Aufgaben:Exercise 2.1: Election Demand - Revision history

Guenter at 12:31, 3 December 2021

2021-12-03T12:31:24Z

Guenter at 12:26, 3 December 2021

2021-12-03T12:26:54Z

Noah at 21:13, 30 November 2021

2021-11-30T21:13:51Z

Noah at 21:13, 30 November 2021

2021-11-30T21:13:39Z

Noah at 21:03, 30 November 2021

2021-11-30T21:03:51Z

Noah at 21:03, 30 November 2021

2021-11-30T21:03:19Z

Noah at 21:01, 30 November 2021

2021-11-30T21:01:55Z

Guenter at 15:43, 25 November 2021

2021-11-25T15:43:14Z

Guenter: Guenter moved page Aufgaben:Aufgabe 2.1: Wahlnachfrage to Aufgaben:Exercise 2.1: Election Demand

2021-11-25T15:40:11Z

Guenter moved page Aufgabe 2.1: Wahlnachfrage to Exercise 2.1: Election Demand

Javier: Text replacement - "Category:Aufgaben zu Stochastische Signaltheorie" to "Category:Theory of Stochastic Signals: Exercises"

2021-03-23T12:44:14Z

Text replacement - "Category:Aufgaben zu Stochastische Signaltheorie" to "Category:Theory of Stochastic Signals: Exercises"

@@ Line 63: / Line 63: @@
-'''(2)'''&nbsp; The probability that the demand&nbsp; $(h_{\rm A})$&nbsp; differs from the final outcome&nbsp; $(p_{\rm A})$&nbsp; by more than&nbsp; $2\%$&nbsp; is, according to Bernoulli's law of large numbers, with&nbsp; $N = 2000$:
+'''(2)'''&nbsp; The probability that the demand&nbsp; $(h_{\rm A})$&nbsp; differs from the final outcome&nbsp; $(p_{\rm A})$&nbsp; by more than&nbsp; $2\%$&nbsp; is, according to Bernoulli's law of large numbers,&nbsp; with&nbsp; $N = 2000$:
 $${\rm Pr}(|h_{\rm A} - p_{\rm A}| \geq 0.02) \leq \frac{1}{4 \cdot 2000\cdot 0.02^2} = 0.3125.$$
-*This probability includes the two equally likely cases that&nbsp; $p_{\rm A} \le 46\%$&nbsp; and&nbsp; $p_{\rm A} \le 50\%$&nbsp; is.
+*This probability includes the two equally likely cases that&nbsp; $p_{\rm A} \le 46\%$&nbsp; and&nbsp; $p_{\rm A} \ge 50\%$&nbsp; is.
 *Only in the last case there is no runoff:
 :$${\rm Pr(no\hspace{0.1cm}runoff)} \le 0.156 \hspace{0.15cm}\underline{=15.6 \%}.$$

@@ Line 4: / Line 4: @@
 [[File:EN_Sto_A_2_1_.png|right|frame|Result of election demand]]
-In an election, the three candidates&nbsp; $A$,&nbsp; $B$&nbsp; and&nbsp; $C$&nbsp; are running for mayor.
+In an election,&nbsp; the three candidates&nbsp; $A$,&nbsp; $B$,&nbsp; $C$&nbsp; are running for mayor.
 *The candidate who receives more than&nbsp; $50\%$&nbsp; of the votes cast is elected.
-*If none of the three candidates succeeds in the first ballot, a runoff election shall be held between the two candidates with the most votes.
+*If none of the three candidates succeeds in the first ballot,&nbsp; a runoff election shall be held between the two candidates with the most votes.
-Immediately after the closing of the polling stations, the result of an election demand shall be presented:
+Immediately after the closing of the polling stations,&nbsp; the result of an election demand shall be presented:
 :Candidate&nbsp; $A$: &nbsp; $48\%$, &nbsp; &nbsp; &nbsp; Candidate&nbsp; $B$: &nbsp; $30\%$, &nbsp; &nbsp; &nbsp; Candidate&nbsp; $C$: &nbsp; $22\%$.
@@ Line 15: / Line 15: @@
 This demand is based on a survey of only&nbsp; $N = 2000$&nbsp; of the total&nbsp; $N' = 800 \hspace{0.05cm}000$&nbsp; voters.
-In answering the following questions, assume the following:
+In answering the following questions,&nbsp; assume the following:
 *The actual (percentage) votes obtained in the election by candidates&nbsp; $A$,&nbsp; $B$&nbsp; and&nbsp; $C$&nbsp; can be taken as the probabilities&nbsp; $p_{\rm A}$,&nbsp; $p_{\rm B}$&nbsp; and&nbsp; $p_{\rm C}$&nbsp; although these are also themselves determined as relative frequencies&nbsp; $($related to&nbsp; $N')$&nbsp; .
@@ Line 30: / Line 30: @@
 *The exercise belongs to the chapter&nbsp; [[Theory_of_Stochastic_Signals/From_Random_Experiment_to_Random_Variable|From Random Experiment to Random Variable]].
-*The topic of this chapter is illustrated with examples in the&nbsp;  (German language)&nbsp;  learning video&nbsp; [[Bernoullisches_Gesetz_der_großen_Zahlen_(Lernvideo)|Bernoullisches Gesetz der großen Zahlen]]&nbsp; $\Rightarrow$ &nbsp; Bernoulli's Law of Large Numbers.
+*The topic of this chapter is illustrated with examples in the&nbsp;  (German language)&nbsp;  learning video&nbsp;
@@ Line 39: / Line 40: @@
 {Who do you expect as future mayor after this demand?
 |type="[]"}
-+ candidate&nbsp; $A$,
++ Candidate&nbsp; $A$,
-- Candidate&nbsp; $B$,
+- candidate&nbsp; $B$,
-- Candidate&nbsp; $C$.
+- candidate&nbsp; $C$.
@@ Line 49: / Line 50: @@
-{We now assume that candidate&nbsp; $A$&nbsp; actually receives exactly&nbsp; $48\%$&nbsp; of the votes. <br>What is the probability (at most) that candidate $C$ will reach the runoff?
+{We now assume that candidate&nbsp; $A$&nbsp; actually receives exactly&nbsp; $48\%$&nbsp; of the votes. <br>What is the probability&nbsp; (at most)&nbsp; that candidate&nbsp; $C$&nbsp; will reach the runoff?
 |type="{}"}
 $\text{Maximum: Pr(}C \ \text{in runoff)}\ = \ $ { 3.9 3% } $\ \rm \%$

@@ Line 57: / Line 57: @@
 </quiz>
-===Musterlösung===
+===Solution===
 {{ML-Kopf}}
 '''(1)'''&nbsp; One should at least believe this demand that&nbsp; $\underline{\text{candidate} \ A}$&nbsp; is likely to win.

@@ Line 6: / Line 6: @@
 In an election, the three candidates&nbsp; $A$,&nbsp; $B$&nbsp; and&nbsp; $C$&nbsp; are running for mayor.
 *The candidate who receives more than&nbsp; $50\%$&nbsp; of the votes cast is elected.
-*If none of the three candidates succeeds in the first ballot, a run-off election shall be held between the two candidates with the most votes.
+*If none of the three candidates succeeds in the first ballot, a runoff election shall be held between the two candidates with the most votes.
@@ Line 59: / Line 59: @@
 ===Musterlösung===
 {{ML-Kopf}}
-'''(1)'''&nbsp; Man sollte dieser Nachfrage zumindest glauben, dass&nbsp; $\underline{\text{Kandidat} \ A}$&nbsp; wahrscheinlich gewinnt.
+'''(1)'''&nbsp; One should at least believe this demand that&nbsp; $\underline{\text{candidate} \ A}$&nbsp; is likely to win.
-'''(2)'''&nbsp; Die Wahrscheinlichkeit, dass die Nachfrage&nbsp; $(h_{\rm A})$&nbsp; vom endg&uuml;ltigen Ergebnis&nbsp; $(p_{\rm A})$&nbsp; betragsm&auml;&szlig;ig um mehr als&nbsp; $2\%$&nbsp; abweicht, ist nach dem Bernouillischen Gesetz der gro&szlig;en Zahlen mit&nbsp; $N = 2000$:
+'''(2)'''&nbsp; The probability that the demand&nbsp; $(h_{\rm A})$&nbsp; differs from the final outcome&nbsp; $(p_{\rm A})$&nbsp; by more than&nbsp; $2\%$&nbsp; is, according to Bernoulli's law of large numbers, with&nbsp; $N = 2000$:
-:$${\rm Pr}(|h_{\rm A} - p_{\rm A}| \geq 0.02) \leq \frac{1}{4 \cdot 2000\cdot 0.02^2} = 0.3125.$$
+$${\rm Pr}(|h_{\rm A} - p_{\rm A}| \geq 0.02) \leq \frac{1}{4 \cdot 2000\cdot 0.02^2} = 0.3125.$$
-*Diese Wahrscheinlichkeit beinhaltet die beiden gleichwahrscheinlichen F&auml;lle, dass&nbsp; $p_{\rm A} \le 46\%$&nbsp; und&nbsp; $p_{\rm A} \ge 50\%$&nbsp; ist.
+*This probability includes the two equally likely cases that&nbsp; $p_{\rm A} \le 46\%$&nbsp; and&nbsp; $p_{\rm A} \le 50\%$&nbsp; is.
-*Nur im letzten Fall gibt es keine Stichwahl:
+*Only in the last case there is no runoff:
-:$${\rm Pr(keine\hspace{0.1cm}Stichwahl)} \le 0.156 \hspace{0.15cm}\underline{=15.6 \%}.$$
+:$${\rm Pr(no\hspace{0.1cm}runoff)} \le 0.156 \hspace{0.15cm}\underline{=15.6 \%}.$$
-'''(3)'''&nbsp; Mit&nbsp; $\varepsilon  = 4\%$&nbsp; $($ergibt sich aus&nbsp; $0.26 -0.22)$&nbsp; liefert das Gesetz der gro&szlig;en Zahlen:
+'''(3)'''&nbsp; With&nbsp; $\varepsilon = 4\%$&nbsp; $($is given by&nbsp; $0.26 -0.22)$&nbsp; yields the law of large numbers:
 :$${\rm Pr}\left(|h_{\rm C}-p_{\rm C}|\ge 0.04\right)\le\rm\frac{1}{4\cdot 2000\cdot 0.04^2}=0.078.$$
-Daraus folgt:
+It follows that:
-*Die Wahrscheinlichkeit, dass Kandidat&nbsp; $C$&nbsp; mindestens&nbsp; $26\%$&nbsp; der Stimmen erh&auml;lt, ist nicht gr&ouml;&szlig;er als&nbsp; $3.9\%$.
+*The probability that candidate&nbsp; $C$&nbsp; receives at least&nbsp; $26\%$&nbsp; of the votes is not greater than&nbsp; $3.9\%$.
-*Da&nbsp; $p_{\rm A} = 0.48$&nbsp; fest vorausgesetzt wurde, gilt in diesem Fall gleichzeitig&nbsp; $p_{\rm B} \le 0.26$.
+*Since&nbsp; $p_{\rm A} = 0.48$&nbsp; was assumed fixed, in this case simultaneously&nbsp; $p_{\rm B} \le 0.26$.
-*Da es sich hier um kontinuierliche Zufallsgr&ouml;&szlig;en handelt, sind&nbsp;  $(p_{\rm C} \ge 0.26, \; p_{\rm B} \le 0.26)$&nbsp;  und&nbsp; $(p_{\rm C} > 0.26, \; p_{\rm B}< 0.26)$&nbsp; gleich.
+*Since these are continuous random variables,&nbsp; $(p_{\rm C} \ge 0.26, \; p_{\rm B} \le 0.26)$&nbsp; and&nbsp; $(p_{\rm C} > 0.26, \; p_{\rm B}< 0.26)$&nbsp; are equal.
-*Damit ist die Wahrscheinlichkeit, dass&nbsp; $C$&nbsp; die Stichwahl erreicht,  ebenfalls auf&nbsp; $3.9\%$&nbsp; beschr&auml;nkt:
+*Thus, the probability that&nbsp; $C$&nbsp; reaches the runoff is also limited to&nbsp; $3.9\%$&nbsp; :
-:$${\rm Pr(}C\rm \hspace{0.1cm}erreicht\hspace{0.1cm}Stichwahl)\le 0.039 \hspace{0.15cm}\underline{= 3.9 \%}.$$
+:$${\rm Pr(}C\rm \hspace{0.1cm}reaches\hspace{0.1cm}runoff \hspace{0.1cm}election)\le 0.039 \hspace{0.15cm}\underline{= 3.9 \%}.$$
 {{ML-Fuß}}
 [[Category:Theory of Stochastic Signals: Exercises|^2.1 From Experiment to Random Variable^]]

@@ Line 49: / Line 49: @@
-{We now assume that candidate&nbsp; $A$&nbsp; actually receives exactly&nbsp; $48\%$&nbsp; of the votes. <br>What is the probability (at most) that candidate $C will reach the runoff?
+{We now assume that candidate&nbsp; $A$&nbsp; actually receives exactly&nbsp; $48\%$&nbsp; of the votes. <br>What is the probability (at most) that candidate $C$ will reach the runoff?
 |type="{}"}
 $\text{Maximum: Pr(}C \ \text{in runoff)}\ = \ $ { 3.9 3% } $\ \rm \%$

@@ Line 37: / Line 37: @@
 <quiz display=simple>
-{Wen erwarten Sie nach dieser Nachfrage als zuk&uuml;nftigen B&uuml;rgermeister?
+{Who do you expect as future mayor after this demand?
 |type="[]"}
-+ Kandidat&nbsp; $A$,
++ candidate&nbsp; $A$,
-- Kandidat&nbsp; $B$,
+- Candidate&nbsp; $B$,
-- Kandidat&nbsp; $C$.
+- Candidate&nbsp; $C$.
-{Wie gro&szlig; ist die Wahrscheinlichkeit, dass keine Stichwahl erforderlich sein wird?&nbsp; Geben Sie hierf&uuml;r die obere Schranke an.
+{What is the probability that no runoff will be required?&nbsp; Specify the upper bound here.
 |type="{}"}
-$\text{Maximum:   Pr(keine Stichwahl)} \ = \ $  { 15.6 3% } $\ \rm \%$
+$\text{Maximum: Pr(no runoff)} \ = \ $ { 15.6 3% } $\ \rm \%$
-{Wir setzen nun voraus, dass Kandidat&nbsp; $A$&nbsp; tats&auml;chlich genau&nbsp; $48\%$&nbsp; der Stimmen erh&auml;lt. <br>Wie gro&szlig; ist damit (h&ouml;chstens) die Wahrscheinlichkeit, dass Kandidat&nbsp; $C$&nbsp; die Stichwahl erreicht?
+{We now assume that candidate&nbsp; $A$&nbsp; actually receives exactly&nbsp; $48\%$&nbsp; of the votes. <br>What is the probability (at most) that candidate $C will reach the runoff?
 |type="{}"}
-$\text{Maximum:    Pr(}C \ \text{in Stichwahl)}\ = \ $ { 3.9 3% } $\ \rm \%$
+$\text{Maximum: Pr(}C \ \text{in runoff)}\ = \ $ { 3.9 3% } $\ \rm \%$

@@ Line 3: / Line 3: @@
 }}
-[[File:EN_Sto_A_2_1_.png|right|frame|Ergebnis der Wahlnachfrage]]
+[[File:EN_Sto_A_2_1_.png|right|frame|Result of election demand]]
-Zu einer B&uuml;rgermeisterwahl treten die drei Kandidaten&nbsp; $A$,&nbsp; $B$&nbsp; und&nbsp; $C$&nbsp; an.
+In an election, the three candidates&nbsp; $A$,&nbsp; $B$&nbsp; and&nbsp; $C$&nbsp; are running for mayor.
-*Gew&auml;hlt ist derjenige Kandidat, der mehr als&nbsp; $50\%$&nbsp; der abgegebenen Stimmen erh&auml;lt.
+*The candidate who receives more than&nbsp; $50\%$&nbsp; of the votes cast is elected.
-*Gelingt dies im ersten Wahlgang keinem der drei Bewerber, so kommt es zwischen den beiden Kandidaten mit den meisten Stimmen zu einer Stichwahl.
+*If none of the three candidates succeeds in the first ballot, a run-off election shall be held between the two candidates with the most votes.
-Direkt nach Schlie&szlig;ung der Wahllokale wird das Ergebnis einer Wahlnachfrage vorgelegt:
+Immediately after the closing of the polling stations, the result of an election demand shall be presented:
-:Kandidat&nbsp; $A$: &nbsp; $48\%$,	 &nbsp; &nbsp; &nbsp; &nbsp; Kandidat&nbsp; $B$: &nbsp; $30\%$,	 &nbsp; &nbsp; &nbsp; &nbsp;   	Kandidat&nbsp; $C$: &nbsp; $22\%$.
+:Candidate&nbsp; $A$: &nbsp; $48\%$, &nbsp; &nbsp; &nbsp; Candidate&nbsp; $B$: &nbsp; $30\%$, &nbsp; &nbsp; &nbsp; Candidate&nbsp; $C$: &nbsp; $22\%$.
-Diese Nachfrage basiert auf einer Umfrage unter lediglich&nbsp; $N = 2000$&nbsp; der insgesamt&nbsp; $N' = 800 \hspace{0.05cm}000$&nbsp; W&auml;hlerinnen und W&auml;hler.
+This demand is based on a survey of only&nbsp; $N = 2000$&nbsp; of the total&nbsp; $N' = 800 \hspace{0.05cm}000$&nbsp; voters.
-Gehen Sie bei der Beantwortung der nachfolgenden Fragen von folgenden Voraussetzungen aus:
+In answering the following questions, assume the following:
-*Die bei der Wahl von den Kandidaten&nbsp; $A$,&nbsp; $B$&nbsp; und&nbsp; $C$&nbsp; tats&auml;chlich erzielten (prozentualen) Stimmen k&ouml;nnen als die Wahrscheinlichkeiten&nbsp; $p_{\rm A}$,&nbsp; $p_{\rm B}$&nbsp; und&nbsp; $p_{\rm C}$&nbsp; aufgefasst werden, obwohl auch diese selbst als relative H&auml;ufigkeiten&nbsp; $($bezogen auf&nbsp; $N')$&nbsp; ermittelt werden.
+*The actual (percentage) votes obtained in the election by candidates&nbsp; $A$,&nbsp; $B$&nbsp; and&nbsp; $C$&nbsp; can be taken as the probabilities&nbsp; $p_{\rm A}$,&nbsp; $p_{\rm B}$&nbsp; and&nbsp; $p_{\rm C}$&nbsp; although these are also themselves determined as relative frequencies&nbsp; $($related to&nbsp; $N')$&nbsp; .
-*Die&nbsp; $2000$&nbsp; ausgew&auml;hlten W&auml;hler repr&auml;sentieren die gesamte W&auml;hlerschaft im statistischen Sinne ideal und haben bei der Wahlnachfrage wahrheitsgem&auml;&szlig; geantwortet.
+*The&nbsp; $2000$&nbsp; selected voters ideally represent the entire electorate in a statistical sense and answered truthfully when asked to vote.
-*Nach dem&nbsp; [[Theory_of_Stochastic_Signals/Vom_Zufallsexperiment_zur_Zufallsgröße#Bernoullisches_Gesetz_der_gro.C3.9Fen_Zahlen|Bernoullischen Gesetz der gro&szlig;en Zahlen]]&nbsp; sollen die Ergebnisse dieser Nachfrage als relative H&auml;ufigkeiten aufgefasst werden:
+*According to [[Theory_of_Stochastic_Signals/From_Random_Experiment_to_Random_Variable#Bernoulli.27s_law_of_large_numbers|Bernoulli's Law of Large Numbers]]&nbsp; the results of this demand are to be understood as relative frequencies:
 :$$h_{\rm A} = 0.48,\hspace{0.8cm}h_{\rm B} = 0.30,\hspace{0.9cm} h_{\rm C} = 0.22.$$
@@ Line 27: / Line 27: @@
-''Hinweise:''
+Hints:
-*Die Aufgabe gehört zum  Kapitel&nbsp; [[Theory_of_Stochastic_Signals/Vom_Zufallsexperiment_zur_Zufallsgröße|Vom Zufallsexperiment zur Zufallsgröße]].
+*The exercise belongs to the chapter&nbsp; [[Theory_of_Stochastic_Signals/From_Random_Experiment_to_Random_Variable|From Random Experiment to Random Variable]].
-*Die Thematik wird im Lernvideo&nbsp; [[Bernoullisches_Gesetz_der_großen_Zahlen_(Lernvideo)|Bernoullisches Gesetz der gro&szlig;en Zahlen]]&nbsp; zusammengefasst.
+*The topic of this chapter is illustrated with examples in the&nbsp;  (German language)&nbsp;  learning video&nbsp; [[Bernoullisches_Gesetz_der_großen_Zahlen_(Lernvideo)|Bernoullisches Gesetz der großen Zahlen]]&nbsp; $\Rightarrow$ &nbsp; Bernoulli's Law of Large Numbers.
-===Fragebogen===
+===Questions===
 <quiz display=simple>

← Older revision		Revision as of 15:43, 25 November 2021
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−	[[Category:Theory of Stochastic Signals: Exercises\|^2.1 ~~Vom~~ Experiment ~~zur Zufallsgröße~~^]]	+	[[Category:Theory of Stochastic Signals: Exercises\|^2.1 From Experiment to Random Variable^]]

← Older revision		Revision as of 12:44, 23 March 2021
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−	[[Category:~~Aufgaben zu Stochastische Signaltheorie~~\|^2.1 Vom Experiment zur Zufallsgröße^]]	+	[[Category:Theory of Stochastic Signals: Exercises\|^2.1 Vom Experiment zur Zufallsgröße^]]