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

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	[[Category:Theory of Stochastic Signals: Exercises\|^3.6 Exponentially Distributed Random Variables^]]		[[Category:Theory of Stochastic Signals: Exercises\|^3.6 Exponentially Distributed Random Variables^]]
	[[de:~~Zusatzaufgaben~~:3.~~8_Kreis~~(ring)fläche]]		[[de:Aufgaben:Aufgabe 3.8Z: Kreis(ring)fläche]]

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	*With  $g(r) = \pi \cdot r^2$  and  $f_r(r) = 1/2$  in the range of  $6$ ... $8$  obtains:		*With  $g(r) = \pi \cdot r^2$  and  $f_r(r) = 1/2$  in the range of  $6$ ... $8$  obtains:
	:$$m_{\rm A}=\int_{\rm 6}^{\rm 8}1/2 \cdot\pi\cdot r^{\rm 2}\, {\rm d} \it r=\frac{\pi}{\rm 6}\cdot \rm ( 8^3-6^3)		:$$m_{\rm A}=\int_{\rm 6}^{\rm 8}1/2 \cdot\pi\cdot r^{\rm 2}\, {\rm d} \it r=\frac{\pi}{\rm 6}\cdot \rm ( 8^3-6^3)\hspace{0.15cm}\underline{=\rm 154.98}.$$
	\hspace{0.15cm}\underline{=\rm 154.98}.$$

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	[[Category:Theory of Stochastic Signals: Exercises\|^3.6 Exponentially Distributed Random Variables^]]		[[Category:Theory of Stochastic Signals: Exercises\|^3.6 Exponentially Distributed Random Variables^]]
			[[de:Zusatzaufgaben:3.8_Kreis(ring)fläche]]

Guenter at 14:03, 2 February 2022

2022-02-02T14:03:14Z

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	{{ML-Kopf}}		{{ML-Kopf}}
	'''(1)'''  The equation of the circular area is at the same time the transformation characteristic:   $A = \pi \cdot r^2$.		'''(1)'''  The equation of the circular area is at the same time the transformation characteristic:   $A = \pi \cdot r^2$.
	*From this, with  $r = 6$  for the minimum value:		*From this,  with  $r = 6$  for the minimum value:
	:$$A_\text{min} \hspace{0.15cm}\underline {= 113.09}.$$		:$$A_\text{min} \hspace{0.15cm}\underline {= 113.09}.$$


	'''(2)'''  Correspondingly, with  $r = 8$  for the maximum value:		'''(2)'''  Correspondingly,  with  $r = 8$  for the maximum value:
	:$$A_\text{max} \hspace{0.15cm}\underline {= 201.06}.$$		:$$A_\text{max} \hspace{0.15cm}\underline {= 201.06}.$$

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	'''(4)'''  The PDF of the transformed ~~randomgrö&transform;e~~  $A$  is:		'''(4)'''  The PDF of the transformed random variable  $A$  is:
	:$$f_A(A)=\frac{f_r(r)}{\|g\hspace{0.05cm}'(r)\|}\Bigg\|_{r=h(y) = \sqrt{A/ \pi }}.$$		:$$f_A(A)=\frac{f_r(r)}{\|g\hspace{0.05cm}'(r)\|}\Bigg\|_{r=h(y) = \sqrt{A/ \pi }}.$$

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	*There is thus a linear relationship between  $R$  and  $r$  .		*There is thus a linear relationship between  $R$  and  $r$  .
	*That is:  $R$  is also uniformly distributed independently of the width  $b$ as long as  $b \ll r$~~ ~~.		*That is:  $R$  is also uniformly distributed independently of the width  $b$  as long as  $b \ll r$.
	*For the minimum value holds:		*For the minimum value holds:
	:$$R_{\rm min}=\rm 2\pi\cdot 6\cdot 0.1\hspace{0.15cm}\underline{\approx3.77}. $$		:$$R_{\rm min}=\rm 2\pi\cdot 6\cdot 0.1\hspace{0.15cm}\underline{\approx3.77}. $$


	'''(6)'''  Accordingly, the maximum value is:		'''(6)'''  Accordingly,  the maximum value is:
	:$$R_{\rm max}=\rm 2\pi\cdot 8\cdot 0.1\hspace{0.15cm}\underline{\approx 5.03}.$$		:$$R_{\rm max}=\rm 2\pi\cdot 8\cdot 0.1\hspace{0.15cm}\underline{\approx 5.03}.$$

Guenter at 13:56, 2 February 2022

2022-02-02T13:56:45Z

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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 ~~universally~~ 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 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  '''(5)'''  narrow circular rings with center radius  $r$  and width  $b$  are considered (lower sketch):		From subtask  '''(5)'''  narrow circular rings with center radius  $r$  and width  $b$  are considered  $($lower sketch$)$:
	*The area of such a circular ring is denoted by  $R$~~ ~~.		*The area of such a circular ring is denoted by  $R$.
	*The possible center radii  $r$  are again ~~universally~~ distributed between  $6$  and  $8$.		*The possible center radii  $r$  are again uniformly distributed between  $6$  and  $8$.
	*The circular ring width is  $b = 0.1$.		*The circular ring width is  $b = 0.1$.





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	Hints:		Hints:
	*The exercise belongs to the chapter  [[Theory_of_Stochastic_Signals/Exponentially_Distributed_Random_Variables]].		*The exercise belongs to the chapter  [[Theory_of_Stochastic_Signals/Exponentially_Distributed_Random_Variables\|Exponentially distributed random variables]].
	*In particular, reference is made to the page  [[Theory_of_Stochastic_Signals/Exponentially_Distributed_Random_Variables#Transformation_of_random_variables\|Transformation of random variables]].		*In particular, reference is made to the page  [[Theory_of_Stochastic_Signals/Exponentially_Distributed_Random_Variables#Transformation_of_random_variables\|Transformation of random variables]].

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	<quiz display=simple>		<quiz display=simple>
	{Give the transformation characteristic curve  $A = g(r)$  analytically.  What is the minimum value of the random variable~~ ~~ $A$?		{Give the transformation characteristic curve  $A = g(r)$  analytically.  What is the minimum value of the random variable $A$?
	\|type="{}"}		\|type="{}"}
	$A_\text{min} \ = \ $ { 113.09 3% }		$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="{}"}		\|type="{}"}
	$A_\text{max} \ = \ $ { 201.06 3% }		$A_\text{max} \ = \ $ { 201.06 3% }


	{What value  $m_{ A} = {\rm E}\big[A\big]$  results for the "mean" circular area?		{What value  $m_{A} = {\rm E}\big[A\big]$  results for the  "mean"  circular area?
	\|type="{}"}		\|type="{}"}
	$m_{ A} \ = \ $ { 154.98 3% }		$m_{ A} \ = \ $ { 154.98 3% }


	{Calculate the probability density function of the random variable~~ ~~ $A$.  What is the probability that the area  $A> 150$ ?		{Calculate the probability density function of the random variable $A$.  What is the probability that the area  $A> 150$ ?
	\|type="{}"}		\|type="{}"}
	${\rm Pr}(A > 150) \ = \ $ { 54.5 3% } $\ \%$		${\rm Pr}(A > 150) \ = \ $ { 54.5 3% } $\ \%$


	{What is the PDF of the random variable  $R$  (area of the circular rings according to the sketch below)?  What is its minimum value?  Let  $b = 0.1$.		{What is the PDF of the random variable  $R$  $($area of the circular rings according to the sketch below$)$?  What is its minimum value?  Let  $b = 0.1$.
	\|type="{}"}		\|type="{}"}
	$R_\text{min} \ = \ $ { 3.77 3% }		$R_\text{min} \ = \ $ { 3.77 3% }


	{It continues to apply  $b = 0.1$. What is the maximum value of the random variable  $R$?		{It continues to apply  $b = 0.1$.  What is the maximum value of the random variable  $R$?
	\|type="{}"}		\|type="{}"}
	$R_\text{max} \ = \ $ { 5.03 3% }		$R_\text{max} \ = \ $ { 5.03 3% }

Noah at 18:44, 15 January 2022

2022-01-15T18:44:30Z

← Older revision		Revision as of 20:44, 15 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 universally distributed:
	:$$f_r(r)=\left\{ \begin{array}{*{4}{c}} 0.5 & \rm for\hspace{0.2cm}{\rm 6\le \it r \le \rm 8}, \rm 0 & \rm else. \end{array} \right.$$		:$$f_r(r)=\left\{ \begin{array}{*{4}{c}} 0.5 & \rm for\hspace{0.2cm}{\rm 6\le \it r \le \rm 8}, \rm 0 & \rm else. \end{array} \right.$$

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	From subtask  '''(5)'''  narrow circular rings with center radius  $r$  and width  $b$  are considered (lower sketch):		From subtask  '''(5)'''  narrow circular rings with center radius  $r$  and width  $b$  are considered (lower sketch):
	*The area of such a circular ring is denoted by  $R$ .		*The area of such a circular ring is denoted by  $R$ .
	*The possible center radii  $r$  are again ~~equally~~ distributed between  $6$  and  $8$.		*The possible center radii  $r$  are again universally distributed between  $6$  and  $8$.
	*The circular ring width is  $b = 0.1$.		*The circular ring width is  $b = 0.1$.

Noah at 11:23, 12 January 2022

2022-01-12T11:23:35Z

← Older revision		Revision as of 13:23, 12 January 2022
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	Hints:		Hints:
	*The ~~task~~ belongs to the chapter  [[Theory_of_Stochastic_Signals/Exponentially_Distributed_Random_Variables]].		*The exercise belongs to the chapter  [[Theory_of_Stochastic_Signals/Exponentially_Distributed_Random_Variables]].
	*In particular, reference is made to the page  [[Theory_of_Stochastic_Signals/Exponentially_Distributed_Random_Variables#Transformation_of_random_variables\|Transformation of random variables]].		*In particular, reference is made to the page  [[Theory_of_Stochastic_Signals/Exponentially_Distributed_Random_Variables#Transformation_of_random_variables\|Transformation of random variables]].

Noah at 20:02, 11 January 2022

2022-01-11T20:02:15Z

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	}}		}}

	[[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:		We consider circles of different sizes:
	*The radius  $r$  and the area  $A$  can be thought of as interdependent random variables.		*The radius  $r$  and the area  $A$  can be thought of as interdependent random variables.
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	===Solution===		===Solution===
	{{ML-Kopf}}		{{ML-Kopf}}
	'''(1)'''  The equation of the circular ~~surface~~ is at the same time the transformation characteristic:   $A = \pi \cdot r^2$.		'''(1)'''  The equation of the circular area is at the same time the transformation characteristic:   $A = \pi \cdot r^2$.
	*From this, with  $r = 6$  for the minimum value:		*From this, with  $r = 6$  for the minimum value:
	:$$A_\text{min} \hspace{0.15cm}\underline {= 113.09}.$$		:$$A_\text{min} \hspace{0.15cm}\underline {= 113.09}.$$
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	'''(5)'''  For the circular ring ~~surface~~  $R$  holds for a given radius  $r$:		'''(5)'''  For the circular ring area  $R$  holds for a given radius  $r$:
	:$$R=\left (r+{b}/{\rm 2} \right)^{\rm 2}\cdot \rm\pi-\left ({\it r}-{\it b}/{\rm 2} \right)^{\rm 2}\cdot \rm\pi= \rm2\cdot\pi\cdot\it r \cdot b.$$		:$$R=\left (r+{b}/{\rm 2} \right)^{\rm 2}\cdot \rm\pi-\left ({\it r}-{\it b}/{\rm 2} \right)^{\rm 2}\cdot \rm\pi= \rm2\cdot\pi\cdot\it r \cdot b.$$

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	'''(7)'''  Due to the linear relationship between  $R$  and  $r$  the mean radius  $r = 7$  also leads to the mean circular ring ~~surface~~:		'''(7)'''  Due to the linear relationship between  $R$  and  $r$  the mean radius  $r = 7$  also leads to the mean circular ring area:
	:$${\rm E}\big[R\big]=\rm 2\pi\cdot 7\cdot 0.1\hspace{0.15cm}\underline{\approx 4.4}.$$		:$${\rm E}\big[R\big]=\rm 2\pi\cdot 7\cdot 0.1\hspace{0.15cm}\underline{\approx 4.4}.$$
	{{ML-Fuß}}		{{ML-Fuß}}

Noah at 20:01, 11 January 2022

2022-01-11T20:01:18Z

← Older revision		Revision as of 22:01, 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:
	:$$f_r(r)=\left\{ \begin{array}{*{4}{c}} 0.5 & \rm ~~f\ddot{u}r~~\hspace{0.2cm}{\rm 6\le \it r \le \rm 8}, \rm 0 & \rm else. \end{array} \right.$$		:$$f_r(r)=\left\{ \begin{array}{*{4}{c}} 0.5 & \rm for\hspace{0.2cm}{\rm 6\le \it r \le \rm 8}, \rm 0 & \rm else. \end{array} \right.$$

Noah at 20:01, 11 January 2022

2022-01-11T20:01:07Z

← Older revision		Revision as of 22:01, 11 January 2022
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	Hints:		Hints:
	*The task belongs to the chapter  [[~~Theory_of_Stochastic_Signals/~~Theory_of_Stochastic_Signals/Exponentially_Distributed_Random_Variables]].		*The task belongs to the chapter  [[Theory_of_Stochastic_Signals/Exponentially_Distributed_Random_Variables]].
	*In particular, reference is made to the page  [[Theory_of_Stochastic_Signals/Exponentially_Distributed_Random_Variables#Transformation_of_random_variables\|Transformation of random variables]].		*In particular, reference is made to the page  [[Theory_of_Stochastic_Signals/Exponentially_Distributed_Random_Variables#Transformation_of_random_variables\|Transformation of random variables]].