Aufgaben:Exercise 4.3Z: Dirac-shaped "2D-PDF" - Revision history

Guenter at 15:41, 10 April 2022

2022-04-10T15:41:10Z

Guenter at 16:21, 31 March 2022

2022-03-31T16:21:52Z

Guenter at 15:20, 7 February 2022

2022-02-07T15:20:24Z

Guenter at 15:06, 7 February 2022

2022-02-07T15:06:21Z

Guenter: Guenter moved page Aufgaben:Exercise 4.3Z: Dirac-shaped 2D PDF to Aufgaben:Exercise 4.3Z: Dirac-shaped "2D-PDF"

2022-01-24T09:05:12Z

Guenter moved page Exercise 4.3Z: Dirac-shaped 2D PDF to Exercise 4.3Z: Dirac-shaped "2D-PDF"

Noah at 14:26, 19 January 2022

2022-01-19T14:26:05Z

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Guenter at 10:52, 12 January 2022

2022-01-12T10:52:31Z

Guenter: Guenter moved page Aufgaben:Aufgabe 4.3Z: Diracförmige 2D-WDF to Aufgaben:Exercise 4.3Z: Dirac-shaped 2D PDF

2022-01-12T10:49:14Z

Guenter moved page Aufgabe 4.3Z: Diracförmige 2D-WDF to Exercise 4.3Z: Dirac-shaped 2D PDF

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

2021-03-23T12:43:41Z

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

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

2020-07-09T09:51:00Z

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

@@ Line 111: / Line 111: @@
-[[File:EN_Sto_Z_4_3_f.png|right|frame|2D&ndash;PDF and correlation line&nbsp; $y = K(x)$]]
+[[File:EN_Sto_Z_4_3_f.png|right|frame|2D&ndash;PDF and regression line&nbsp; $\rm (RL)$]]
 '''(6)'''&nbsp; For the correlation coefficient:
 :$$\rho_{xy} = \frac{\mu_{xy}}{\sigma_x\cdot \sigma_y} = \frac{1.2}{\sqrt{2}\cdot\sqrt{1.4}}\hspace{0.15cm}\underline{=0.717}.$$
@@ Line 120: / Line 120: @@
 :$$y=\frac{\sigma_y}{\sigma_x}\cdot \rho_{xy}\cdot x = \frac{\mu_{xy}}{\sigma_x^{\rm 2}}\cdot x = \rm 0.6\cdot \it x.$$
-*See sketch on the right.&nbsp; The angle between the correlation straight line&nbsp; $K(x)$&nbsp;  and the&nbsp; $x$-axis is
+*See sketch on the right.&nbsp; The angle between the regression line&nbsp; $\rm (RL)$&nbsp;  and the&nbsp; $x$-axis is
 :$$\theta_{y\hspace{0.05cm}→\hspace{0.05cm} x} = \arctan(0.6) \hspace{0.15cm}\underline{=31^\circ}.$$

@@ Line 111: / Line 111: @@
-[[File:P_ID259__Sto_Z_4_3_f.png|right|frame|2D&ndash;PDF and correlation line&nbsp; $y = K(x)$]]
+[[File:EN_Sto_Z_4_3_f.png|right|frame|2D&ndash;PDF and correlation line&nbsp; $y = K(x)$]]
 '''(6)'''&nbsp; For the correlation coefficient:
 :$$\rho_{xy} = \frac{\mu_{xy}}{\sigma_x\cdot \sigma_y} = \frac{1.2}{\sqrt{2}\cdot\sqrt{1.4}}\hspace{0.15cm}\underline{=0.717}.$$

@@ Line 72: / Line 72: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; Correct are the <u>first two answers</u>:
+'''(1)'''&nbsp; Correct are the&nbsp; <u>first two answers</u>:
-*The marginal probability density function $f_{x}(x)$&nbsp; is obtained from the 2D PDF $f_{xy}(x, y)$&nbsp; by integration over&nbsp; $y$.
+*The marginal probability density function&nbsp; $f_{x}(x)$&nbsp; is obtained from the 2D&ndash;PDF&nbsp; $f_{xy}(x, y)$&nbsp; by integration over&nbsp; $y$.
 *For all possible values&nbsp; $ x \in \{-2, -1, \ 0, +1, +2\}$&nbsp; the probabilities are equal&nbsp; $0.2$.
-*It holds&nbsp; ${\rm Pr}(x \le 1)= 0.8$&nbsp; and the mean is&nbsp; $m_x = 0$.
+*It holds&nbsp; ${\rm Pr}(x \le 1)= 0.8$.&nbsp; The mean is&nbsp; $m_x = 0$.
-[[File:P_ID258__Sto_Z_4_3_b.png|right|frame|Discrete marginal PDF $f_{y}(y)$]]
+[[File:P_ID258__Sto_Z_4_3_b.png|right|frame|Discrete marginal PDF&nbsp; $f_{y}(y)$]]
-'''(2)'''&nbsp; Correct are <u>the proposed solutions 2 and 3</u>:
+'''(2)'''&nbsp; Correct are&nbsp; <u>the proposed solutions 2 and 3</u>:
-*By integration over&nbsp; $x$&nbsp; one obtains the PDF $f_{y}(y)$ sketched on the right.
+*By integration over&nbsp; $x$&nbsp; one obtains the PDF&nbsp; $f_{y}(y)$ sketched on the right.
-*Due to symmetry, the mean value&nbsp; $m_y = 0$ is obtained.
+*Due to symmetry,&nbsp; the mean value&nbsp; $m_y = 0$&nbsp; is obtained.
 *The probability we are looking for is&nbsp; ${\rm Pr}(y \le 1)= 0.9$.
@@ Line 92: / Line 92: @@
 *For&nbsp; $r_x = r_y = 1$&nbsp; it follows:
 :$$F_{xy}(+1, +1) = {\rm Pr}\big [(x \le 1)\cap(y\le 1)\big ].$$
-*As can be seen from the 2D PDF on the details page, this probability is&nbsp; ${\rm Pr}\big [(x \le 1)\cap(y\le 1)\big ]\hspace{0.15cm}\underline{=0.8}$.
+*As can be seen from the 2D&ndash;PDF on the information page,&nbsp; this probability is&nbsp; ${\rm Pr}\big [(x \le 1)\cap(y\le 1)\big ]\hspace{0.15cm}\underline{=0.8}$.
-'''(4)'''&nbsp; For this, Bayes' theorem can also be used to write:
+'''(4)'''&nbsp; For this,&nbsp; Bayes' theorem can also be used to write:
 :$$ \rm Pr(\it x \le \rm 1)\hspace{0.05cm} | \hspace{0.05cm} \it y \le \rm 1) = \frac{ \rm Pr\big [(\it x \le \rm 1)\cap(\it y\le \rm 1)\big ]}{ \rm Pr(\it y\le \rm 1)} = \it \frac{F_{xy} \rm (1, \rm 1)}{F_{y}\rm (1)}.$$
@@ Line 106: / Line 106: @@
 :$$m_{xy} = {\rm E}\big[x\cdot y \big] = \sum\limits_{i} {\rm Pr}( x_i \cap y_i)\cdot x_i\cdot y_i. $$
-*There remain five Dirac functions with&nbsp; $x_i \cdot y_i \ne 0$:
+*There remain five Dirac delta functions with&nbsp; $x_i \cdot y_i \ne 0$:
 :$$m_{xy} = \rm 0.1\cdot (-2) (-1) + 0.2\cdot(-1) (-1)+ 0.2\cdot 1\cdot 1 + 0.1\cdot 2\cdot 1+ 0.1\cdot 2\cdot 2\hspace{0.15cm}\underline{=\rm 1.2}.$$
-[[File:P_ID259__Sto_Z_4_3_f.png|right|frame|2D PDF and correlation line&nbsp; $y = K(x)$]]
+[[File:P_ID259__Sto_Z_4_3_f.png|right|frame|2D&ndash;PDF and correlation line&nbsp; $y = K(x)$]]
 '''(6)'''&nbsp; For the correlation coefficient:
 :$$\rho_{xy} = \frac{\mu_{xy}}{\sigma_x\cdot \sigma_y} = \frac{1.2}{\sqrt{2}\cdot\sqrt{1.4}}\hspace{0.15cm}\underline{=0.717}.$$
@@ Line 120: / Line 120: @@
 :$$y=\frac{\sigma_y}{\sigma_x}\cdot \rho_{xy}\cdot x = \frac{\mu_{xy}}{\sigma_x^{\rm 2}}\cdot x = \rm 0.6\cdot \it x.$$
-*In the picture the straight line&nbsp; $y = K(x)$&nbsp; is drawn.&nbsp; The angle between correlation straight line and&nbsp; $x$-axis amounts to.
+*See sketch on the right.&nbsp; The angle between the correlation straight line&nbsp; $K(x)$&nbsp;  and the&nbsp; $x$-axis is
 :$$\theta_{y\hspace{0.05cm}→\hspace{0.05cm} x} = \arctan(0.6) \hspace{0.15cm}\underline{=31^\circ}.$$
-'''(7)'''&nbsp; The correct solutions are <u>solutions 2 and 3</u>:
+'''(7)'''&nbsp; The correct solutions are&nbsp; <u>solutions 2 and 3</u>:
-*If statistically independent, $f_{xy}(x, y) = f_{x}(x) \cdot f_{y}(y)$&nbsp; should hold, which is not done here.
+*If statistically independent,&nbsp; $f_{xy}(x, y) = f_{x}(x) \cdot f_{y}(y)$&nbsp; should hold,&nbsp; which is not done here.
 *From correlatedness&nbsp; $($follows from&nbsp; $\rho_{xy} \ne 0)$&nbsp; it is possible to directly infer statistical dependence,
-*because correlation means a special form of statistical dependence,
+*because correlation means a special form of statistical dependence,&nbsp;  namely linear statistical dependence.
 {{ML-Fuß}}

@@ Line 3: / Line 3: @@
 }}
-[[File:P_ID257__Sto_Z_4_3.png|right|frame|Dirac-shaped 2D PDF]]
+[[File:P_ID257__Sto_Z_4_3.png|right|frame|Dirac-shaped 2D&ndash; PDF]]
-The graph shows the two-dimensional probability density function $f_{xy}(x, y)$&nbsp; of the two discrete random variables&nbsp; $x$&nbsp; and&nbsp; $y$&nbsp;.
+The graph shows the two-dimensional probability density function $f_{xy}(x, y)$&nbsp; of two discrete random variables&nbsp; $x$,&nbsp;  $y$.
-*This 2D PDF consists of eight Dirac points, marked by crosses.&nbsp; The numerical values indicate the corresponding probabilities.
+*This 2D&ndash;PDF consists of eight Dirac points,&nbsp;  marked by crosses.&nbsp;
-*It can be seen that both&nbsp; $x$&nbsp; and&nbsp; $y$&nbsp; can take all integer values between the limits&nbsp; $-2$&nbsp; and&nbsp; $+2$&nbsp; .
+*The numerical values indicate the corresponding probabilities.
 *The variances of the two random variables are given as follows:  &nbsp; $\sigma_x^2 = 2$, &nbsp; $\sigma_y^2 = 1.4$.
@@ Line 18: / Line 16: @@
 Hints:
 *The exercise belongs to the chapter&nbsp; [[Theory_of_Stochastic_Signals/Two-Dimensional_Random_Variables|Two-Dimensional Random Variables]].
-*Reference is made to the chapter&nbsp; [[Theory_of_Stochastic_Signals/Moments_of_a_Discrete_Random_Variable|Moments of a Discrete Random Variable]]
+*Reference is also made to the chapter&nbsp; [[Theory_of_Stochastic_Signals/Moments_of_a_Discrete_Random_Variable|Moments of a Discrete Random Variable]]
@@ Line 28: / Line 26: @@
 <quiz display=simple>
-{Which of the following statements are true regarding the random variable $x$&nbsp;?
+{Which of the following statements are true regarding the random variable&nbsp; $x$?
 |type="[]"}
 + The probabilities for&nbsp; $-2$,&nbsp; $-1$,&nbsp; &nbsp; $0$,&nbsp; $+1$&nbsp; and&nbsp; $+2$&nbsp; are equal.
 + The random variable&nbsp; $x$&nbsp; is mean-free&nbsp; $(m_x = 0)$.
-- The probability&nbsp; ${\rm Pr}(x \le 1)$&nbsp; is&nbsp; $0.9$.
+- The probability&nbsp; ${\rm Pr}(x \le 1)=0.9$.
-{Which of the following statements are true with respect to the random variable&nbsp; $y$&nbsp;?
+{Which of the following statements are true with respect to the random variable&nbsp; $y$?
 |type="[]"}
 - The probabilities for&nbsp; $-2$,&nbsp; $-1$,&nbsp; &nbsp; $0$,&nbsp; $+1$&nbsp; and&nbsp; $+2$&nbsp; are equal.
 + The random variable&nbsp; $y$&nbsp; is mean-free&nbsp; $(m_y = 0)$.
-+ The probability&nbsp; ${\rm Pr}(y \le 1)$&nbsp; is&nbsp; $0.9$.
++ The probability&nbsp; ${\rm Pr}(y \le 1)=0.9$.
-{Calculate the value of the two-dimensional CDF at location&nbsp; $(+1, +1)$.
+{Calculate the value of the two-dimensional cumulative distribution function&nbsp; $\rm (CDF)$&nbsp; at location&nbsp; $(+1, +1)$.
 |type="{}"}
 $F_{xy}(+1, +1) \ = \ $ { 0.8 3% }
-{Calculate the probability that&nbsp; $x \le 1$&nbsp; holds, conditioned on&nbsp; $y \le 1$&nbsp; simultaneously.
+{Calculate the probability that&nbsp; $x \le 1$&nbsp; holds,&nbsp; conditioned on&nbsp; $y \le 1$&nbsp; simultaneously.
 |type="{}"}
 ${\rm Pr}(x ≤ 1\hspace{0.05cm} | \hspace{0.05cm}y ≤ 1)\ = \ $ { 0.889 3% }
@@ Line 56: / Line 54: @@
-{Calculate the correlation coefficient&nbsp; $\rho_{xy}$&nbsp; and give the equation of the correlation line&nbsp; $K(x)$&nbsp; What is its angle to&nbsp; $x$ axis?
+{Calculate the correlation coefficient&nbsp; $\rho_{xy}$.&nbsp; Give the equation of the correlation line&nbsp; $K(x)$&nbsp; What is its angle to the&nbsp; $x$&ndash;axis?
 |type="{}"}
 $\rho_{xy}\ = \ $ { 0.707 3% }
@@ Line 65: / Line 63: @@
 |type="[]"}
 - The random variables&nbsp; $x$&nbsp; and&nbsp; $y$&nbsp; are statistically independent.
-+ It can already be seen from the given 2D PDF that&nbsp; $x$&nbsp; and&nbsp; $y$&nbsp; are statistically dependent on each other.
++ It can already be seen from the given 2D&ndash;PDF that&nbsp; $x$&nbsp; and&nbsp; $y$&nbsp; are statistically dependent on each other.
 + From the calculated correlation coefficient&nbsp; $\rho_{xy}$&nbsp; one can conclude the statistical dependence between&nbsp; $x$&nbsp; and&nbsp; $y$&nbsp;.

← 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^]]

← Older revision	Revision as of 09:05, 24 January 2022
(No difference)

← Older revision		Revision as of 12:43, 23 March 2021
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−	[[Category:~~Aufgaben zu Stochastische Signaltheorie~~\|^4.1 Zweidimensionale Zufallsgrößen^]]	+	[[Category:Theory of Stochastic Signals: Exercises\|^4.1 Zweidimensionale Zufallsgrößen^]]

@@ Line 17: / Line 17: @@
 ''Hinweise:''
-*Die Aufgabe gehört zum  Kapitel&nbsp; [[Stochastische_Signaltheorie/Zweidimensionale_Zufallsgrößen|Zweidimensionale Zufallsgrößen]].
+*Die Aufgabe gehört zum  Kapitel&nbsp; [[Theory_of_Stochastic_Signals/Zweidimensionale_Zufallsgrößen|Zweidimensionale Zufallsgrößen]].
-*Bezug genommen wird auch auf das Kapitel&nbsp; [[Stochastische_Signaltheorie/Momente_einer_diskreten_Zufallsgröße|Momente einer diskreten Zufallsgröße]]
+*Bezug genommen wird auch auf das Kapitel&nbsp; [[Theory_of_Stochastic_Signals/Momente_einer_diskreten_Zufallsgröße|Momente einer diskreten Zufallsgröße]]