Aufgaben:Exercise 3.9: Characteristic Curve for Cosine PDF - Revision history

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Guenter at 10:26, 20 December 2021

2021-12-20T10:26:26Z

Guenter: Guenter moved page Aufgaben:Aufgabe 3.9: Kennlinie für Cosinus-WDF to Aufgaben:Exercise 3.9: Characteristic Curve for Cosine PDF

2021-12-20T10:22:57Z

Guenter moved page Aufgabe 3.9: Kennlinie für Cosinus-WDF to Exercise 3.9: Characteristic Curve for Cosine PDF

Javier: Text replacement - "”" to """

2021-05-28T14:33:25Z

Text replacement - "”" to """

Javier: Text replacement - "„" to """

2021-05-28T14:30:45Z

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@@ Line 56: / Line 56: @@
 {{ML-Kopf}}
-'''(1)'''&nbsp; Correct are <u>statements 1 and 3</u>:
+'''(1)'''&nbsp; Correct are the&nbsp; <u>statements 1 and 3</u>:
-*Since&nbsp; $x$&nbsp; can only take values between&nbsp; $\pm 1$&nbsp;, the course of the characteristic curve outside of this range is irrelevant for the random variable&nbsp; $y$&nbsp;.
+*Since&nbsp; $x$&nbsp; can only take values between&nbsp; $\pm 1$,&nbsp; the course of the characteristic curve outside of this range is irrelevant for the random variable&nbsp; $y$.
 *The condition&nbsp; $g(-x) = g(x)$&nbsp; does not have to be met.&nbsp; There are any number of characteristic curves that can generate the desired PDF.
-*For example, the characteristic curve calculated in point&nbsp; '''(5)'''&nbsp; is point symmetric: &nbsp; $g(-x) = -g(x)$.
+*For example,&nbsp; the characteristic curve calculated in point&nbsp; '''(5)'''&nbsp; is point-symmetric: &nbsp; $g(-x) = -g(x)$.
-*The graphical representations of the two density functions already show that&nbsp; $\sigma_y^2 < \sigma_x^2$&nbsp; is.
+*The graphical representations of the two density functions already show that&nbsp; $\sigma_y^2 < \sigma_x^2$.
@@ Line 69: / Line 69: @@
-'''(3)'''&nbsp; The transformation formula can be transformed as follows:
+'''(3)'''&nbsp; The transformation formula can be described as follows:
 :$$f_y(y)=\frac{f_x(x)}{| g'(x)|}\Big|_{\, x=h(y)}=f_x(x)\cdot |h'(y)| \Big|_{\, x=h(y)}.$$
@@ Line 86: / Line 86: @@
 :$$h(y) = \sin({\pi}/{2}\cdot y) \hspace{0.5cm} \rightarrow\hspace{0.5cm}
 h(y = 1) \hspace{0.15cm}\underline{= +1}.$$
@@ Line 92: / Line 91: @@
 :$$y=g(x)={\rm 2}/{\rm \pi}\cdot \rm arcsin({\it x}).$$
-*This characteristic increases monotonically in the range&nbsp; $-1 \le x \le +1$&nbsp; from &nbsp;$y = -1$&nbsp; to &nbsp;$y = +1$&nbsp;.
+*This characteristic curve increases monotonically in the range&nbsp; $-1 \le x \le +1$&nbsp; from &nbsp;$y = -1$&nbsp; to &nbsp;$y = +1$&nbsp;.
 *So the value we are looking for is&nbsp; $g(x= 1) \hspace{0.15cm}\underline{= +1}$.

@@ Line 4: / Line 4: @@
 [[File:P_ID136__Sto_A_3_9.png|right|frame|Rectangular and cosine PDF]]
-We are looking for a continuous, monotonically increasing nonlinear characteristic&nbsp; $y =g(x)$, which generates a new random variable&nbsp; with "cosine" PDF from a between&nbsp; $-1$&nbsp; and&nbsp; $+1$&nbsp; uniformly distributed random variable&nbsp; $x$&nbsp;:
+We are looking for a continuous,&nbsp; monotonically increasing nonlinear characteristic&nbsp; $y =g(x)$,&nbsp; which generates a new random variable&nbsp; with&nbsp; "cosine"&nbsp; PDF from a between&nbsp; $-1$&nbsp; and&nbsp; $+1$&nbsp; uniformly distributed random variable&nbsp; $x$:
 :$$f_y(y)=A\cdot\cos({\pi}/{2}\cdot y).$$
-*The random variable&nbsp; $y$&nbsp; can also only take values between&nbsp; $-1$&nbsp; and&nbsp; $+1$&nbsp;.
+*The random variable&nbsp; $y$&nbsp; can also only take values between&nbsp; $-1$&nbsp; and&nbsp; $+1$.
-*The two density functions&nbsp; $f_x(x)$&nbsp; and&nbsp; $f_y(y)$&nbsp; are sketched opposite.
+*The two density functions&nbsp; $f_x(x)$&nbsp; and&nbsp; $f_y(y)$&nbsp; are sketched on the right.
@@ Line 28: / Line 25: @@
 {Which of the following statements are true?
 |type="[]"}
-+ Outside the range&nbsp; $-1 \le x \le +1$&nbsp; can be&nbsp; $g(x)$&nbsp; arbitrary.
++ Outside the range&nbsp; $-1 \le x \le +1$&nbsp; &rArr; &nbsp; $g(x)$&nbsp; can be arbitrary.
 - The characteristic curve must be symmetrical about&nbsp; $x= 0$&nbsp; : &nbsp; $g(-x) = g(x)$.
 + The random variable&nbsp; $y$&nbsp; has a smaller variance than&nbsp; $x$.
-{Calculate the&nbsp; $f_y(y)$&ndash;value at&nbsp; $y = 0$: &nbsp; $A = f_y(0)$.
+{Calculate the&nbsp; $f_y(y)$ value at&nbsp; $y = 0$: &nbsp; $A = f_y(0)$.
 |type="{}"}
 $A \ = \ $ { 0.785 3% }
-{Determine the slope&nbsp; $h\hspace{0.05cm}'(y)$&nbsp; of the inverse function&nbsp; $x = h(y)$, where for&nbsp; $|y| \le 1$&nbsp; always&nbsp; $h\hspace{0.05cm}'(y) > 0$&nbsp; should hold?&nbsp; What slope holds for&nbsp; $y = 0$&nbsp;?
+{Determine the slope&nbsp; $h\hspace{0.05cm}'(y)$&nbsp; of the inverse function&nbsp; $x = h(y)$,&nbsp; where for&nbsp; $|y| \le 1$&nbsp; always&nbsp; $h\hspace{0.05cm}'(y) > 0$&nbsp; should hold?&nbsp; What slope holds for&nbsp; $y = 0$&nbsp;?
 |type="{}"}
 $h'(y = 0) \ = \ $ { 1.571 3% }
@@ Line 48: / Line 45: @@
-{Determine the function progression&nbsp; $y = g(x)$&nbsp; of the characteristic we are looking for.&nbsp; What is the function value at the point&nbsp; $x = 1$&nbsp;?
+{Determine the function&nbsp; $y = g(x)$&nbsp; of the characteristic we are looking for.&nbsp; What is the function value at the point&nbsp; $x = 1$&nbsp;?
 |type="{}"}
 $g(x = 1) \ = \ $ { 1 3% }

@@ Line 76: / Line 76: @@
 *The inverse function&nbsp; $x = h(y)$&nbsp; of a monotonically increasing characteristic&nbsp; $y = g(x)$&nbsp; 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:
 :$$h\hspace{0.05cm}'(y)=\frac{f_y(y)}{f_x(x)\Big|_{\, x=h(y)}}={\pi}/{\rm 2}\cdot \cos({\pi}/{2}\cdot y).$$

@@ Line 17: / Line 17: @@
 Hints:
-*The exercise belongs to the chapter&nbsp; [[Theory_of_Stochastic_Signals/Exponentially_Distributed_Random_Variables/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 17: / Line 17: @@
 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]].

← Older revision		Revision as of 10:26, 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^]]

@@ Line 4: / Line 4: @@
 [[File:P_ID136__Sto_A_3_9.png|right|frame|Rechteck&ndash; und Cosinus&ndash;WDF]]
-Gesucht ist eine stetige, monoton steigende nichtlineare Kennlinie&nbsp; $y =g(x)$, die aus einer zwischen&nbsp; $-1$&nbsp; und&nbsp; $+1$&nbsp; gleichverteilten Zufallsgr&ouml;&szlig;e&nbsp; $x$&nbsp; eine neue Zufallsgr&ouml;&szlig;e&nbsp; $y$&nbsp; mit "cosinusf&ouml;rmiger&rdquo; WDF generiert:
+Gesucht ist eine stetige, monoton steigende nichtlineare Kennlinie&nbsp; $y =g(x)$, die aus einer zwischen&nbsp; $-1$&nbsp; und&nbsp; $+1$&nbsp; gleichverteilten Zufallsgr&ouml;&szlig;e&nbsp; $x$&nbsp; eine neue Zufallsgr&ouml;&szlig;e&nbsp; $y$&nbsp; mit "cosinusf&ouml;rmiger" WDF generiert:
 :$$f_y(y)=A\cdot\cos({\pi}/{2}\cdot y).$$