Aufgaben:Exercise 2.1Z: Different Signal Courses - Revision history

Guenter at 13:02, 3 December 2021

2021-12-03T13:02:31Z

Guenter at 12:56, 3 December 2021

2021-12-03T12:56:45Z

Guenter at 12:54, 3 December 2021

2021-12-03T12:54:07Z

Guenter at 12:52, 3 December 2021

2021-12-03T12:52:27Z

Noah at 21:41, 30 November 2021

2021-11-30T21:41:00Z

Noah at 21:35, 30 November 2021

2021-11-30T21:35:10Z

Guenter at 15:43, 25 November 2021

2021-11-25T15:43:46Z

Guenter: Guenter moved page Aufgaben:Aufgabe 2.1Z: Signalverläufe to Aufgaben:Exercise 2.1Z: Different Signal Courses

2021-11-25T15:41:45Z

Guenter moved page Aufgabe 2.1Z: Signalverläufe to Exercise 2.1Z: Different Signal Courses

Javier: Text replacement - "„" to """

2021-05-28T14:31:49Z

Text replacement - "„" to """

Javier: Text replacement - "”" to """

2021-05-28T14:28:36Z

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@@ Line 76: / Line 76: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; Correct are <u>suggested solutions 3, 4, and 5</u>:
+'''(1)'''&nbsp; Correct are&nbsp; <u>suggested solutions 3, 4, and 5</u>:
 *The random variables&nbsp; $\rm (C)$&nbsp; and&nbsp; $\rm (D)$&nbsp; are binary&nbsp; $(M= 2)$,
-*whilst the random variable&nbsp; $\rm (E)$&nbsp; is trivalent&nbsp; $(M= 3)$.
+*while the random variable&nbsp; $\rm (E)$&nbsp; is trivalent&nbsp; $(M= 3)$.
-'''(2)'''&nbsp; The <u>proposed solution 1</u> alone is correct:
+'''(2)'''&nbsp; The&nbsp; <u>proposed solution 1</u>&nbsp; alone is correct:
 *The random variable&nbsp; $\rm (A)$&nbsp; is continuous in value and can take all values between&nbsp; $\pm 2 \hspace{0.03cm} \rm V$&nbsp; with equal probability.
 *All other random variables are discrete in value.
@@ Line 88: / Line 88: @@
-'''(3)'''&nbsp; The <u>proposed solution 2</u> alone is correct:
+'''(3)'''&nbsp; The&nbsp; <u>proposed solution 2</u>&nbsp; alone is correct:
-*Only the random variable&nbsp; $\rm (B)$&nbsp; has a discrete part at&nbsp; $0\hspace{0.03cm}\rm V$, and
+*Only the random variable&nbsp; $\rm (B)$&nbsp; has a discrete part at&nbsp; $0\hspace{0.03cm}\rm V$,&nbsp; and
 *also has a continuous component&nbsp; (between&nbsp; $0\hspace{0.03cm} \rm V$&nbsp; and&nbsp; $+2\hspace{0.03cm}\rm V)$.
@@ Line 96: / Line 96: @@
 :$$\rm Pr\left(|\it h_{\rm 0} - \it p_{\rm 0}|\ge\it\varepsilon\right)\le\frac{\rm 1}{\rm 4\cdot \it N\cdot\it\varepsilon^{\rm 2}} = {\it p}_{\rm \hspace{0.01cm}Bernouilli}.$$
-*Thus, the probability that the relative frequency $h_0$ deviates from the probability $p_0 = 0.5$ by more than $0.01$ can be calculated as $\varepsilon = 0.01$:
+*Thus,&nbsp; the probability that the relative frequency&nbsp; $h_0$&nbsp; deviates from the probability&nbsp; $p_0 = 0.5$&nbsp; by more than&nbsp; $0.01$&nbsp; can be calculated as&nbsp; $\varepsilon = 0.01$:
 :$${\it p}_{\rm \hspace{0.01cm}Bernoulli} = \rm\frac{1}{4\cdot 100000\cdot 0.01^2}=\rm 2.5\% \hspace{0.5cm}\Rightarrow \hspace{0.5cm}
-{\rm Min}\big[({\rm Pr}(0.49 \le h_0 \le 0.51)\big] \hspace{0.15cm}\underline{= 0.975}.$$
+{\rm Min}\big[({\rm Pr}(0.49 \le h_0 \le 0.51)\big] \hspace{0.15cm}\underline{= 97.5\%}.$$
@@ Line 104: / Line 104: @@
 '''(5)'''&nbsp; With&nbsp; $p_{\rm Bernoulli} = 1 - 0.99 = 0.01$&nbsp; and&nbsp; $\varepsilon = 0.001$&nbsp; holds again by the law of large numbers:
 :$${\it p}_{\rm \hspace{0.01cm}Bernoulli}\le\frac{\rm 1}{\rm 4\cdot \it N\cdot\it \varepsilon^{\rm 2}}.$$
-*Solved for&nbsp; $N$&nbsp;, one gets:
+*Solved for&nbsp; $N$,&nbsp; one gets:
 :$$N\ge\frac{\rm 1}{\rm 4\cdot\it p_{\rm \hspace{0.01cm}Bernoulli}\cdot\varepsilon^{\rm 2}}=\rm \frac{1}{4\cdot 0.01\cdot 0.001^{2}}=\rm 0.25\cdot 10^8
 \hspace{0.5cm}\Rightarrow \hspace{0.5cm}

@@ Line 63: / Line 63: @@
 {For the signal&nbsp; $\rm (D)$&nbsp; the relative frequency&nbsp; $h_0$&nbsp; is determined empirically over $100\hspace{0.03cm}000$ binary symbols. <br>Name a lower bound for the probability that the determined value lies between&nbsp; $0.49$&nbsp; and&nbsp; $0.51$&nbsp;?
 |type="{}"}
-${\rm Min\big[\ Pr(0.49}≤h_0≤0.51)\ \big] \ = \ $ { 0.975 3% }
+${\rm Min\big[\ Pr(0.49}≤h_0≤0.51)\ \big] \ = \ $ { 97.5 3% } &nbsp; $\%$

← Older revision		Revision as of 12:54, 3 December 2021
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−	{{quiz-Header\|Buchseite=~~Stochastische Signaltheorie~~/~~Wahrscheinlichkeit und relative Häufigkeit~~	+	{{quiz-Header\|Buchseite=Theory_of_Stochastic_Signals/From_Random_Experiment_to_Random_Variable
	}}		}}

@@ Line 3: / Line 3: @@
 }}
-[[File:P_ID59__Sto_Z_2_1.png|right|frame|Discrete value or continuous value?]]
+[[File:P_ID59__Sto_Z_2_1.png|right|frame|Discrete-value or continuous-value?]]
-On the right are shown five signals.&nbsp; The first three signals&nbsp; $\rm (A)$,&nbsp; $\rm (B)$&nbsp; and&nbsp; $\rm (C)$&nbsp; are periodic and thus also deterministic, the two lower signals have stochastic character. The current value of these signals&nbsp; $x(t)$&nbsp; is taken as a random quantity in each case.
+On the right are shown five signals.&nbsp; The first three signals&nbsp; $\rm (A)$,&nbsp; $\rm (B)$&nbsp; and&nbsp; $\rm (C)$&nbsp; are periodic and thus also deterministic,&nbsp; the two lower signals have stochastic character.&nbsp; The current value of these signals&nbsp; $x(t)$&nbsp; is taken as a random variable in each case.
 Shown in detail are:
-$\rm (A)$: &nbsp;&nbsp;a triangular-shaped periodic signal,
+$\rm (A)$: &nbsp;&nbsp;A triangular-shaped periodic signal,
 $\rm (B)$: &nbsp;&nbsp;the signal&nbsp; $\rm (A)$&nbsp; after one-way rectification,
@@ Line 16: / Line 16: @@
 $\rm (D)$: &nbsp;&nbsp;a rectangular random signal,
-$\rm (E)$: &nbsp;&nbsp;the random signal&nbsp; $\rm (D)$&nbsp; according to &nbsp;AMI coding; &nbsp; <br>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; here the "zero" is preserved, while each "one" is alternately encoded with "$+2\hspace{0.03cm}\rm V$" and "$-2\hspace{0.03cm} \rm V$".
+$\rm (E)$: &nbsp;&nbsp;the random signal&nbsp; $\rm (D)$&nbsp; according to &nbsp;AMI coding; &nbsp; <br>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; here the&nbsp; "zero"&nbsp; is preserved,&nbsp; while each&nbsp; "one" is&nbsp; alternately encoded with&nbsp; $+2\hspace{0.03cm}\rm V$&nbsp; and&nbsp; $-2\hspace{0.03cm} \rm V$.
@@ Line 34: / Line 34: @@
 <quiz display=simple>
-{For which signals does the current value describe a discrete random variable? <br>Consider also the respective number of steps&nbsp; $M$.
+{For which signals does the current value describe a discrete random variable? <br>Consider also the respective number of steps &nbsp; &rArr; &nbsp; $M$.
 |type="[]"}
 - Signal $\rm (A)$,
@@ Line 43: / Line 43: @@
-{For which signals is the current value (exclusively) a continuous random variable?
+{For which signals is the current value&nbsp; (exclusively)&nbsp; a continuous random variable?
 |type="[]"}
-+ signal $\rm (A)$,
++ Signal $\rm (A)$,
 - signal $\rm (B)$,
 - signal $\rm (C)$,
@@ Line 66: / Line 66: @@
-{How many symbols&nbsp; $(N_\min)$&nbsp; would you need to use for this investigation to ensure <br>that the probability for the event "The frequency so determined is between&nbsp; $0.499$&nbsp; and&nbsp; $0.501$" is greater than&nbsp; $99\%$&nbsp;?
+{How many symbols&nbsp; $(N_\min)$&nbsp; would you need to use for this investigation to ensure <br>that the probability for the event&nbsp; "The frequency so determined is between&nbsp; $0.499$&nbsp; and&nbsp; $0.501$"&nbsp; is greater than&nbsp; $99\%$&nbsp;?
 |type="{}"}
 $N_\min \ = \ $ { 2.5 3% } $\ \cdot 10^9$

@@ Line 76: / Line 76: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; Richtig sind die <u>Lösungsvorschläge 3, 4 und 5</u>:
+'''(1)'''&nbsp; Correct are <u>suggested solutions 3, 4, and 5</u>:
-*Die Zufallsgrößen&nbsp; $\rm (C)$&nbsp; und&nbsp; $\rm (D)$&nbsp; sind binär&nbsp; $(M= 2)$,
+*The random variables&nbsp; $\rm (C)$&nbsp; and&nbsp; $\rm (D)$&nbsp; are binary&nbsp; $(M= 2)$,
-*w&auml;hrend die Zufallsgr&ouml;&szlig;e&nbsp; $\rm (E)$&nbsp; dreiwertig ist &nbsp; $(M= 3)$.
+*whilst the random variable&nbsp; $\rm (E)$&nbsp; is trivalent&nbsp; $(M= 3)$.
-'''(2)'''&nbsp; Richtig ist allein der <u>Lösungsvorschlag 1</u>:
+'''(2)'''&nbsp; The <u>proposed solution 1</u> alone is correct:
-*Die Zufallsgr&ouml;&szlig;e&nbsp; $\rm (A)$&nbsp; ist wertkontinuierlich und kann alle Werte zwischen&nbsp; $\pm 2 \hspace{0.03cm} \rm V$&nbsp; mit gleicher Wahrscheinlichkeit annehmen.
+*The random variable&nbsp; $\rm (A)$&nbsp; is continuous in value and can take all values between&nbsp; $\pm 2 \hspace{0.03cm} \rm V$&nbsp; with equal probability.
-*Alle anderen Zufallsgrößen sind wertdiskret.
+*All other random variables are discrete in value.
-'''(3)'''&nbsp; Richtig ist allein der <u>Lösungsvorschlag 2</u>:
+'''(3)'''&nbsp; The <u>proposed solution 2</u> alone is correct:
-*Nur die Zufallsgr&ouml;&szlig;e&nbsp; $\rm (B)$&nbsp; hat einen diskreten Anteil bei&nbsp; $0\hspace{0.03cm}\rm V$ und
+*Only the random variable&nbsp; $\rm (B)$&nbsp; has a discrete part at&nbsp; $0\hspace{0.03cm}\rm V$, and
-*au&szlig;erdem noch eine kontinuierliche Komponente&nbsp; (zwischen&nbsp; $0\hspace{0.03cm} \rm V$&nbsp; und&nbsp; $+2\hspace{0.03cm}\rm V)$.
+*also has a continuous component&nbsp; (between&nbsp; $0\hspace{0.03cm} \rm V$&nbsp; and&nbsp; $+2\hspace{0.03cm}\rm V)$.
-'''(4)'''&nbsp; Nach dem Bernoullischen Gesetz der gro&szlig;en Zahlen gilt:
+'''(4)'''&nbsp; According to Bernoulli's law of large numbers:
 :$$\rm Pr\left(|\it h_{\rm 0} - \it p_{\rm 0}|\ge\it\varepsilon\right)\le\frac{\rm 1}{\rm 4\cdot \it N\cdot\it\varepsilon^{\rm 2}} = {\it p}_{\rm \hspace{0.01cm}Bernouilli}.$$
-*Damit ist die Wahrscheinlichkeit, dass die relative H&auml;ufigkeit $h_0$ von der Wahrscheinlichkeit $p_0 = 0.5$ betragsm&auml;&szlig;ig um mehr als $0.01$ abweicht, mit $\varepsilon = 0.01$ berechenbar:
+*Thus, the probability that the relative frequency $h_0$ deviates from the probability $p_0 = 0.5$ by more than $0.01$ can be calculated as $\varepsilon = 0.01$:
 :$${\it p}_{\rm \hspace{0.01cm}Bernoulli} = \rm\frac{1}{4\cdot 100000\cdot 0.01^2}=\rm 2.5\% \hspace{0.5cm}\Rightarrow \hspace{0.5cm}
 {\rm Min}\big[({\rm Pr}(0.49 \le h_0 \le 0.51)\big] \hspace{0.15cm}\underline{= 0.975}.$$
@@ Line 102: / Line 102: @@
-'''(5)'''&nbsp; Mit&nbsp; $p_{\rm Bernoulli} = 1 - 0.99 = 0.01$&nbsp; und&nbsp; $\varepsilon = 0.001$&nbsp; gilt wiederum nach dem Gesetz der gro&szlig;en Zahlen:
+'''(5)'''&nbsp; With&nbsp; $p_{\rm Bernoulli} = 1 - 0.99 = 0.01$&nbsp; and&nbsp; $\varepsilon = 0.001$&nbsp; holds again by the law of large numbers:
-:$${\it p}_{\rm \hspace{0.01cm}Bernoulli}\le\frac{\rm 1}{\rm 4\cdot  \it N\cdot\it \varepsilon^{\rm 2}}.$$
+:$${\it p}_{\rm \hspace{0.01cm}Bernoulli}\le\frac{\rm 1}{\rm 4\cdot \it N\cdot\it \varepsilon^{\rm 2}}.$$
-*Aufgel&ouml;st nach&nbsp; $N$&nbsp; erh&auml;lt man:
+*Solved for&nbsp; $N$&nbsp;, one gets:
-:$$N\ge\frac{\rm 1}{\rm 4\cdot\it p_{\rm \hspace{0.01cm}Bernoulli}\cdot\it\varepsilon^{\rm 2}}=\rm \frac{1}{4\cdot 0.01\cdot 0.001^{2}}=\rm 0.25\cdot 10^8
+:$$N\ge\frac{\rm 1}{\rm 4\cdot\it p_{\rm \hspace{0.01cm}Bernoulli}\cdot\varepsilon^{\rm 2}}=\rm \frac{1}{4\cdot 0.01\cdot 0.001^{2}}=\rm 0.25\cdot 10^8
 \hspace{0.5cm}\Rightarrow \hspace{0.5cm}
 {\it N}_{\rm min} \hspace{0.15cm}\underline{= 2.5\cdot 10^9}.$$

@@ Line 3: / Line 3: @@
 }}
-[[File:P_ID59__Sto_Z_2_1.png|right|frame|Wertdiskret oder wertkontinuierlich?]]
+[[File:P_ID59__Sto_Z_2_1.png|right|frame|Discrete value or continuous value?]]
-Rechts sind f&uuml;nf Signalverl&auml;ufe dargestellt.&nbsp; Die ersten drei Signale&nbsp; $\rm (A)$,&nbsp; $\rm (B)$&nbsp; und&nbsp; $\rm (C)$&nbsp; sind periodisch und damit auch deterministisch, die beiden unteren Signale haben stochastischen Charakter. Der Momentanwert dieser Signale&nbsp; $x(t)$&nbsp; wird jeweils als eine Zufallsgr&ouml;&szlig;e aufgefasst.
+On the right are shown five signals.&nbsp; The first three signals&nbsp; $\rm (A)$,&nbsp; $\rm (B)$&nbsp; and&nbsp; $\rm (C)$&nbsp; are periodic and thus also deterministic, the two lower signals have stochastic character. The current value of these signals&nbsp; $x(t)$&nbsp; is taken as a random quantity in each case.
-Im Einzelnen sind dargestellt:
+Shown in detail are:
-$\rm (A)$: &nbsp;&nbsp;ein dreieckf&ouml;rmiges periodisches Signal,
+$\rm (A)$: &nbsp;&nbsp;a triangular-shaped periodic signal,
-$\rm (B)$: &nbsp;&nbsp;das Signal&nbsp; $\rm (A)$&nbsp; nach Einweggleichrichtung,
+$\rm (B)$: &nbsp;&nbsp;the signal&nbsp; $\rm (A)$&nbsp; after one-way rectification,
-$\rm (C)$: &nbsp;&nbsp;ein rechteckf&ouml;rmiges periodisches Signal,
+$\rm (C)$: &nbsp;&nbsp;a rectangular periodic signal,
-$\rm (D)$: &nbsp;&nbsp;ein rechteckf&ouml;rmiges Zufallssignal,
+$\rm (D)$: &nbsp;&nbsp;a rectangular random signal,
-$\rm (E)$: &nbsp;&nbsp;das Zufallssignal&nbsp; $\rm (D)$&nbsp; nach &nbsp;AMI-Codierung; &nbsp; <br>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; hierbei bleibt die "Null" erhalten, w&auml;hrend eine jede "Eins" alternierend mit "$+2\hspace{0.03cm}\rm V$" und "$-2\hspace{0.03cm} \rm V$" codiert wird.
+$\rm (E)$: &nbsp;&nbsp;the random signal&nbsp; $\rm (D)$&nbsp; according to &nbsp;AMI coding; &nbsp; <br>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; here the "zero" is preserved, while each "one" is alternately encoded with "$+2\hspace{0.03cm}\rm V$" and "$-2\hspace{0.03cm} \rm V$".
@@ Line 24: / Line 24: @@
-''Hinweis:''
+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]].
@@ Line 31: / Line 31: @@
-===Fragebogen===
+===Questions===
 <quiz display=simple>
-{Bei welchen Signalen beschreibt der Momentanwert eine diskrete Zufallsgr&ouml;&szlig;e? <br>&Uuml;berlegen Sie sich auch die jeweilige Stufenzahl&nbsp; $M$.
+{For which signals does the current value describe a discrete random variable? <br>Consider also the respective number of steps&nbsp; $M$.
 |type="[]"}
 - Signal $\rm (A)$,
-- Signal $\rm (B)$,
+- signal $\rm (B)$,
-+ Signal $\rm (C)$,
++ signal $\rm (C)$,
-+ Signal $\rm (D)$,
++ signal $\rm (D)$,
-+ Signal $\rm (E)$.
++ signal $\rm (E)$.
-{Bei welchen Signalen ist der Momentanwert eine (ausschlie&szlig;lich) kontinuierliche Zufallsgr&ouml;&szlig;e?
+{For which signals is the current value (exclusively) a continuous random variable?
 |type="[]"}
-+ Signal $\rm (A)$,
++ signal $\rm (A)$,
-- Signal $\rm (B)$,
+- signal $\rm (B)$,
-- Signal $\rm (C)$,
+- signal $\rm (C)$,
-- Signal $\rm (D)$,
+- signal $\rm (D)$,
-- Signal $\rm (E)$.
+- signal $\rm (E)$.
-{Welche Zufallsgr&ouml;&szlig;en besitzen einen diskreten und einen kontinuierlichen Anteil?
+{Which random variables have a discrete and a continuous part?
 |type="[]"}
 - Signal $\rm (A)$,
-+ Signal $\rm (B)$,
++ signal $\rm (B)$,
-- Signal $\rm (C)$,
+- signal $\rm (C)$,
-- Signal $\rm (D)$,
+- signal $\rm (D)$,
-- Signal $\rm (E)$.
+- signal $\rm (E)$.
-{F&uuml;r das Signal&nbsp; $\rm (D)$&nbsp; wird die relative H&auml;ufigkeit&nbsp; $h_0$&nbsp; empirisch &uuml;ber $100\hspace{0.03cm}000$ Binärsymbole ermittelt. <br>Benennen Sie eine untere Schranke f&uuml;r die Wahrscheinlichkeit, dass der ermittelte Wert zwischen&nbsp; $0.49$&nbsp; und&nbsp; $0.51$&nbsp; liegt?
+{For the signal&nbsp; $\rm (D)$&nbsp; the relative frequency&nbsp; $h_0$&nbsp; is determined empirically over $100\hspace{0.03cm}000$ binary symbols. <br>Name a lower bound for the probability that the determined value lies between&nbsp; $0.49$&nbsp; and&nbsp; $0.51$&nbsp;?
 |type="{}"}
 ${\rm Min\big[\ Pr(0.49}≤h_0≤0.51)\ \big] \ = \ $ { 0.975 3% }
-{Wieviele Symbole&nbsp; $(N_\min)$&nbsp; m&uuml;sste man f&uuml;r diese Untersuchung heranziehen, damit sichergestellt wird, <br>dass die Wahrscheinlichkeit f&uuml;r das Ereignis "Die so ermittelte H&auml;ufigkeit liegt zwischen&nbsp; $0.499$&nbsp; und&nbsp; $0.501$" größer als&nbsp; $99\%$&nbsp; ist?
+{How many symbols&nbsp; $(N_\min)$&nbsp; would you need to use for this investigation to ensure <br>that the probability for the event "The frequency so determined is between&nbsp; $0.499$&nbsp; and&nbsp; $0.501$" is greater than&nbsp; $99\%$&nbsp;?
 |type="{}"}
-$N_\min \ =  \ $  { 2.5 3% } $\ \cdot 10^9$
+$N_\min \ = \ $ { 2.5 3% } $\ \cdot 10^9$
@@ Line 74: / Line 74: @@
 </quiz>
-===Musterlösung===
+===Solution===
 {{ML-Kopf}}
 '''(1)'''&nbsp; Richtig sind die <u>Lösungsvorschläge 3, 4 und 5</u>:

← 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^]]