Aufgaben:Exercise 2.5Z: Flower Meadow - Revision history

Hwang at 12:20, 18 January 2023

2023-01-18T12:20:21Z

Bene: Text replacement - " rms " to " standard deviation "

2022-02-17T11:34:31Z

Text replacement - " rms " to " standard deviation "

Bene: Text replacement - "rms value" to "standard deviation"

2022-02-17T11:11:29Z

Text replacement - "rms value" to "standard deviation"

Guenter at 14:26, 18 December 2021

2021-12-18T14:26:39Z

Guenter at 14:17, 18 December 2021

2021-12-18T14:17:17Z

Noah at 21:46, 15 December 2021

2021-12-15T21:46:45Z

Noah at 20:56, 15 December 2021

2021-12-15T20:56:14Z

Guenter at 10:08, 1 December 2021

2021-12-01T10:08:50Z

Guenter: Guenter moved page Aufgaben:Aufgabe 2.5Z: Blumenwiese to Aufgaben:Exercise 2.5Z: Flower Meadow

2021-12-01T10:06:55Z

Guenter moved page Aufgabe 2.5Z: Blumenwiese to Exercise 2.5Z: Flower Meadow

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

2021-03-23T12:44:46Z

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

← Older revision		Revision as of 12:20, 18 January 2023
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−	'''(2)'''  For the ~~quadratic mean~~ of the random variable  $z$  applies accordingly:	+	'''(2)'''  For the second moment of the random variable  $z$  applies accordingly:
	:$$m_{\rm 2\it z}=\frac{1}{10}\cdot (0^2+1^2+ 2\cdot 2^2+ 2\cdot 3^2+2\cdot 4^2+ 5^2+6^2)=12.$$		:$$m_{\rm 2\it z}=\frac{1}{10}\cdot (0^2+1^2+ 2\cdot 2^2+ 2\cdot 3^2+2\cdot 4^2+ 5^2+6^2)=12.$$

@@ Line 76: / Line 76: @@
 '''(3)'''&nbsp; Correct are&nbsp; <u>solutions 1, 2, and 4</u>:
-*Mean and rms agree here.&nbsp; This is indicative of the Poisson distribution with rate&nbsp; $\lambda = 3$&nbsp; <br>(equal to the mean and equal to the variance,&nbsp; not equal to the standard deviation).
+*Mean and standard deviation agree here.&nbsp; This is indicative of the Poisson distribution with rate&nbsp; $\lambda = 3$&nbsp; <br>(equal to the mean and equal to the variance,&nbsp; not equal to the standard deviation).
 *Naturally,&nbsp; it is questionable to make this statement on the basis of only ten values.&nbsp;
 *However,&nbsp; in the case of moments,&nbsp; a smaller sample number is less serious than,&nbsp; for example,&nbsp; in the case of probabilities.

@@ Line 34: / Line 34: @@
-{Determine the rms value of the random variable&nbsp; $z$.
+{Determine the standard deviation of the random variable&nbsp; $z$.
 |type="{}"}
 $\sigma_z\ = \ $ { 1.732 3% }
@@ Line 43: / Line 43: @@
 + Actually,&nbsp; one would have to use considerably more than ten random numbers&nbsp; (squares)&nbsp; for the moment calculation.
 + The random variable&nbsp; $z$&nbsp; is in fact Poisson distributed.
-- The rate&nbsp; $\lambda$&nbsp; of the Poisson distribution is equal to the rms value&nbsp; $\sigma_z$.
+- The rate&nbsp; $\lambda$&nbsp; of the Poisson distribution is equal to the standard deviation&nbsp; $\sigma_z$.
 + The rate&nbsp; $\lambda$&nbsp; of the Poisson distribution is equal to the mean&nbsp; $m_z$.
@@ Line 71: / Line 71: @@
 *According to Steiner's theorem, the variance is:
 :$$\sigma_z^2 =12 -3^2 = 3,$$
-:and thus the rms value:
+:and thus the standard deviation:
 :$$\underline{\sigma_z \approx 1.732}.$$
 '''(3)'''&nbsp; Correct are&nbsp; <u>solutions 1, 2, and 4</u>:
-*Mean and rms agree here.&nbsp; This is indicative of the Poisson distribution with rate&nbsp; $\lambda = 3$&nbsp; <br>(equal to the mean and equal to the variance,&nbsp; not equal to the rms value).
+*Mean and rms agree here.&nbsp; This is indicative of the Poisson distribution with rate&nbsp; $\lambda = 3$&nbsp; <br>(equal to the mean and equal to the variance,&nbsp; not equal to the standard deviation).
 *Naturally,&nbsp; it is questionable to make this statement on the basis of only ten values.&nbsp;
 *However,&nbsp; in the case of moments,&nbsp; a smaller sample number is less serious than,&nbsp; for example,&nbsp; in the case of probabilities.

@@ Line 17: / Line 17: @@
 Hints:
-*This exercise belongs to the chapter&nbsp; [[Theory_of_Stochastic_Signals/Poisson_Distribution|Poisson distribution]].
+*This exercise belongs to the chapter&nbsp; [[Theory_of_Stochastic_Signals/Poisson_Distribution|Poisson Distribution]].
 *Reference is also made to the chapter&nbsp; [[Theory_of_Stochastic_Signals/Moments_of_a_Discrete_Random_Variable|Moments of a Discrete Random Variable]].
@@ Line 60: / Line 60: @@
 </quiz>
-===Musterlösung===
+===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; The linear mean of these ten numbers gives&nbsp; $\underline{m_z = 3}$.
+'''(1)'''&nbsp; The linear mean of these ten numbers gives&nbsp;
@@ Line 68: / Line 69: @@
 :$$m_{\rm 2\it z}=\frac{1}{10}\cdot (0^2+1^2+ 2\cdot 2^2+ 2\cdot 3^2+2\cdot 4^2+ 5^2+6^2)=12.$$
-*According to Steiner's theorem, the variance is thus equal to.
+*According to Steiner's theorem, the variance is:
-:$$\sigma_z^2 =12 -3^2 = 3$$
+:$$\sigma_z^2 =12 -3^2 = 3,$$
-:and accordingly the rms
+:and thus the rms value:
 :$$\underline{\sigma_z \approx 1.732}.$$
+'''(4)'''&nbsp; In total,&nbsp; there&nbsp; are $80000$&nbsp; such squares,&nbsp; each with three flowers in the mean.&nbsp;
 *This suggests a total of &nbsp; $\underline{B = 240}$&nbsp; thousand flowers.
-'''(5)'''&nbsp; According to the Poisson distribution, this probability results in
+'''(5)'''&nbsp; According to the Poisson distribution,&nbsp; this probability results in
-$${\rm Pr}(z = 0) = \frac{3^0}{0!} \cdot{\rm e}^{-3}\hspace{0.15cm}\underline{\approx 5\%}.$$
+:$${\rm Pr}(z = 0) = \frac{3^0}{0!} \cdot{\rm e}^{-3}\hspace{0.15cm}\underline{\approx 5\%}.$$
-*However, the small sample size&nbsp; $N = 10$&nbsp; on which this task was based would have indicated probability&nbsp; ${\rm Pr}(z = 0) = { 10\%}$&nbsp; since only in a single square no single flower was counted.
+*However,&nbsp; the small sample size&nbsp; $N = 10$&nbsp; on which this task was based would have indicated probability&nbsp; ${\rm Pr}(z = 0) = { 10\%}$&nbsp; <br>since only in a single square no single flower was counted.
 {{ML-Fuß}}

@@ Line 3: / Line 3: @@
 }}
-[[File:P_ID124__Sto_Z_2_5.gif|right|frame|flower meadow &ndash; another <br>example of the Poisson distribution]]
+[[File:P_ID124__Sto_Z_2_5.gif|right|frame|Flower meadow &ndash; another example of the Poisson distribution]]
 A farmer is happy about the splendor of flowers on his land and wants to know how many dandelions are currently blooming on his meadow.
-*He knows that the meadow has an area of $5000$&nbsp; square meters and he also knows from the agricultural school that the number of flowers in a small area is always poisson distributed.
+*He knows that the meadow has an area of&nbsp; $5000$&nbsp; square meters and he also knows from the agricultural school that the number of flowers in a small area is always Poisson distributed.
-*He stakes out ten squares, each with an edge length of &nbsp; $\text{25 cm}$&nbsp;, randomly distributed over the entire meadow and counts the flowers in each of these squares:
+*He stakes out ten squares,&nbsp; each with an edge length of&nbsp; $\text{25 cm}$,&nbsp; randomly distributed over the entire meadow and counts the flowers in each of these squares:
 ::$$\rm 3, \ 4, \ 1, \ 5, \ 0, \ 3, \ 2, \ 4, \ 2, \ 6.$$
 Consider these numerical values as random results of the discrete random variable&nbsp; $z$.
-It is obvious that the sample size is very small at&nbsp; $10$&nbsp; but &ndash; this much is revealed &ndash; the farmer is lucky.&nbsp; First consider how you would proceed to solve this task, and then answer the following questions.
+It is obvious that the sample size&nbsp; $(10)$&nbsp;  is very small but &ndash; this much is revealed &ndash; the farmer is lucky.&nbsp; First consider how you would proceed to solve this task,&nbsp; and then answer the following questions.
@@ Line 33: / Line 29: @@
 <quiz display=simple>
-{Find the mean of&nbsp; $z$,&nbsp; that is, the mean number of flowers counted in the ten squares.
+{Find the mean of&nbsp; $z$,&nbsp; that is,&nbsp; the mean number of flowers counted in each of the ten squares.
 |type="{}"}
 $m_z \ =$ { 3 3% }
-{Determine the rms of the random variable&nbsp; $z$.
+{Determine the rms value of the random variable&nbsp; $z$.
 |type="{}"}
 $\sigma_z\ = \ $ { 1.732 3% }
@@ Line 45: / Line 41: @@
 {Which of the following statements are true?
 |type="[]"}
-+ Actually, one would have to use considerably more than ten random numbers (squares) for the moment calculation.
++ Actually,&nbsp; one would have to use considerably more than ten random numbers&nbsp; (squares)&nbsp; for the moment calculation.
-+ The random size $z$&nbsp; is in fact Poisson distributed.
++ The random variable&nbsp; $z$&nbsp; is in fact Poisson distributed.
-- The rate&nbsp; $\lambda$&nbsp; of the Poisson distribution is equal to the rms&nbsp; $\sigma_z$.
+- The rate&nbsp; $\lambda$&nbsp; of the Poisson distribution is equal to the rms value&nbsp; $\sigma_z$.
 + The rate&nbsp; $\lambda$&nbsp; of the Poisson distribution is equal to the mean&nbsp; $m_z$.

@@ Line 1: / Line 1: @@
-{{quiz-Header|Buchseite=Stochastische Signaltheorie/Poissonverteilung
+{{quiz-Header|Buchseite=Theory_of_Stochastic_Signals/Poisson_Distribution
 }}
-[[File:P_ID124__Sto_Z_2_5.gif|right|frame|Blumenwiese &ndash; ein weiteres <br>Beispiel der Poissonverteilung]]
+[[File:P_ID124__Sto_Z_2_5.gif|right|frame|flower meadow &ndash; another <br>example of the Poisson distribution]]
-Ein Bauer freut sich &uuml;ber die Bl&uuml;tenpracht auf seinem Grund und m&ouml;chte wissen, wie viele L&ouml;wenzahn gerade auf seiner Wiese bl&uuml;hen.
+A farmer is happy about the splendor of flowers on his land and wants to know how many dandelions are currently blooming on his meadow.
-*Er wei&szlig;, dass die Wiese eine Fl&auml;che von&nbsp; $5000$&nbsp; Quadratmeter hat und au&szlig;erdem wei&szlig; er noch von der Landwirtschaftsschule, dass die Anzahl der Blumen in einem kleinen Gebiet stets&nbsp; ''poissonverteilt''&nbsp; ist.
+*He knows that the meadow has an area of $5000$&nbsp; square meters and he also knows from the agricultural school that the number of flowers in a small area is always poisson distributed.
-*Er steckt &uuml;ber der gesamten Wiese &ndash; zuf&auml;llig verteilt &ndash; zehn Quadrate mit einer jeweiligen Kantenl&auml;nge von&nbsp; $\text{25 cm}$&nbsp; ab und z&auml;hlt in jedem dieser Quadrate die Blumen.&nbsp; Dabei kommt er zu folgendem Ergebnis:
+*He stakes out ten squares, each with an edge length of &nbsp; $\text{25 cm}$&nbsp;, randomly distributed over the entire meadow and counts the flowers in each of these squares:
 ::$$\rm 3, \ 4, \ 1, \ 5, \ 0, \ 3, \ 2, \ 4, \ 2, \ 6.$$
-Betrachten Sie diese Zahlenwerte als zuf&auml;llige Ergebnisse der diskreten Zufallsgr&ouml;&szlig;e&nbsp; $z$.
+Consider these numerical values as random results of the discrete random variable&nbsp; $z$.
-Es ist offensichtlich, dass die Stichprobenmenge mit&nbsp; $10$&nbsp; sehr klein ist, aber &ndash; soviel sei verraten &ndash; der Bauer hat Gl&uuml;ck.&nbsp; &Uuml;berlegen Sie sich zun&auml;chst, wie Sie zur L&ouml;sung dieser Aufgabe vorgehen w&uuml;rden, und beantworten Sie dann die folgenden Fragen.
+It is obvious that the sample size is very small at&nbsp; $10$&nbsp; but &ndash; this much is revealed &ndash; the farmer is lucky.&nbsp; First consider how you would proceed to solve this task, and then answer the following questions.
@@ Line 19: / Line 19: @@
-''Hinweise:''
-*Die Aufgabe gehört zum  Kapitel&nbsp; [[Theory_of_Stochastic_Signals/Poissonverteilung|Poissonverteilung]].
+Hints:
-*Bezug genommen wird auch auf das  Kapitel&nbsp; [[Theory_of_Stochastic_Signals/Momente_einer_diskreten_Zufallsgröße|Momente einer diskreten Zufallsgröße]].
+*This exercise belongs to the chapter&nbsp; [[Theory_of_Stochastic_Signals/Poisson_Distribution|Poisson distribution]].
@@ Line 29: / Line 30: @@
-===Fragebogen===
+===Questions===
 <quiz display=simple>
-{Ermitteln Sie den Mittelwert von&nbsp; $z$,&nbsp; das heißt die mittlere Anzahl der in den zehn Quadraten  abgez&auml;hlten Blumen.
+{Find the mean of&nbsp; $z$,&nbsp; that is, the mean number of flowers counted in the ten squares.
 |type="{}"}
 $m_z \ =$ { 3 3% }
-{Bestimmen Sie die Streuung der Zufallsgr&ouml;&szlig;e&nbsp; $z$.
+{Determine the dispersion of the random variable&nbsp; $z$.
 |type="{}"}
 $\sigma_z\ = \ $ { 1.732 3% }
-{Welche der folgenden Aussagen sind zutreffend?
+{Which of the following statements are true?
 |type="[]"}
-+ Eigentlich m&uuml;sste man deutlich mehr als zehn Zufallszahlen (Quadrate) zur Momentenberechnung heranziehen.
++ Actually, one would have to use considerably more than ten random numbers (squares) for the moment calculation.
-+ Die Zufallsgr&ouml;&szlig;e&nbsp; $z$&nbsp; ist tats&auml;chlich poissonverteilt.
++ The random size $z$&nbsp; is in fact Poisson distributed.
-- Die Rate&nbsp; $\lambda$&nbsp; der Poissonverteilung ist gleich der Streuung&nbsp; $\sigma_z$.
+- The rate&nbsp; $\lambda$&nbsp; of the Poisson distribution is equal to the dispersion&nbsp; $\sigma_z$.
-+ Die Rate&nbsp; $\lambda$&nbsp; der Poissonverteilung ist gleich dem Mittelwert&nbsp; $m_z$.
++ The rate&nbsp; $\lambda$&nbsp; of the Poisson distribution is equal to the mean&nbsp; $m_z$.
-{Sagen Sie die Gesamtzahl&nbsp; $B$&nbsp; aller Blumen auf der Wiese voraus.
+{Predict the total number&nbsp; $B$&nbsp; of all flowers in the meadow.
 |type="{}"}
-$B\ =  \ $  { 240 3% }  $\ \text{Tausend}$
+$B\ = \ $ { 240 3% } $\ \text{thousand}$
-{Wie gro&szlig; ist die Wahrscheinlichkeit f&uuml;r ein Quadrat ganz ohne Blumen?
+{What is the probability of a square without any flowers?
 |type="{}"}
 ${\rm Pr}(z = 0) \ = \ $ { 5 3% } $\ \%$

← Older revision		Revision as of 10:08, 1 December 2021
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−	[[Category:Theory of Stochastic Signals: Exercises\|^2.4 ~~Poissonverteilung~~^]]	+	[[Category:Theory of Stochastic Signals: Exercises\|^2.4 Poisson Distribution^]]

← Older revision	Revision as of 10:06, 1 December 2021
(No difference)

← Older revision		Revision as of 12:44, 23 March 2021
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−	[[Category:~~Aufgaben zu Stochastische Signaltheorie~~\|^2.4 Poissonverteilung^]]	+	[[Category:Theory of Stochastic Signals: Exercises\|^2.4 Poissonverteilung^]]