Aufgaben:Exercise 3.9: Characteristic Curve for Cosine PDF - 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:Aufgaben:3.~~9_Kennlinie_für_Cosinus~~-WDF]]		[[de:Aufgaben:Aufgabe 3.9: Kennlinie für Cosinus-WDF]]

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	*The constraint  $h(y= 0) = 0$  leads to the constant  $C = 0$  and thus to the result:		*The constraint  $h(y= 0) = 0$  leads to the constant  $C = 0$  and thus to the result:
	:$$h(y) = \sin({\pi}/{2}\cdot y) \hspace{0.5cm} \rightarrow\hspace{0.5cm}		:$$h(y) = \sin({\pi}/{2}\cdot y) \hspace{0.5cm} \rightarrow\hspace{0.5cm}h(y = 1) \hspace{0.15cm}\underline{= +1}.$$
	h(y = 1) \hspace{0.15cm}\underline{= +1}.$$

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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:Aufgaben:3.9_Kennlinie_für_Cosinus-WDF]]

Guenter at 14:27, 2 February 2022

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Guenter at 14:16, 2 February 2022

2022-02-02T14:16:49Z

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	[[File:P_ID136__Sto_A_3_9.png\|right\|frame\|Rectangular and cosine PDF]]		[[File:P_ID136__Sto_A_3_9.png\|right\|frame\|Rectangular and cosine PDF]]
	We are looking for a continuous, monotonically increasing nonlinear characteristic  $y =g(x)$, which generates a new random variable  with "cosine" PDF from a between  $-1$  and  $+1$  uniformly distributed random variable  $x$~~ ~~:		We are looking for a continuous,  monotonically increasing nonlinear characteristic  $y =g(x)$,  which generates a new random variable  with  "cosine"  PDF from a between  $-1$  and  $+1$  uniformly distributed random variable  $x$:
	:$$f_y(y)=A\cdot\cos({\pi}/{2}\cdot y).$$		:$$f_y(y)=A\cdot\cos({\pi}/{2}\cdot y).$$

	*The random variable  $y$  can also only take values between  $-1$  and  $+1$~~ ~~.		*The random variable  $y$  can also only take values between  $-1$  and  $+1$.
	*The two density functions  $f_x(x)$  and  $f_y(y)$  are sketched ~~opposite~~.		*The two density functions  $f_x(x)$  and  $f_y(y)$  are sketched on the right.





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	{Which of the following statements are true?		{Which of the following statements are true?
	\|type="[]"}		\|type="[]"}
	+ Outside the range  $-1 \le x \le +1$  ~~can be~~  $g(x)$  arbitrary.		+ Outside the range  $-1 \le x \le +1$  ⇒   $g(x)$  can be arbitrary.
	- The characteristic curve must be symmetrical about  $x= 0$  :   $g(-x) = g(x)$.		- The characteristic curve must be symmetrical about  $x= 0$  :   $g(-x) = g(x)$.
	+ The random variable  $y$  has a smaller variance than  $x$.		+ The random variable  $y$  has a smaller variance than  $x$.


	{Calculate the  $f_y(y)$~~–~~value at  $y = 0$:   $A = f_y(0)$.		{Calculate the  $f_y(y)$ value at  $y = 0$:   $A = f_y(0)$.
	\|type="{}"}		\|type="{}"}
	$A \ = \ $ { 0.785 3% }		$A \ = \ $ { 0.785 3% }


	{Determine the slope  $h\hspace{0.05cm}'(y)$  of the inverse function  $x = h(y)$, where for  $\|y\| \le 1$  always  $h\hspace{0.05cm}'(y) > 0$  should hold?  What slope holds for  $y = 0$ ?		{Determine the slope  $h\hspace{0.05cm}'(y)$  of the inverse function  $x = h(y)$,  where for  $\|y\| \le 1$  always  $h\hspace{0.05cm}'(y) > 0$  should hold?  What slope holds for  $y = 0$ ?
	\|type="{}"}		\|type="{}"}
	$h'(y = 0) \ = \ $ { 1.571 3% }		$h'(y = 0) \ = \ $ { 1.571 3% }
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	{Determine the function ~~progression~~  $y = g(x)$  of the characteristic we are looking for.  What is the function value at the point  $x = 1$ ?		{Determine the function  $y = g(x)$  of the characteristic we are looking for.  What is the function value at the point  $x = 1$ ?
	\|type="{}"}		\|type="{}"}
	$g(x = 1) \ = \ $ { 1 3% }		$g(x = 1) \ = \ $ { 1 3% }

Noah at 11:57, 12 January 2022

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	*The inverse function  $x = h(y)$  of a monotonically increasing characteristic  $y = g(x)$  also increases monotonically.		*The inverse function  $x = h(y)$  of a monotonically increasing characteristic  $y = g(x)$  also increases monotonically.
	*Therefore the ~~magnitude formation can be omitted~~ and ~~one~~ obtains:		*Therefore one does not need to make use of the absolute value and subsequently obtains:
	:$$h\hspace{0.05cm}'(y)=\frac{f_y(y)}{f_x(x)\Big\|_{\, x=h(y)}}={\pi}/{\rm 2}\cdot \cos({\pi}/{2}\cdot y).$$		:$$h\hspace{0.05cm}'(y)=\frac{f_y(y)}{f_x(x)\Big\|_{\, x=h(y)}}={\pi}/{\rm 2}\cdot \cos({\pi}/{2}\cdot y).$$

Noah at 11:54, 12 January 2022

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	Hints:		Hints:
	*The exercise belongs to the chapter  [[Theory_of_Stochastic_Signals/Exponentially_Distributed_Random_Variables/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]].

Noah at 11:54, 12 January 2022

2022-01-12T11:54:04Z

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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]].

Noah at 11:52, 12 January 2022

2022-01-12T11:52:37Z

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Guenter at 10:26, 20 December 2021

2021-12-20T10:26:26Z

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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^]]

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	{{ML-Kopf}}		{{ML-Kopf}}

	'''(1)'''  Correct are <u>statements 1 and 3</u>:		'''(1)'''  Correct are the  <u>statements 1 and 3</u>:
	*Since  $x$  can only take values between  $\pm 1$ , the course of the characteristic curve outside of this range is irrelevant for the random variable  $y$~~ ~~.		*Since  $x$  can only take values between  $\pm 1$,  the course of the characteristic curve outside of this range is irrelevant for the random variable  $y$.
	*The condition  $g(-x) = g(x)$  does not have to be met.  There are any number of characteristic curves that can generate the desired PDF.		*The condition  $g(-x) = g(x)$  does not have to be met.  There are any number of characteristic curves that can generate the desired PDF.
	*For example, the characteristic curve calculated in point  '''(5)'''  is point symmetric:   $g(-x) = -g(x)$.		*For example,  the characteristic curve calculated in point  '''(5)'''  is point-symmetric:   $g(-x) = -g(x)$.
	*The graphical representations of the two density functions already show that  $\sigma_y^2 < \sigma_x^2$~~  is~~.		*The graphical representations of the two density functions already show that  $\sigma_y^2 < \sigma_x^2$.


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	'''(3)'''  The transformation formula can be ~~transformed~~ as follows:		'''(3)'''  The transformation formula can be described as follows:
	:$$f_y(y)=\frac{f_x(x)}{\| g'(x)\|}\Big\|_{\, x=h(y)}=f_x(x)\cdot \|h'(y)\| \Big\|_{\, x=h(y)}.$$		:$$f_y(y)=\frac{f_x(x)}{\| g'(x)\|}\Big\|_{\, x=h(y)}=f_x(x)\cdot \|h'(y)\| \Big\|_{\, x=h(y)}.$$

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	:$$h(y) = \sin({\pi}/{2}\cdot y) \hspace{0.5cm} \rightarrow\hspace{0.5cm}		:$$h(y) = \sin({\pi}/{2}\cdot y) \hspace{0.5cm} \rightarrow\hspace{0.5cm}
	h(y = 1) \hspace{0.15cm}\underline{= +1}.$$		h(y = 1) \hspace{0.15cm}\underline{= +1}.$$



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	:$$y=g(x)={\rm 2}/{\rm \pi}\cdot \rm arcsin({\it x}).$$		:$$y=g(x)={\rm 2}/{\rm \pi}\cdot \rm arcsin({\it x}).$$

	*This characteristic increases monotonically in the range  $-1 \le x \le +1$  from  $y = -1$  to  $y = +1$ .		*This characteristic curve increases monotonically in the range  $-1 \le x \le +1$  from  $y = -1$  to  $y = +1$ .
	*So the value we are looking for is  $g(x= 1) \hspace{0.15cm}\underline{= +1}$.		*So the value we are looking for is  $g(x= 1) \hspace{0.15cm}\underline{= +1}$.