Aufgaben:Exercise 3.8Z: Circle (Ring) Area - Revision history

Guenter at 14:03, 2 February 2022

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Noah at 20:02, 11 January 2022

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Noah at 20:01, 11 January 2022

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Noah at 20:01, 11 January 2022

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Guenter at 10:25, 20 December 2021

2021-12-20T10:25:49Z

Guenter: Guenter moved page Aufgaben:Aufgabe 3.8Z: Kreis(ring)fläche to Aufgaben:Exercise 3.8Z: Circle (Ring) Area

2021-12-20T10:21:41Z

Guenter moved page Aufgabe 3.8Z: Kreis(ring)fläche to Exercise 3.8Z: Circle (Ring) Area

@@ Line 71: / Line 71: @@
 {{ML-Kopf}}
 '''(1)'''&nbsp; The equation of the circular area is at the same time the transformation characteristic: &nbsp; $A = \pi \cdot r^2$.
-*From this, with&nbsp; $r = 6$&nbsp; for the minimum value: &nbsp;
+*From this,&nbsp; with&nbsp; $r = 6$&nbsp; for the minimum value: &nbsp;
 :$$A_\text{min} \hspace{0.15cm}\underline {= 113.09}.$$
-'''(2)'''&nbsp; Correspondingly, with&nbsp; $r = 8$&nbsp; for the maximum value: &nbsp;
+'''(2)'''&nbsp; Correspondingly,&nbsp; with&nbsp; $r = 8$&nbsp; for the maximum value: &nbsp;
 :$$A_\text{max} \hspace{0.15cm}\underline {= 201.06}.$$
@@ Line 87: / Line 87: @@
-'''(4)'''&nbsp; The PDF of the transformed randomgr&ouml;&transform;e&nbsp; $A$&nbsp; is:
+'''(4)'''&nbsp; The PDF of the transformed random variable&nbsp; $A$&nbsp; is:
 :$$f_A(A)=\frac{f_r(r)}{|g\hspace{0.05cm}'(r)|}\Bigg|_{r=h(y) = \sqrt{A/ \pi }}.$$
@@ Line 104: / Line 104: @@
 *There is thus a linear relationship between&nbsp; $R$&nbsp; and&nbsp; $r$&nbsp; .
-*That is:&nbsp; $R$&nbsp; is also uniformly distributed independently of the width&nbsp; $b$ as long as&nbsp; $b \ll r$&nbsp;.
+*That is:&nbsp; $R$&nbsp; is also uniformly distributed independently of the width&nbsp; $b$&nbsp; as long as&nbsp; $b \ll r$.
 *For the minimum value holds:
 :$$R_{\rm min}=\rm 2\pi\cdot 6\cdot 0.1\hspace{0.15cm}\underline{\approx3.77}. $$
-'''(6)'''&nbsp; Accordingly, the maximum value is:
+'''(6)'''&nbsp; Accordingly,&nbsp; the maximum value is:
 :$$R_{\rm max}=\rm 2\pi\cdot 8\cdot 0.1\hspace{0.15cm}\underline{\approx 5.03}.$$

@@ Line 9: / Line 9: @@
-In the sketch above, the area in which such circles&nbsp; (all with center at coordinate origin)&nbsp; can lie is marked in yellow. Furthermore, it can be assumed that the radius in this interval is universally distributed:
+In the sketch above,&nbsp; the area in which such circles&nbsp; $($all with center at coordinate origin$)$&nbsp; can lie is marked in yellow.&nbsp; Furthermore,&nbsp; it can be assumed that the radius in this interval is uniformly distributed:
-:$$f_r(r)=\left\{ \begin{array}{*{4}{c}} 0.5 & \rm for\hspace{0.2cm}{\rm 6\le \it r \le \rm 8}, \rm 0 & \rm else. \end{array} \right.$$
+:$$f_r(r)=\left\{ \begin{array}{*{4}{c}} 0.5 & \rm for\hspace{0.2cm}{\rm 6\le \it r \le \rm 8}, \\\rm 0 & \rm else. \end{array} \right.$$
-From subtask&nbsp; '''(5)'''&nbsp; narrow circular rings with center radius&nbsp; $r$&nbsp; and width&nbsp; $b$&nbsp; are considered (lower sketch):
+From subtask&nbsp; '''(5)'''&nbsp; narrow circular rings with center radius&nbsp; $r$&nbsp; and width&nbsp; $b$&nbsp; are considered&nbsp; $($lower sketch$)$:
-*The area of such a circular ring is denoted by&nbsp; $R$&nbsp;.
+*The area of such a circular ring is denoted by&nbsp; $R$.
-*The possible center radii&nbsp; $r$&nbsp; are again universally distributed between&nbsp; $6$&nbsp; and&nbsp; $8$.
+*The possible center radii&nbsp; $r$&nbsp; are again uniformly distributed between&nbsp; $6$&nbsp; and&nbsp; $8$.
 *The circular ring width is&nbsp; $b = 0.1$.
@@ Line 26: / Line 23: @@
 Hints:
-*The exercise belongs to the chapter&nbsp; [[Theory_of_Stochastic_Signals/Exponentially_Distributed_Random_Variables]].
+*The exercise belongs to the chapter&nbsp; [[Theory_of_Stochastic_Signals/Exponentially_Distributed_Random_Variables|Exponentially distributed random variables]].
 *In particular, reference is made to the page&nbsp; [[Theory_of_Stochastic_Signals/Exponentially_Distributed_Random_Variables#Transformation_of_random_variables|Transformation of random variables]].
@@ Line 35: / Line 32: @@
 <quiz display=simple>
-{Give the transformation characteristic curve&nbsp; $A = g(r)$&nbsp; analytically.&nbsp; What is the minimum value of the random variable&nbsp; $A$?
+{Give the transformation characteristic curve&nbsp; $A = g(r)$&nbsp; analytically.&nbsp; What is the minimum value of the random variable $A$?
 |type="{}"}
 $A_\text{min} \ = \ $ { 113.09 3% }
-{what is the maximum value of the random variable of $A$?
+{What is the maximum value of the random variable $A$?
 |type="{}"}
 $A_\text{max} \ = \ $ { 201.06 3% }
-{What value&nbsp; $m_{ A} = {\rm E}\big[A\big]$&nbsp; results for the "mean" circular area?
+{What value&nbsp; $m_{A} = {\rm E}\big[A\big]$&nbsp; results for the&nbsp; "mean"&nbsp; circular area?
 |type="{}"}
 $m_{ A} \ = \ $ { 154.98 3% }
-{Calculate the probability density function of the random variable&nbsp; $A$.&nbsp; What is the probability that the area&nbsp; $A> 150$&nbsp;?
+{Calculate the probability density function of the random variable $A$.&nbsp; What is the probability that the area&nbsp; $A> 150$&nbsp;?
 |type="{}"}
 ${\rm Pr}(A > 150) \ = \ $ { 54.5 3% } $\ \%$
-{What is the PDF of the random variable&nbsp; $R$&nbsp; (area of the circular rings according to the sketch below)?&nbsp; What is its minimum value?&nbsp; Let&nbsp; $b = 0.1$.
+{What is the PDF of the random variable&nbsp; $R$&nbsp; $($area of the circular rings according to the sketch below$)$?&nbsp; What is its minimum value?&nbsp; Let&nbsp; $b = 0.1$.
 |type="{}"}
 $R_\text{min} \ = \ $ { 3.77 3% }
-{It continues to apply&nbsp; $b = 0.1$. What is the maximum value of the random variable&nbsp; $R$?
+{It continues to apply&nbsp; $b = 0.1$.&nbsp; What is the maximum value of the random variable&nbsp; $R$?
 |type="{}"}
 $R_\text{max} \ = \ $ { 5.03 3% }

@@ Line 9: / Line 9: @@
-In the sketch above, the area in which such circles&nbsp; (all with center at coordinate origin)&nbsp; can lie is marked in yellow. Furthermore, it can be assumed that the radius in this interval is equally distributed:
+In the sketch above, the area in which such circles&nbsp; (all with center at coordinate origin)&nbsp; can lie is marked in yellow. Furthermore, it can be assumed that the radius in this interval is universally distributed:
 :$$f_r(r)=\left\{ \begin{array}{*{4}{c}} 0.5 & \rm for\hspace{0.2cm}{\rm 6\le \it r \le \rm 8}, \rm 0 & \rm else. \end{array} \right.$$
@@ Line 15: / Line 15: @@
 From subtask&nbsp; '''(5)'''&nbsp; narrow circular rings with center radius&nbsp; $r$&nbsp; and width&nbsp; $b$&nbsp; are considered (lower sketch):
 *The area of such a circular ring is denoted by&nbsp; $R$&nbsp;.
-*The possible center radii&nbsp; $r$&nbsp; are again equally distributed between&nbsp; $6$&nbsp; and&nbsp; $8$.
+*The possible center radii&nbsp; $r$&nbsp; are again universally distributed between&nbsp; $6$&nbsp; and&nbsp; $8$.
 *The circular ring width is&nbsp; $b = 0.1$.

@@ Line 26: / Line 26: @@
 Hints:
-*The task belongs to the chapter&nbsp; [[Theory_of_Stochastic_Signals/Exponentially_Distributed_Random_Variables]].
+*The exercise belongs to the chapter&nbsp; [[Theory_of_Stochastic_Signals/Exponentially_Distributed_Random_Variables]].
 *In particular, reference is made to the page&nbsp; [[Theory_of_Stochastic_Signals/Exponentially_Distributed_Random_Variables#Transformation_of_random_variables|Transformation of random variables]].

@@ Line 3: / Line 3: @@
 }}
-[[File:P_ID133__Sto_Z_3_8.png|right|frame|To the circular ring areas]]
+[[File:P_ID133__Sto_Z_3_8.png|right|frame|About the circular ring area]]
 We consider circles of different sizes:
 *The radius&nbsp; $r$&nbsp; and the area&nbsp; $A$&nbsp; can be thought of as interdependent random variables.
@@ Line 73: / Line 73: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; The equation of the circular surface is at the same time the transformation characteristic: &nbsp; $A = \pi \cdot r^2$.
+'''(1)'''&nbsp; The equation of the circular area is at the same time the transformation characteristic: &nbsp; $A = \pi \cdot r^2$.
 *From this, with&nbsp; $r = 6$&nbsp; for the minimum value: &nbsp;
 :$$A_\text{min} \hspace{0.15cm}\underline {= 113.09}.$$
@@ Line 103: / Line 103: @@
-'''(5)'''&nbsp; For the circular ring surface&nbsp; $R$&nbsp; holds for a given radius&nbsp; $r$:
+'''(5)'''&nbsp; For the circular ring area&nbsp; $R$&nbsp; holds for a given radius&nbsp; $r$:
 :$$R=\left (r+{b}/{\rm 2} \right)^{\rm 2}\cdot \rm\pi-\left ({\it r}-{\it b}/{\rm 2} \right)^{\rm 2}\cdot \rm\pi= \rm2\cdot\pi\cdot\it r \cdot b.$$
@@ Line 116: / Line 116: @@
-'''(7)'''&nbsp; Due to the linear relationship between&nbsp; $R$&nbsp; and&nbsp; $r$&nbsp; the mean radius&nbsp; $r = 7$&nbsp; also leads to the mean circular ring surface:
+'''(7)'''&nbsp; Due to the linear relationship between&nbsp; $R$&nbsp; and&nbsp; $r$&nbsp; the mean radius&nbsp; $r = 7$&nbsp; also leads to the mean circular ring area:
 :$${\rm E}\big[R\big]=\rm 2\pi\cdot 7\cdot 0.1\hspace{0.15cm}\underline{\approx 4.4}.$$
 {{ML-Fuß}}

← Older revision		Revision as of 20:01, 11 January 2022
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	In the sketch above, the area in which such circles  (all with center at coordinate origin)  can lie is marked in yellow. Furthermore, it can be assumed that the radius in this interval is equally distributed:		In the sketch above, the area in which such circles  (all with center at coordinate origin)  can lie is marked in yellow. Furthermore, it can be assumed that the radius in this interval is equally distributed:
−	:$$f_r(r)=\left\{ \begin{array}{*{4}{c}} 0.5 & \rm ~~f\ddot{u}r~~\hspace{0.2cm}{\rm 6\le \it r \le \rm 8}, \rm 0 & \rm else. \end{array} \right.$$	+	:$$f_r(r)=\left\{ \begin{array}{*{4}{c}} 0.5 & \rm for\hspace{0.2cm}{\rm 6\le \it r \le \rm 8}, \rm 0 & \rm else. \end{array} \right.$$

← Older revision		Revision as of 10:25, 20 December 2021
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−	[[Category:Theory of Stochastic Signals: Exercises\|^3.6 ~~Exponentialverteilte Zufallsgrößen~~^]]	+	[[Category:Theory of Stochastic Signals: Exercises\|^3.6 Exponentially Distributed Random Variables^]]