Aufgaben:Exercise 4.7: Weighted Sum and Difference - Revision history

Guenter at 16:33, 25 February 2022

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Guenter at 13:42, 25 February 2022

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Guenter at 13:22, 25 February 2022

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Guenter at 12:44, 25 February 2022

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Noah at 15:51, 24 February 2022

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Noah at 14:36, 24 February 2022

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Guenter at 17:54, 24 January 2022

2022-01-24T17:54:01Z

Guenter: Guenter moved page Aufgaben:Aufgabe 4.7: Gewichtete Summe und Differenz to Aufgaben:Exercise 4.7: Weighted Sum and Difference

2022-01-24T17:49:25Z

Guenter moved page Aufgabe 4.7: Gewichtete Summe und Differenz to Exercise 4.7: Weighted Sum and Difference

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

2021-03-23T12:44:09Z

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

Javier: Text replacement - "[[Stochastische_Signaltheorie/" to "[[Theory_of_Stochastic_Signals/"

2020-07-09T09:52:07Z

Text replacement - "[[Stochastische_Signaltheorie/" to "[[Theory_of_Stochastic_Signals/"

@@ Line 22: / Line 22: @@
-Note:
+Note:&nbsp; The exercise belongs to the chapter&nbsp; [[Theory_of_Stochastic_Signals/Linear_Combinations_of_Random_Variables|Linear Combinations of Random Variables]].
@@ Line 116: / Line 115: @@
 *The second statement is not true: &nbsp; For&nbsp; $A = 1$&nbsp; and&nbsp; $B= -2$&nbsp; also results&nbsp; $\rho_{xy} = -0.6$.
 *So the sign of the quotient does not matter because in the equation calculated in subtask&nbsp; '''(4)''''&nbsp; the quotient&nbsp; $B/A$&nbsp; occurs only quadratically.
-*If&nbsp; $B \gg A$,&nbsp; both&nbsp; $x$&nbsp; and&nbsp; $y$&nbsp; are determined almost exclusively by the random variable&nbsp; $v$&nbsp; and it is&nbsp; $ y \approx -x$.&nbsp; This corresponds to the correlation coefficient&nbsp; $\rho_{xy} = -1$.
+*If&nbsp; $B \gg A$,&nbsp; both&nbsp; $x$&nbsp; and&nbsp; $y$&nbsp; are determined almost exclusively by the random variable&nbsp; $v$&nbsp; and it is&nbsp; $ y \approx -x$.&nbsp; This corresponds to the correlation coefficient&nbsp; $\rho_{xy} \approx -1$.
 *In contrast,&nbsp; $B/A = 1$&nbsp; always yields the correlation coefficient&nbsp; $\rho_{xy} = 0$&nbsp; and thus the uncorrelatedness between&nbsp; $x$&nbsp; and&nbsp; $y$.

@@ Line 77: / Line 77: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; Since the random variables&nbsp; $u$&nbsp; and&nbsp; $v$&nbsp; are zero mean&nbsp; $(m = 0)$, the random variable&nbsp; $x$&nbsp; is also zero mean:
+'''(1)'''&nbsp; Since the random variables&nbsp; $u$&nbsp; and&nbsp; $v$&nbsp; are zero mean&nbsp; $(m = 0)$,&nbsp; the random variable&nbsp; $x$&nbsp; is also zero mean:
 :$$m_x = (A +B) \cdot m \hspace{0.15cm}\underline{ =0}.$$
 *For the variance and standard deviation:
@@ Line 84: / Line 84: @@
-'''(2)'''&nbsp; Since&nbsp; $u$&nbsp; and&nbsp; $v$&nbsp; have the same standard deviation, so does&nbsp; $\sigma_y =\sigma_x \hspace{0.15cm}\underline{ \approx 2.236}$.
+'''(2)'''&nbsp; Since&nbsp; $u$&nbsp; and&nbsp; $v$&nbsp; have the same standard deviation,&nbsp; so does&nbsp; $\sigma_y =\sigma_x \hspace{0.15cm}\underline{ \approx 2.236}$.
-*Because&nbsp; $m=0$&nbsp; also&nbsp; $m_y = m_x \hspace{0.15cm}\underline{ =0}.$.
+*Because&nbsp; $m=0$&nbsp; also&nbsp; $m_y = m_x \hspace{0.15cm}\underline{ =0}$.
-*For mean-valued random variable&nbsp; $u$&nbsp; and&nbsp; $v$&nbsp; on the other hand, for&nbsp; $m_y = (A -B) \cdot m$&nbsp; adds up to a different value than for&nbsp; $m_x = (A +B) \cdot m$.
+*For mean-valued random variable&nbsp; $u$&nbsp; and&nbsp; $v$&nbsp; on the other hand,&nbsp; for&nbsp; $m_y = (A -B) \cdot m$&nbsp; adds up to a different value than for&nbsp; $m_x = (A +B) \cdot m$.
-'''(3)'''&nbsp; We assume here in the sample solution the more general case&nbsp; $m \ne 0$&nbsp; Then for the common moment holds:
+'''(3)'''&nbsp; We assume here in the sample solution the more general case&nbsp; $m \ne 0$.&nbsp; Then,&nbsp; for the common moment holds:
 :$$m_{xy} = {\rm E} \big[x \cdot y \big] = {\rm E} \big[(A \cdot u + B \cdot v) (A \cdot u - B \cdot v)\big] . $$
-*According to the general calculation rules for expected values, it follows:
+*According to the general calculation rules for expected values,&nbsp; it follows:
 :$$m_{xy} = A^2 \cdot {\rm E} \big[u^2 \big] - B^2 \cdot {\rm E} \big[v^2 \big] = (A^2 - B^2)(m^2 + \sigma^2).$$
-*This gives the covariance to.
+*This gives the covariance to
 :$$\mu_{xy} = m_{xy} - m_{x} \cdot m_{y}= (A^2 - B^2)(m^2 + \sigma^2) - (A + B)(A-B) \cdot m^2 = (A^2 - B^2) \cdot \sigma^2.$$
-*With&nbsp; $\sigma = 1$,&nbsp; $A = 1$&nbsp; and&nbsp; $B = 2$&nbsp; we get&nbsp; $\mu_{xy}  \hspace{0.15cm}\underline{ =-3}$&nbsp; and this is independent of the mean&nbsp; $m$&nbsp; of the variables&nbsp; $u$&nbsp; and&nbsp; $v$.
+*With&nbsp; $\sigma = 1$,&nbsp; $A = 1$&nbsp; and&nbsp; $B = 2$&nbsp; we get&nbsp; $\mu_{xy}  \hspace{0.15cm}\underline{ =-3}$.&nbsp; Tthis is independent of the mean&nbsp; $m$&nbsp; of the variables&nbsp; $u$&nbsp; and&nbsp; $v$.
-[[File:P_ID403__Sto_A_4_7_d_neu.png|right|frame|correlation coefficient as a function of quotient&nbsp; $B/A$]]
+[[File:P_ID403__Sto_A_4_7_d_neu.png|right|frame|correlation coefficient as a function of the quotient&nbsp; $B/A$]]
-'''(4)'''&nbsp; The correlation coefficient is obtained as.
+'''(4)'''&nbsp; The correlation coefficient is obtained as
 :$$\rho_{xy} =\frac{\mu_{xy}}{\sigma_x \cdot \sigma_y} = \frac{(A^2 - B^2) \cdot \sigma^2}{(A^2 +B^2) \cdot \sigma^2}
-\hspace{0.5 cm}\rightarrow \hspace{0.5 cm}\rho_{xy} =\frac{1 - (B/A)^2} {1 +(B/A)^2}.$$
+\hspace{0.5 cm}\Rightarrow \hspace{0.5 cm}\rho_{xy} =\frac{1 - (B/A)^2} {1 +(B/A)^2}.$$
 *With&nbsp; $B/A = 2$&nbsp; it follows&nbsp; $\rho_{xy}  \hspace{0.15cm}\underline{ =-0.6}$.
@@ Line 113: / Line 113: @@
 '''(5)'''&nbsp; Correct are <u>statements 1, 3, and 4</u>:
-*From&nbsp; $B= 0$&nbsp; follows&nbsp; $\rho_{xy} = 1$&nbsp; (strict correlation).&nbsp; It can be further seen that in this case&nbsp; $x = u$&nbsp; and&nbsp; $y = u$&nbsp; are identical random variables.
+*From&nbsp; $B= 0$&nbsp; follows&nbsp; $\rho_{xy} = 1$&nbsp; ("strict correlation").&nbsp; It can be further seen that in this case&nbsp; $x = u$&nbsp; and&nbsp; $y = u$&nbsp; are identical random variables.
 *The second statement is not true: &nbsp; For&nbsp; $A = 1$&nbsp; and&nbsp; $B= -2$&nbsp; also results&nbsp; $\rho_{xy} = -0.6$.
 *So the sign of the quotient does not matter because in the equation calculated in subtask&nbsp; '''(4)''''&nbsp; the quotient&nbsp; $B/A$&nbsp; occurs only quadratically.
-*If&nbsp; $B \gg A$, both&nbsp; $x$&nbsp; and&nbsp; $y$&nbsp; are determined almost exclusively by the random variable&ouml;&ahead;e&nbsp; $v$&nbsp; and it is&nbsp; $ y \approx -x$.&nbsp; This corresponds to the correlation coefficient&nbsp; $\rho_{xy} = -1$.
+*If&nbsp; $B \gg A$,&nbsp; both&nbsp; $x$&nbsp; and&nbsp; $y$&nbsp; are determined almost exclusively by the random variable&nbsp; $v$&nbsp; and it is&nbsp; $ y \approx -x$.&nbsp; This corresponds to the correlation coefficient&nbsp; $\rho_{xy} = -1$.
 *In contrast,&nbsp; $B/A = 1$&nbsp; always yields the correlation coefficient&nbsp; $\rho_{xy} = 0$&nbsp; and thus the uncorrelatedness between&nbsp; $x$&nbsp; and&nbsp; $y$.
-'''(6)'''&nbsp; <u>Both statements correct</u> are true:
+'''(6)'''&nbsp; <u>Both statements</u> are true:
-*When&nbsp; $A=B$&nbsp; are&nbsp; $x$&nbsp; and&nbsp; $y$&nbsp; always&nbsp; $($so for any PDF of the variables $u$&nbsp; and&nbsp; $v)$&nbsp; uncorrelated.
+*When&nbsp; $A=B$&nbsp; &rArr; &nbsp; $x$&nbsp; and&nbsp; $y$&nbsp; are always  uncorrelated&nbsp; $($for any PDF of the variables $u$&nbsp; and&nbsp; $v)$.
 *The new random variables&nbsp; $x$&nbsp; and&nbsp; $y$&nbsp; are therefore also distributed randomly.
-*For Gaussian randomness, however, statistical independence follows from uncorrelatedness, and vice versa.
+*For Gaussian randomness,&nbsp; however,&nbsp; statistical independence follows from uncorrelatedness,&nbsp; and vice versa.
-[[File:P_ID404__Sto_A_4_7_g.png|right|frame|2D-PDF and Edge-PDF]]
+[[File:P_ID404__Sto_A_4_7_g.png|right|frame|Joint PDF and edge PDFs]]
-'''(7)'''&nbsp; Here, only <u>statement 1</u> is true:
+'''(7)'''&nbsp; Here,&nbsp; only <u>statement 1</u> is true:
-*The correlation coefficient results with&nbsp; $A=B= 1$&nbsp; also here to&nbsp; $\rho_{xy} = 0$.&nbsp; That is:&nbsp; $x$&nbsp; and&nbsp; $y$&nbsp; are uncorrelated also here.
+*The correlation coefficient results with&nbsp; $A=B= 1$&nbsp; to&nbsp; $\rho_{xy} = 0$.&nbsp; That is:&nbsp; $x$&nbsp; and&nbsp; $y$&nbsp; are uncorrelated.
-*On the other hand, it can be seen from the sketched two dimensional PDF that the condition of statistical independence no longer applies in the present case:
+*But it can be seen from the sketched two-dimensional PDF that the condition of statistical independence no longer applies in the present case:
-$$f_{xy}(x, y) \ne f_{x}(x) \cdot f_{y}(y).$$
+:$$f_{xy}(x, y) \ne f_{x}(x) \cdot f_{y}(y).$$

@@ Line 5: / Line 5: @@
 [[File:P_ID400__Sto_A_4_7.png|right|frame|Sum and difference of random variables]]
 Let the random variables&nbsp; $u$&nbsp; and&nbsp; $v$&nbsp; be statistically independent of each other, each with mean&nbsp; $m$&nbsp; and variance&nbsp; $\sigma^2$.
-*Both variables have equal PDF and CDF.
+*Both variables have equal probability density function&nbsp; $\rm (PDF)$&nbsp; and cumulative distribution function&nbsp; $\rm (CDF)$.
 *Nothing is known about the course of these functions for the time being.
@@ Line 13: / Line 13: @@
 :$$y= A \cdot u - B \cdot v.$$
-Here&nbsp; $A$&nbsp; and&nbsp; $B$&nbsp; denote (any) constant values.
+Here,&nbsp; $A$&nbsp; and&nbsp; $B$&nbsp; denote&nbsp; (any)&nbsp; constant values.
 *For the subtasks&nbsp; '''(1)'''&nbsp; to&nbsp; '''(4)'''&nbsp; let &nbsp; $m= 0$, &nbsp; $\sigma = 1$, &nbsp; $A = 1$&nbsp; and&nbsp; $B = 2$.
-*In the sub-task&nbsp; '''(6)''''&nbsp; it is assumed that&nbsp; $u$&nbsp; and&nbsp; $v$&nbsp; are each uniformly distributed with mean&nbsp; $m= 1$&nbsp; and standard deviation&nbsp; $\sigma = 0.5$&nbsp; . For the constants, $A = B = 1$.
+*In subtask&nbsp; '''(6)'''&nbsp;  $u$&nbsp; and&nbsp; $v$&nbsp; are each uniformly distributed with&nbsp; $m= 1$&nbsp; and&nbsp; $\sigma = 0.5$. For the constants,&nbsp; $A = B = 1$.
-*For the exercise&nbsp; '''(7)'''&nbsp; it is still valid&nbsp; $A = B = 1$.&nbsp; Here the random variables&nbsp; $u$&nbsp; and&nbsp; $v$&nbsp; are symmetrically two-point distributed on&nbsp; $\pm$1:
+*For subtask&nbsp; '''(7)'''&nbsp; it is still valid&nbsp; $A = B = 1$.&nbsp; Here the random variables&nbsp; $u$&nbsp; and&nbsp; $v$&nbsp; are symmetrically two-point distributed on&nbsp; $\pm$1:
 :$${\rm Pr}(u=+1) = {\rm Pr}(u=-1) = {\rm Pr}(v=+1) = {\rm Pr}(v=-1) =0.5.$$
@@ Line 31: / Line 31: @@
 <quiz display=simple>
-{What is the mean and standard deviation of &nbsp;$x$&nbsp; for&nbsp; $A = 1$&nbsp; and&nbsp; $B = 2$?
+{What is the mean and the standard deviation of &nbsp;$x$&nbsp; for&nbsp; $A = 1$&nbsp; and&nbsp; $B = 2$?
 |type="{}"}
 $m_x \ = \ $ { 0. }
@@ Line 37: / Line 37: @@
-{What is the mean and standard deviation of &nbsp;$y$&nbsp; for&nbsp; $A = 1$&nbsp; and&nbsp; $B = 2$?
+{What is the mean and the standard deviation of &nbsp;$y$&nbsp; for&nbsp; $A = 1$&nbsp; and&nbsp; $B = 2$?
 |type="{}"}
 $m_y \ = \ $ { 0. }
@@ Line 61: / Line 61: @@
-{Which statements are true if&nbsp; $A =B = 1$&nbsp; holds and &nbsp;$x$&nbsp; and &nbsp;$y$&nbsp; are each gauged;distributed with mean &nbsp;$m = 1$&nbsp; and standard deviation &nbsp;$\sigma = 0.5$&nbsp;?
+{Which statements are true if&nbsp; $A =B = 1$&nbsp; holds and &nbsp;$x$&nbsp; and &nbsp;$y$&nbsp; are each Gaussian distributed with mean &nbsp;$m = 1$&nbsp; and standard deviation &nbsp;$\sigma = 0.5$?
 |type="[]"}
 + The random variables&nbsp;$x$&nbsp; and &nbsp;$y$&nbsp; are uncorrelated.

← Older revision		Revision as of 12:44, 25 February 2022
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−		+	Note:
−
−
−
−
−	''Note:''
	*The exercise belongs to the chapter  [[Theory_of_Stochastic_Signals/Linear_Combinations_of_Random_Variables\|Linear Combinations of Random Variables]].		*The exercise belongs to the chapter  [[Theory_of_Stochastic_Signals/Linear_Combinations_of_Random_Variables\|Linear Combinations of Random Variables]].

@@ Line 104: / Line 104: @@
 :$$\mu_{xy} = m_{xy} - m_{x} \cdot m_{y}= (A^2 - B^2)(m^2 + \sigma^2) - (A + B)(A-B) \cdot m^2 = (A^2 - B^2) \cdot \sigma^2.$$
-*With&nbsp; $\sigma = 1$,&nbsp; $A = 1$&nbsp; and&nbsp; $B = 2$&nbsp; we get&nbsp; $\mu_{xy}  \hspace{0.15cm}\underline{ =-3}$&nbsp; and this is independent of the mean&nbsp; $m$&nbsp; of the sizes&nbsp; $u$&nbsp; and&nbsp; $v$.
+*With&nbsp; $\sigma = 1$,&nbsp; $A = 1$&nbsp; and&nbsp; $B = 2$&nbsp; we get&nbsp; $\mu_{xy}  \hspace{0.15cm}\underline{ =-3}$&nbsp; and this is independent of the mean&nbsp; $m$&nbsp; of the variables&nbsp; $u$&nbsp; and&nbsp; $v$.
@@ Line 118: / Line 118: @@
 '''(5)'''&nbsp; Correct are <u>statements 1, 3, and 4</u>:
-*From&nbsp; $B= 0$&nbsp; follows&nbsp; $\rho_{xy} = 1$&nbsp; (strict correlation).&nbsp; It can be further seen that in this case&nbsp; $x = u$&nbsp; and&nbsp; $y = u$&nbsp; are identical random sizes.
+*From&nbsp; $B= 0$&nbsp; follows&nbsp; $\rho_{xy} = 1$&nbsp; (strict correlation).&nbsp; It can be further seen that in this case&nbsp; $x = u$&nbsp; and&nbsp; $y = u$&nbsp; are identical random variables.
 *The second statement is not true: &nbsp; For&nbsp; $A = 1$&nbsp; and&nbsp; $B= -2$&nbsp; also results&nbsp; $\rho_{xy} = -0.6$.
 *So the sign of the quotient does not matter because in the equation calculated in subtask&nbsp; '''(4)''''&nbsp; the quotient&nbsp; $B/A$&nbsp; occurs only quadratically.
@@ Line 127: / Line 127: @@
 '''(6)'''&nbsp; <u>Both statements correct</u> are true:
-*When&nbsp; $A=B$&nbsp; are&nbsp; $x$&nbsp; and&nbsp; $y$&nbsp; always&nbsp; $($so for any PDF of the sizes&ouml;&aeries;$u$&nbsp; and&nbsp; $v)$&nbsp; uncorrelated.
+*When&nbsp; $A=B$&nbsp; are&nbsp; $x$&nbsp; and&nbsp; $y$&nbsp; always&nbsp; $($so for any PDF of the variables $u$&nbsp; and&nbsp; $v)$&nbsp; uncorrelated.
 *The new random variables&nbsp; $x$&nbsp; and&nbsp; $y$&nbsp; are therefore also distributed randomly.
 *For Gaussian randomness, however, statistical independence follows from uncorrelatedness, and vice versa.

← Older revision		Revision as of 17:54, 24 January 2022
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−	[[Category:Theory of Stochastic Signals: Exercises\|^4.3 ~~Linearkombinationen~~^]]	+	[[Category:Theory of Stochastic Signals: Exercises\|^4.3 Linear Combinations^]]

← Older revision		Revision as of 12:44, 23 March 2021
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−	[[Category:~~Aufgaben zu Stochastische Signaltheorie~~\|^4.3 Linearkombinationen^]]	+	[[Category:Theory of Stochastic Signals: Exercises\|^4.3 Linearkombinationen^]]

← Older revision		Revision as of 09:52, 9 July 2020
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	''Hinweis:''		''Hinweis:''
−	*Die Aufgabe gehört zum Kapitel  [[~~Stochastische_Signaltheorie~~/Linearkombinationen_von_Zufallsgrößen\|Linearkombinationen von Zufallsgrößen]].	+	*Die Aufgabe gehört zum Kapitel  [[Theory_of_Stochastic_Signals/Linearkombinationen_von_Zufallsgrößen\|Linearkombinationen von Zufallsgrößen]].