Aufgaben:Exercise 4.3: Algebraic and Modulo Sum - Revision history

Hwang at 12:49, 17 November 2022

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Guenter at 15:42, 22 February 2022

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Guenter at 14:39, 7 February 2022

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Guenter at 13:48, 7 February 2022

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Noah at 14:14, 19 January 2022

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Guenter at 10:52, 12 January 2022

2022-01-12T10:52:04Z

Guenter: Guenter moved page Aufgaben:Aufgabe 4.3: Algebraische und Modulo-Summe to Aufgaben:Exercise 4.3: Algebraic and Modulo Sum

2022-01-12T10:48:16Z

Guenter moved page Aufgabe 4.3: Algebraische und Modulo-Summe to Exercise 4.3: Algebraic and Modulo Sum

Javier: Text replacement - "„" to """

2021-05-28T14:45:58Z

Text replacement - "„" to """

@@ Line 66: / Line 66: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; It can be seen from the table on the information page that for the modulo&ndash;2 sum,&nbsp; the two values&nbsp; $0$&nbsp; and&nbsp; $1$&nbsp; have equal probability:
+'''(1)'''&nbsp; It can be seen from the table in the information section that for the modulo&ndash;2 sum,&nbsp; the two values&nbsp; $0$&nbsp; and&nbsp; $1$&nbsp; have equal probability:
 :$${\rm Pr}(m_\nu = 0) = {\rm Pr}(m_\nu = 1)\hspace{0.15cm}\underline{=0.5}.$$
@@ Line 80: / Line 80: @@
 '''(3)'''&nbsp; Correct are&nbsp; <u>the second and the last suggested solutions</u>.
 *The 2D&ndash;PDF consists of four Dirac delta functions,&nbsp; each with weight&nbsp; $1/4$.
-*One obtains this result,&nbsp; for example,&nbsp; by evaluating the table on the data page.
+*One obtains this result,&nbsp; for example,&nbsp; by evaluating the table in the data section.
 *Since&nbsp; $f_{xm}(x_\nu, m_\nu)=f_{x}(x_\nu) \cdot f_{m}(m_\nu)$,&nbsp; the quantities&nbsp; $x_\nu$&nbsp; and&nbsp; $m_\nu$&nbsp; are statistically independent.
 *Statistically independent random variables,&nbsp; however,&nbsp; are also linearly statistically independent,&nbsp; so they are certainly uncorrelated.

@@ Line 3: / Line 3: @@
 }}
-[[File:EN_Sto_A_4_3.png|right|frame|Algebraic & modulo&ndash;2 sum]]
+[[File:EN_Sto_A_4_3_neu2.png|right|frame|Algebraic & modulo&ndash;2 sum]]
 [[File:P_ID254__Sto_A_4_3Tab.png|right|frame|Table for moment calculation]]

@@ Line 71: / Line 71: @@
-'''(2)'''&nbsp; The table shows that for each preassignment &nbsp; &rArr; &nbsp; $( x_{\nu-1}, x_{\nu-2}) = (0,0), (0,1), (1,0), (1,1)$ &nbsp; the values&nbsp; $m_\nu = 0$&nbsp; and&nbsp; $m_\nu = 1$&nbsp; resp. are equally likely.
+'''(2)'''&nbsp; The table shows that for each preassignment &nbsp; &rArr; &nbsp; $( x_{\nu-1}, x_{\nu-2}) = (0,0), (0,1), (1,0), (1,1)$,&nbsp; the values&nbsp; $m_\nu = 0$&nbsp; and&nbsp; $m_\nu = 1$&nbsp; resp. are equally likely.
 *Expressed differently: &nbsp; ${\rm Pr}(m_{\nu}\hspace{0.05cm}|\hspace{0.05cm}m_{\nu-1}) = {\rm Pr}( m_{\nu}).$
-*This exactly matches the definition of "statistical independence" &nbsp; &rArr; &nbsp; <u>Answer 1</u>.
+*This exactly matches the definition of&nbsp; "statistical independence" &nbsp; &rArr; &nbsp; <u>Answer 1</u>.
-[[File:P_ID224__Sto_A_4_3_c.png|right|frame|2D PDF of&nbsp; $x$&nbsp; and&nbsp; $m$]]
+[[File:P_ID224__Sto_A_4_3_c.png|right|frame|2D&ndash;PDF of&nbsp; $x$&nbsp; and&nbsp; $m$]]
-'''(3)'''&nbsp; Correct are <u>the second and the last suggested solutions</u>.
+'''(3)'''&nbsp; Correct are&nbsp; <u>the second and the last suggested solutions</u>.
-*The 2D PDF consists of four Dirac functions, each with weight&nbsp; $1/4$.
+*The 2D&ndash;PDF consists of four Dirac delta functions,&nbsp; each with weight&nbsp; $1/4$.
-*One obtains this result, for example, by evaluating the table on the data page.
+*One obtains this result,&nbsp; for example,&nbsp; by evaluating the table on the data page.
-*Since $f_{xm}(x_\nu, m_\nu)$&nbsp; is equal to the product $f_{x}(x_\nu) \cdot f_{m}(m_\nu)$&nbsp; the quantities&nbsp; $x_\nu$&nbsp; and&nbsp; $m_\nu$&nbsp; are statistically independent.
+*Since&nbsp; $f_{xm}(x_\nu, m_\nu)=f_{x}(x_\nu) \cdot f_{m}(m_\nu)$,&nbsp; the quantities&nbsp; $x_\nu$&nbsp; and&nbsp; $m_\nu$&nbsp; are statistically independent.
-*Statistically independent random variables, however, are also linearly statistically independent, so they are certainly uncorrelated.
+*Statistically independent random variables,&nbsp; however,&nbsp; are also linearly statistically independent,&nbsp; so they are certainly uncorrelated.
@@ Line 88: / Line 88: @@
 '''(4)'''&nbsp; Within the sequence&nbsp; $\langle a_\nu \rangle$&nbsp; of algebraic sum there are statistical bindings &nbsp; &#8658; &nbsp; <u>Answer 2</u>.
-*You can see this because the unconditional probability&nbsp; $ {\rm Pr}( a_{\nu} = 0) =1/8$&nbsp; is,
+*You can see this because the unconditional probability is&nbsp; $ {\rm Pr}( a_{\nu} = 0) =1/8$,&nbsp;
-*while, for example&nbsp; ${\rm Pr}(a_{\nu} = 0\hspace{0.05cm}|\hspace{0.05cm}a_{\nu-1} = 3) =0$&nbsp; holds.
+*while,&nbsp; for example,&nbsp; ${\rm Pr}(a_{\nu} = 0\hspace{0.05cm}|\hspace{0.05cm}a_{\nu-1} = 3) =0$&nbsp; holds.
-[[File:P_ID225__Sto_A_4_3_e.png|right|frame|2D PDF of&nbsp; $a$&nbsp; and&nbsp; $m$]]
+[[File:P_ID225__Sto_A_4_3_e.png|right|frame|2D&ndash;PDF of&nbsp; $a$&nbsp; and&nbsp; $m$]]
-'''(5)'''&nbsp; Correct are <u>the first and the last suggested solutions</u>:
+'''(5)'''&nbsp; Correct are&nbsp; <u>the first and the last suggested solutions</u>:
-*As in the subtask&nbsp; '''(3)'''&nbsp; there are again four Dirac functions, but this time not with equal momentum weights&nbsp; $1/4$.
+*As in the subtask&nbsp; '''(3)'''&nbsp; there are again four Dirac delta functions,&nbsp; but this time not with equal Dirac weights&nbsp; $1/4$.
 *The two-dimensional PDF thus cannot be written as a product of the two marginal probability densities.
-*But this means that statistical bindings must exist between&nbsp; $a_\nu$&nbsp; and&nbsp; $m_\nu$&nbsp; .
+*But this means that statistical bindings must exist between&nbsp; $a_\nu$&nbsp; and&nbsp; $m_\nu$.
-*For the joint expected value, one obtains:
+*For the joint expected value,&nbsp; one obtains:
 :$${\rm E}\big[a\cdot m \big] = \rm \frac{1}{8}\cdot 0 \cdot 0 +\frac{3}{8}\cdot 2 \cdot 0 +\frac{3}{8}\cdot 1 \cdot 1 + \frac{1}{8}\cdot 3 \cdot 1 = \frac{3}{4}.$$
-*With the linear means&nbsp; ${\rm E}\big[a \big] = 1.5$ &nbsp;and&nbsp; ${\rm E}[m] = 0.5$&nbsp; it thus follows for the covariance:
+*With the linear means&nbsp; ${\rm E}\big[a \big] = 1.5$ &nbsp;and&nbsp; ${\rm E}[m] = 0.5$&nbsp; it follows for the covariance:
 :$$\mu_{am}= {\rm E}\big[ a\cdot m \big] - {\rm E}\big[ a \big]\cdot {\rm E} \big[ m \big] = \rm 0.75-1.5\cdot 0.5 = \rm 0.$$
-*Thus, the correlation coefficient&nbsp; $\rho_{am}= 0$.&nbsp; That is, &nbsp; The dependencies present are nonlinear.
+*Thus,&nbsp; the correlation coefficient&nbsp; $\rho_{am}= 0$.&nbsp; That is: &nbsp; The dependencies present are nonlinear.
-*The quantities&nbsp; $a_\nu$&nbsp; and&nbsp; $m_\nu$&nbsp; are statistically dependent, but still uncorrelated.
+*The quantities&nbsp; $a_\nu$&nbsp; and&nbsp; $m_\nu$&nbsp; are statistically dependent,&nbsp; but still uncorrelated.
 {{ML-Fuß}}

@@ Line 20: / Line 20: @@
+<br><br><br><br><br><br>
 Hints: &nbsp;
 *This exercise belongs to the chapter&nbsp; [[Theory_of_Stochastic_Signals/Two-Dimensional_Random_Variables|Two-Dimensional Random Variables]].
@@ Line 66: / Line 66: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; It can be seen from the table on the information page that for the modulo 2 sum, the two values&nbsp; $0$&nbsp; and&nbsp; $1$&nbsp; have equal probability:
+'''(1)'''&nbsp; It can be seen from the table on the information page that for the modulo&ndash;2 sum,&nbsp; the two values&nbsp; $0$&nbsp; and&nbsp; $1$&nbsp; have equal probability:
 :$${\rm Pr}(m_\nu = 0) = {\rm Pr}(m_\nu = 1)\hspace{0.15cm}\underline{=0.5}.$$
-'''(2)'''&nbsp; The table shows that for each preassignment &nbsp; &rArr; &nbsp; $( x_{\nu-1}, x_{\nu-2}) = (0,0), (0,1), (1,0), (1,1)$ &nbsp; the values&nbsp; $m_\nu = 0$&nbsp; respectively&nbsp; $m_\nu = 1$&nbsp; are equally likely.
+'''(2)'''&nbsp; The table shows that for each preassignment &nbsp; &rArr; &nbsp; $( x_{\nu-1}, x_{\nu-2}) = (0,0), (0,1), (1,0), (1,1)$ &nbsp; the values&nbsp; $m_\nu = 0$&nbsp; and&nbsp; $m_\nu = 1$&nbsp; resp. are equally likely.
 *Expressed differently: &nbsp; ${\rm Pr}(m_{\nu}\hspace{0.05cm}|\hspace{0.05cm}m_{\nu-1}) = {\rm Pr}( m_{\nu}).$
 *This exactly matches the definition of "statistical independence" &nbsp; &rArr; &nbsp; <u>Answer 1</u>.

@@ Line 3: / Line 3: @@
 }}
-[[File:EN_Sto_A_4_3.png|right|frame|Algebraic & modulo 2 sum]]
+[[File:EN_Sto_A_4_3.png|right|frame|Algebraic & modulo&ndash;2 sum]]
 A&nbsp; "clocked"&nbsp; random number generator returns a sequence&nbsp; $\langle x_\nu \rangle$&nbsp; of binary random numbers.
 *It is assumed that the binary numbers&nbsp; $0$&nbsp; and&nbsp; $1$&nbsp; occur with equal probabilities and that the individual random numbers do not depend on each other.
@@ Line 14: / Line 16: @@
 :$$a_\nu=x_\nu+x_{\nu-1}+x_{\nu-2},$$
-*the&nbsp; "modulo 2 sum"&nbsp; $m_\nu$:
+*the&nbsp; "modulo&ndash;2 sum"&nbsp; $m_\nu$:
 :$$m_\nu=x_\nu\oplus x_{\nu-1}\oplus x_{\nu-2}.$$
-Hints: &nbsp; This exercise belongs to the chapter&nbsp; [[Theory_of_Stochastic_Signals/Two-Dimensional_Random_Variables|Two-Dimensional Random Variables]].
+Hints: &nbsp;
+*This exercise belongs to the chapter&nbsp; [[Theory_of_Stochastic_Signals/Two-Dimensional_Random_Variables|Two-Dimensional Random Variables]].
+*Use the following table for moment calculation.
 <br clear=all>
 ===Questions===
@@ Line 43: / Line 39: @@
-{Determine the joint PDF&nbsp; $f_{xm}(x_\nu, m_\nu)$.&nbsp; Based on the result, evaluate the following statements (true or not).
+{Determine the 2D&ndash;PDF&nbsp; $f_{xm}(x_\nu, m_\nu)$.&nbsp; Based on the result,&nbsp; evaluate the following statements.
 |type="[]"}
 - The random variables&nbsp; $x_\nu$&nbsp; and&nbsp; $m_\nu$&nbsp; are statistically dependent.
@@ Line 57: / Line 53: @@
-{Determine the 2D PDF $f_{am}(a_\nu, m_\nu)$&nbsp; and the correlation coefficient&nbsp; $\rho_{am}$.&nbsp; Which of the following statements are true?
+{Determine the 2D&ndash;PDF&nbsp; $f_{am}(a_\nu, m_\nu)$&nbsp; and the correlation coefficient&nbsp; $\rho_{am}$.&nbsp; Which of the following statements are true?
 |type="[]"}
 + The random variables&nbsp; $a_\nu$&nbsp; and&nbsp; $m_\nu$&nbsp; are statistically dependent.

← Older revision		Revision as of 10:52, 12 January 2022
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−	[[Category:Theory of Stochastic Signals: Exercises\|^4.1 ~~Zweidimensionale Zufallsgrößen~~^]]	+	[[Category:Theory of Stochastic Signals: Exercises\|^4.1 Two-Dimensional Random Variables^]]

@@ Line 4: / Line 4: @@
 [[File:EN_Sto_A_4_3.png|right|frame|Algebraische Summe und Modulo-2-Summe]]
-Ein &bdquo;getakteter" Zufallsgenerator liefert eine Folge&nbsp; $\langle x_\nu \rangle$&nbsp; von bin&auml;ren Zufallszahlen.
+Ein "getakteter" Zufallsgenerator liefert eine Folge&nbsp; $\langle x_\nu \rangle$&nbsp; von bin&auml;ren Zufallszahlen.
 *Es wird vorausgesetzt, dass die Bin&auml;rzahlen&nbsp; $0$&nbsp; und&nbsp; $1$&nbsp; mit gleichen Wahrscheinlichkeiten auftreten und dass die einzelnen Zufallszahlen nicht voneinander abh&auml;ngen.
 *Die Zufallszahlen&nbsp; $ x_\nu \in \{0, 1\}$&nbsp; werden in die erste Speicherstelle eines Schieberegisters eingetragen und mit jeden Takt um eine Stelle nach unten verschoben.
@@ Line 71: / Line 71: @@
 '''(2)'''&nbsp; Die Tabelle zeigt, dass bei jeder Vorbelegung &nbsp; &rArr; &nbsp;  $( x_{\nu-1}, x_{\nu-2}) = (0,0), (0,1), (1,0), (1,1)$  &nbsp; die Werte&nbsp; $m_\nu = 0$&nbsp; bzw.&nbsp; $m_\nu = 1$&nbsp;  gleichwahrscheinlich sind.
 *Anders ausgedr&uuml;ckt: &nbsp; ${\rm Pr}(m_{\nu}\hspace{0.05cm}|\hspace{0.05cm}m_{\nu-1}) = {\rm Pr}( m_{\nu}).$
-*Dies entspricht genau der Definition der &bdquo;statistischen Unabh&auml;ngigkeit" &nbsp; &rArr; &nbsp; <u>Antwort 1</u>.
+*Dies entspricht genau der Definition der "statistischen Unabh&auml;ngigkeit" &nbsp; &rArr; &nbsp; <u>Antwort 1</u>.