Aufgaben:Exercise 3.1Z: Drawing Cards - Revision history

Noah at 08:09, 24 September 2021

2021-09-24T08:09:25Z

Guenter at 15:11, 16 August 2021

2021-08-16T15:11:57Z

Guenter at 14:14, 16 August 2021

2021-08-16T14:14:45Z

Guenter at 14:00, 16 August 2021

2021-08-16T14:00:41Z

Noah at 20:06, 10 August 2021

2021-08-10T20:06:44Z

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Guenter: Guenter moved page Aufgaben:Exercise 3.1Z: Drawing cards to Aufgaben:Exercise 3.1Z: Drawing Cards

2021-07-16T14:07:33Z

Guenter moved page Exercise 3.1Z: Drawing cards to Exercise 3.1Z: Drawing Cards

Guenter at 13:36, 16 July 2021

2021-07-16T13:36:46Z

Guenter: Guenter moved page Aufgaben:Aufgabe 3.1Z: Karten ziehen to Aufgaben:Exercise 3.1Z: Drawing cards

2021-07-16T13:08:33Z

Guenter moved page Aufgabe 3.1Z: Karten ziehen to Exercise 3.1Z: Drawing cards

Javier: Text replacement - "”" to """

2021-05-28T14:29:30Z

Text replacement - "”" to """

Javier: Text replacement - "„" to """

2021-05-28T14:28:57Z

Text replacement - "„" to """

← Older revision		Revision as of 08:09, 24 September 2021
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−	{{quiz-Header\|Buchseite=~~Informationstheorie~~/~~Einige Vorbemerkungen zu zweidimensionalen Zufallsgrößen~~	+	{{quiz-Header\|Buchseite=Information_Theory/Some_Preliminary_Remarks_on_Two-Dimensional_Random_Variables
	}}		}}

@@ Line 13: / Line 13: @@
 *In the following, we use&nbsp; $A_i$&nbsp; to denote the event that the card drawn at time&nbsp; $i$&nbsp; is an ace. &nbsp; <br>Here we have to set&nbsp; $i = 1,\ 2,\ 3$&nbsp;.
-*The complementary event then states that at time&nbsp; $i$&nbsp; no ace is drawn, but any other card.
+*The complementary event&nbsp; $\overline{\it A_i}$&nbsp; then states that at time&nbsp; $i$&nbsp; no ace is drawn, but any other card.
@@ Line 59: / Line 59: @@
 '''(1)'''&nbsp; If the cards are put back after being drawn, the probability of an ace is the same at every time&nbsp; $(1/8)$:
-:$$ p_{\rm 1} = \rm Pr (3 \hspace{0.1cm} Asse) = \rm Pr (\it A_{\rm 1} \rm )\cdot \rm Pr (\it A_{\rm 2} \rm )\cdot \rm Pr (\it A_{\rm 3} \rm ) = \rm \big({1}/{8}\big)^3 \hspace{0.15cm}\underline{\approx 0.002}.$$
+:$$ p_{\rm 1} = \rm Pr (3 \hspace{0.1cm} aces) = \rm Pr (\it A_{\rm 1} \rm )\cdot \rm Pr (\it A_{\rm 2} \rm )\cdot \rm Pr (\it A_{\rm 3} \rm ) = \rm \big({1}/{8}\big)^3 \hspace{0.15cm}\underline{\approx 0.002}.$$
@@ Line 74: / Line 74: @@
-'''(3)'''&nbsp; Analogous to sub-task&nbsp; '''(2)'''&nbsp;, we obtain here:
+'''(3)'''&nbsp; Analogous to sub-task&nbsp; '''(2)''',&nbsp; we obtain here:
 :$$p_{\rm 3} = \rm Pr (\overline{\it A_{\rm 1}})\cdot \rm Pr (\overline{\it A_{\rm 2}} \hspace{0.05cm}|\hspace{0.05cm}\overline{\it A_{\rm 1}})\cdot \rm Pr (\overline{\it A_{\rm3}}\hspace{0.05cm}|\hspace{0.05cm}(\overline{\it A_{\rm 1}} \cap \overline{\it A_{\rm 2}} )) =\rm \frac{28}{32}\cdot\frac{27}{31}\cdot\frac{26}{30}\hspace{0.15cm}\underline{\approx 0.6605}.$$
@@ Line 80: / Line 80: @@
-'''(4)'''&nbsp; This probability can be expressed as the sum of three probabilities.  &nbsp; &rArr; &nbsp; $p_{\rm 4} = \rm Pr (\it D_{\rm 1} \cup \it D_{\rm 2} \cup \it D_{\rm 3}) $.
+'''(4)'''&nbsp; This probability can be expressed as the sum of three probabilities  &nbsp; &rArr; &nbsp; $p_{\rm 4} = \rm Pr (\it D_{\rm 1} \cup \it D_{\rm 2} \cup \it D_{\rm 3}) $.
-* The corresponding events&nbsp; ${\rm Pr}(D_1)$,&nbsp;  ${\rm Pr}(D_2)$&nbsp; and&nbsp; ${\rm Pr}(D_3)$&nbsp; are disjoint:
+* The corresponding events&nbsp; $D_1$,&nbsp;  $D_2$&nbsp; and&nbsp; $D_3$&nbsp; are disjoint:
-:$$\rm Pr (\it D_{\rm 1}) = \rm Pr (\it A_{\rm 1} \cap \overline{ \it A_{\rm 2}} \cap \overline{\it A_{\rm 3}}) = \rm \frac{4}{32}\cdot \frac{28}{31}\cdot \frac{27}{30}=\rm 0.1016,$$
+:$${\rm Pr} (D_1) = {\rm Pr} (A_1 \cap \overline{ \it A_{\rm 2}} \cap \overline{\it A_{\rm 3}}) = \rm \frac{4}{32}\cdot \frac{28}{31}\cdot \frac{27}{30}=\rm 0.1016,$$
-:$$\rm Pr (\it D_{\rm 2}) =  \rm Pr ( \overline{\it A_{\rm 1}} \cap \it A_{\rm 2} \cap \overline{\it A_{\rm 3}})  = \rm \frac{28}{32}\cdot \frac{4}{31}\cdot\frac{27}{30}=\rm 0.1016,$$
+:$${\rm Pr} (D_2) =  {\rm Pr} ( \overline{A_1} \cap A_2 \cap \overline{A_3})  = \rm \frac{28}{32}\cdot \frac{4}{31}\cdot\frac{27}{30}=\rm 0.1016,$$
-:$$\rm Pr (\it D_{\rm 3}) =  \rm Pr ( \overline{\it A_{\rm 1}} \cap  \overline{\it A_{\rm 2}} \cap \it A_{\rm 3}) = \rm \frac{28}{32}\cdot \frac{27}{31}\cdot \frac{4}{30}=\rm 0.1016.$$
+:$${\rm Pr} (D_3) =  {\rm Pr} ( \overline{A_1} \cap  \overline{A_2} \cap A_3) = \rm \frac{28}{32}\cdot \frac{27}{31}\cdot \frac{4}{30}=\rm 0.1016.$$
 *These probabilities are all the same - why should it be any different?
@@ Line 91: / Line 91: @@
 *This gives us for the sum:
-:$$p_{\rm 4}= \rm Pr (\it D_{\rm 1} \cup \it D_{\rm 2} \cup \it D_{\rm 3}) \rm \hspace{0.15cm}\underline{= 0.3084}.$$
+:$$p_{\rm 4}= {\rm Pr} (D_1 \cup D_2 \cup D_3) \rm \hspace{0.15cm}\underline{= 0.3084}.$$
-'''(5)'''&nbsp; If one defines the events&nbsp; $E_i =$&nbsp; "Exactly&nbsp; $i$&nbsp; aces are drawn" with the indices&nbsp; $i = 0,\ 1,\ 2,\ 3$,
+'''(5)'''&nbsp; If one defines the events&nbsp; $E_i =$&nbsp; &raquo;Exactly&nbsp; $i$&nbsp; aces are drawn&laquo;&nbsp; with the indices&nbsp; $i = 0,\ 1,\ 2,\ 3$,
-*then&nbsp; $E_0$,&nbsp; $E_1$,&nbsp; $E_2$&nbsp; and $E_3$&nbsp; describe a complete system.
+*then&nbsp; $E_0$,&nbsp; $E_1$,&nbsp; $E_2$&nbsp; and $E_3$&nbsp; describe a&nbsp; "complete system".
 *Therefore:
-:$$p_{\rm 5} = \rm Pr (\it E_{\rm 2}) = \rm 1 - \it p_{\rm 2} -\it p_{\rm 3} - \it p_{\rm 4} \hspace{0.15cm}\underline{= \rm 0.0339}.$$
+:$$p_{\rm 5} = {\rm Pr} (E_2) = 1 - p_{\rm 2} - p_{\rm 3} -  p_{\rm 4} \hspace{0.15cm}\underline{= \rm 0.0339}.$$
 {{ML-Fuß}}

@@ Line 10: / Line 10: @@
-In contrast, for the other sub-questions from&nbsp; '''(2)'''&nbsp; onwards, you should assume that the three cards are drawn all at once&nbsp; <br>("draw without putting back").
+In contrast, for the other sub-questions from&nbsp; '''(2)'''&nbsp; onwards, you should assume that the three cards are drawn all at once&nbsp; <br>("card draw without putting back").
 *In the following, we use&nbsp; $A_i$&nbsp; to denote the event that the card drawn at time&nbsp; $i$&nbsp; is an ace. &nbsp; <br>Here we have to set&nbsp; $i = 1,\ 2,\ 3$&nbsp;.
@@ Line 23: / Line 23: @@
 *The exercise belongs to the chapter&nbsp; [[Information_Theory/Einige_Vorbemerkungen_zu_zweidimensionalen_Zufallsgrößen|Some preliminary remarks on 2D random variables]].
 *In particular, the subject matter of the chapter&nbsp;   [[Theory_of_Stochastic_Signals/Statistische_Abhängigkeit_und_Unabhängigkeit|Statistical Dependence and Independence]] in the book "Stochastic Signal Theory" is repeated here.
-*A summary of the theoretical basics with examples can be found in the (German language) learning video&nbsp;[[Statistische_Abhängigkeit_und_Unabhängigkeit_(Lernvideo)|Statistische Abhängigkeit und Unabhängigkeit ]]&nbsp; &rArr; &nbsp;  "Statistical Dependence and Independence".
+*A summary of the theoretical basics with examples can be found in the (German language) learning video&nbsp;<br> &nbsp; &nbsp; &nbsp; [[Statistische_Abhängigkeit_und_Unabhängigkeit_(Lernvideo)|Statistische Abhängigkeit und Unabhängigkeit ]]&nbsp; &rArr; &nbsp;  "Statistical Dependence and Independence".
@@ Line 34: / Line 34: @@
-First, consider the case of „sampling with replacement“.&nbsp; What is the probability&nbsp; $p_1$, that three aces will be drawn?
+First, consider the case of&nbsp; "card draw with putting back".&nbsp; What is the probability&nbsp; $p_1$, that three aces will be drawn?
 |type="{}"}
   $p_1 \ = \ $  { 0.002 3%  }
@@ Line 42: / Line 42: @@
 $p_2 \ = \ $ { 0.0008 3% }
-{Consider further the case of „sampling without replacement“.&nbsp; What is the probability&nbsp; $p_3$ that not a single ace is drawn?
+{Consider further the case of&nbsp; "card draw without putting back".&nbsp; What is the probability&nbsp; $p_3$ that not a single ace is drawn?
 |type="{}"}
 $p_3 \ = \ $ { 0.6605 3% }
-{What is the probability&nbsp; $p_4$ that exactly one ace is drawn in the case „sampling without replacement“?
+{What is the probability&nbsp; $p_4$ that exactly one ace is drawn in the case&nbsp; "card draw without putting back"?
 |type="{}"}
 $p_4 \ = \ $ { 0.3048 3% }

@@ Line 3: / Line 3: @@
 }}
-[[File:P_ID77__Sto_A_1_5.gif|right|frame|The desired result <br>"Three aces are drawn"]]
+[[File:P_ID77__Sto_A_1_5.gif|right|frame|The desired result <br>&raquo;Three aces are drawn&laquo;]]
-From a deck of&nbsp; $32$&nbsp; cards, including four aces, three cards are drawn in succession.&nbsp; For question&nbsp; '''(1)'''&nbsp; , it is assumed that after a card has been drawn
+From a deck of&nbsp; $32$&nbsp; cards, including four aces, three cards are drawn in succession.&nbsp; For question&nbsp; '''(1)''',&nbsp; it is assumed that after a card has been drawn
 *it is put back into the deck,
 *the deck is reshuffled and
@@ Line 10: / Line 10: @@
-In contrast, for the other sub-questions from&nbsp; '''(2)'''&nbsp; onwards, you should assume that the three cards are drawn all at once&nbsp; („draw without putting back“).
+In contrast, for the other sub-questions from&nbsp; '''(2)'''&nbsp; onwards, you should assume that the three cards are drawn all at once&nbsp; <br>("draw without putting back").
 *In the following, we use&nbsp; $A_i$&nbsp; to denote the event that the card drawn at time&nbsp; $i$&nbsp; is an ace. &nbsp; <br>Here we have to set&nbsp; $i = 1,\ 2,\ 3$&nbsp;.
 *The complementary event then states that at time&nbsp; $i$&nbsp; no ace is drawn, but any other card.
@@ Line 27: / Line 23: @@
 *The exercise belongs to the chapter&nbsp; [[Information_Theory/Einige_Vorbemerkungen_zu_zweidimensionalen_Zufallsgrößen|Some preliminary remarks on 2D random variables]].
 *In particular, the subject matter of the chapter&nbsp;   [[Theory_of_Stochastic_Signals/Statistische_Abhängigkeit_und_Unabhängigkeit|Statistical Dependence and Independence]] in the book "Stochastic Signal Theory" is repeated here.
-*A summary of the theoretical basics with examples can be found in the learning video &nbsp;[[Statistische_Abhängigkeit_und_Unabhängigkeit_(Lernvideo)|Statistical Dependence and Independence]].
+*A summary of the theoretical basics with examples can be found in the (German language) learning video&nbsp;[[Statistische_Abhängigkeit_und_Unabhängigkeit_(Lernvideo)|Statistische Abhängigkeit und Unabhängigkeit ]]&nbsp; &rArr; &nbsp;  "Statistical Dependence and Independence".

← Older revision		Revision as of 13:36, 16 July 2021
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−	[[Category:Information Theory: Exercises\|^3.1 ~~Allgemeines zu~~ 2D~~-Zufallsgrößen~~^]]	+	[[Category:Information Theory: Exercises\|^3.1 General Information on 2D Random Variables^]]

@@ Line 3: / Line 3: @@
 }}
-[[File:P_ID77__Sto_A_1_5.gif|right|frame|Das gewünschte Ergebnis <br>"Drei Asse werden gezogen&rdquo;]]
+[[File:P_ID77__Sto_A_1_5.gif|right|frame|Das gewünschte Ergebnis <br>"Drei Asse werden gezogen"]]
 Aus einem Kartenspiel mit&nbsp; $32$&nbsp; Karten, darunter vier Asse, werden nacheinander drei Karten herausgezogen.&nbsp; Für Frage&nbsp; '''(1)'''&nbsp; wird vorausgesetzt, dass nach dem Ziehen einer Karte
 *diese in den Stapel zurückgelegt wird,
@@ Line 26: / Line 26: @@
 ''Hinweise:''
 *Die Aufgabe gehört zum  Kapitel&nbsp; [[Information_Theory/Einige_Vorbemerkungen_zu_zweidimensionalen_Zufallsgrößen|Einige Vorbemerkungen zu den 2D-Zufallsgrößen]].
-*Wiederholt wird hier insbesondere der Lehrstoff des Kapitels&nbsp;   [[Theory_of_Stochastic_Signals/Statistische_Abhängigkeit_und_Unabhängigkeit|Statistische Abhängigkeit und Unabhängigkeit]] im Buch "Stochastische Signaltheorie&rdquo;.
+*Wiederholt wird hier insbesondere der Lehrstoff des Kapitels&nbsp;   [[Theory_of_Stochastic_Signals/Statistische_Abhängigkeit_und_Unabhängigkeit|Statistische Abhängigkeit und Unabhängigkeit]] im Buch "Stochastische Signaltheorie".
 *Eine Zusammenfassung der theoretischen Grundlagen mit Beispielen bringt das Lernvideo &nbsp;[[Statistische_Abhängigkeit_und_Unabhängigkeit_(Lernvideo)|Statistische Abhängigkeit und Unabhängigkeit]].
@@ Line 54: / Line 54: @@
 $p_4 \ = \ $ { 0.3048 3% }
-{Wie groß ist die Wahrscheinlichkeit, dass zwei der drei gezogenen Karten Asse sind? <br>Hinweis: &nbsp; Die Ereignisse „genau&nbsp; $i$&nbsp; Asse werden gezogen” mit&nbsp;  $i = 0,\ 1,\ 2,\ 3$&nbsp; beschreiben ein so genanntes&nbsp; "vollständiges System&rdquo;.
+{Wie groß ist die Wahrscheinlichkeit, dass zwei der drei gezogenen Karten Asse sind? <br>Hinweis: &nbsp; Die Ereignisse „genau&nbsp; $i$&nbsp; Asse werden gezogen” mit&nbsp;  $i = 0,\ 1,\ 2,\ 3$&nbsp; beschreiben ein so genanntes&nbsp; "vollständiges System".
 |type="{}"}
 $p_5 \ = \ $ { 0.0339 3% }
@@ Line 98: / Line 98: @@
-'''(5)'''&nbsp; Definiert man die Ereignisse&nbsp; $E_i =$&nbsp; "Es werden genau&nbsp; $i$&nbsp; Asse gezogen&rdquo; mit den Indizes&nbsp; $i = 0,\ 1,\ 2,\ 3$,
+'''(5)'''&nbsp; Definiert man die Ereignisse&nbsp; $E_i =$&nbsp; "Es werden genau&nbsp; $i$&nbsp; Asse gezogen" mit den Indizes&nbsp; $i = 0,\ 1,\ 2,\ 3$,
 *so beschreiben&nbsp; $E_0$,&nbsp; $E_1$,&nbsp; $E_2$&nbsp; und $E_3$&nbsp; ein vollständiges System.
 *Deshalb gilt: