchen Treffen zu kommen.
-*Weiter vereinbaren sie, dass jeder von ihnen in diesem Zeitraum (und nur in diesem) auf &bdquo;Gut Gl&uuml;ck&rdquo; eintrifft und bis zu einer Viertelstunde auf den Anderen wartet.
+*Weiter vereinbaren sie, dass jeder von ihnen in diesem Zeitraum (und nur in diesem) auf &bdquo;Gut Gl&uuml;ck" eintrifft und bis zu einer Viertelstunde auf den Anderen wartet.
@@ Line 18: / Line 18: @@
 *Die Aufgabe gehört zum  Kapitel&nbsp; [[Theory_of_Stochastic_Signals/Zweidimensionale_Zufallsgrößen|Zweidimensionale Zufallsgrößen]].
-*Verwenden Sie bei den folgenden Fragen als Zeitangabe die Minute der Ankunftszeit:&nbsp; <br>&bdquo;Minute = 0&rdquo; steht f&uuml;r 8 Uhr,&nbsp; &bdquo;Minute = 60&rdquo; f&uuml;r 9 Uhr.
+*Verwenden Sie bei den folgenden Fragen als Zeitangabe die Minute der Ankunftszeit:&nbsp; <br>&bdquo;Minute = 0" steht f&uuml;r 8 Uhr,&nbsp; &bdquo;Minute = 60" f&uuml;r 9 Uhr.
 *Die Aufgabe entstand vor der Bundestagswahl 2002, als sowohl Dr. Angela Merkel als auch Dr. Edmund Stoiber Kanzlerkandidat(in) der CDU/CSU werden wollten.
 *Bei einem gemeinsamen Frühstück in Wolfratshausen verzichtete Frau Merkel.&nbsp; Die spätere Wahl gewann Gerhard Schröder (SPD).
@@ Line 43: / Line 43: @@
-{Wie gro&szlig; ist die Wahrscheinlichkeit&nbsp; $p_4$&nbsp; f&uuml;r ein Zusammentreffen generell, das heißt, wenn beide tats&auml;chlich auf &bdquo;Gut Gl&uuml;ck&rdquo; erscheinen?
+{Wie gro&szlig; ist die Wahrscheinlichkeit&nbsp; $p_4$&nbsp; f&uuml;r ein Zusammentreffen generell, das heißt, wenn beide tats&auml;chlich auf &bdquo;Gut Gl&uuml;ck" erscheinen?
 |type="{}"}
 $p_4 \ = \  $ { 43.75 1% } $\ \%$
@@ Line 57: / Line 57: @@
-[[File:EN_Sto_Z_4_1_d.png|right|frame|&bdquo;Günstiger Bereich&rdquo; für Zusammentreffen]]
+[[File:EN_Sto_Z_4_1_d.png|right|frame|&bdquo;Günstiger Bereich" für Zusammentreffen]]
 '''(2)'''&nbsp; Kommt Frau M. um 8 Uhr, so trifft sie Herrn S. nur dann, wenn dieser vor 8 Uhr 15 kommt.
 *Erscheint Frau M. um 9 Uhr, dann muss Herr S. nach 8 Uhr 45 angekommen sein, damit sich beide treffen k&ouml;nnen.