Aufgaben:Exercise 2.2Z: Discrete Random Variables - Revision history

Hwang at 12:20, 18 January 2023

2023-01-18T12:20:05Z

Guenter at 13:54, 16 February 2022

2022-02-16T13:54:47Z

Guenter at 13:18, 7 December 2021

2021-12-07T13:18:53Z

Guenter at 16:58, 6 December 2021

2021-12-06T16:58:50Z

Noah at 20:33, 3 December 2021

2021-12-03T20:33:24Z

Noah at 20:33, 3 December 2021

2021-12-03T20:33:05Z

Noah at 20:32, 3 December 2021

2021-12-03T20:32:40Z

Noah at 20:25, 3 December 2021

2021-12-03T20:25:16Z

Noah at 20:25, 3 December 2021

2021-12-03T20:25:05Z

Noah at 20:20, 3 December 2021

2021-12-03T20:20:43Z

@@ Line 52: / Line 52: @@
-{What is the quadratic mean value&nbsp; (second order moment)&nbsp; $m_{2d}$&nbsp; of this random variable?
+{What is the second moment value&nbsp; (second order moment)&nbsp; $m_{2d}$&nbsp; of this random variable?
 |type="{}"}
 $m_{2d}\ = \ $ { 2.5 3% }

@@ Line 32: / Line 32: @@
 <quiz display=simple>
-{What is the rms value of the random variable&nbsp; $a$?
+{What is the standard deviation of the random variable&nbsp; $a$?
 |type="{}"}
 $\sigma_a \ = \ $ { 1 3% }
-{What is the rms value of the random variable&nbsp; $b$?&nbsp; Set&nbsp; $p = 0.25$.
+{What is the standard deviation of the random variable&nbsp; $b$?&nbsp; Set&nbsp; $p = 0.25$.
 |type="{}"}
 $\sigma_b \ = \ $ { 0.433 3% }
-{What is the rms value of the random variable&nbsp; $c$?
+{What is the standard deviation of the random variable&nbsp; $c$?
 |type="{}"}
 $\sigma_c \ = \ $ { 0.707 3% }
@@ Line 57: / Line 57: @@
-{What is the rms value&nbsp; $\sigma_d$?
+{What is the standard deviation&nbsp; $\sigma_d$?
 |type="{}"}
 $\sigma_d\ = \ $ { 1.5 3% }

@@ Line 52: / Line 52: @@
-{What is the root mean square value&nbsp; $m_{2d}$&nbsp; of this random variable?
+{What is the quadratic mean value&nbsp; (second order moment)&nbsp; $m_{2d}$&nbsp; of this random variable?
 |type="{}"}
 $m_{2d}\ = \ $ { 2.5 3% }
-{What is the dispersion&nbsp; $\sigma_d$?
+{What is the rms value&nbsp; $\sigma_d$?
 |type="{}"}
 $\sigma_d\ = \ $ { 1.5 3% }
@@ Line 71: / Line 71: @@
 *From this one obtains with Steiner's theorem:
 :$$\it\sigma_a^{\rm 2} = \rm\sqrt{1-0^2}=1 \hspace{0.5cm}or \hspace{0.5cm}\it\sigma_a\hspace{0.15cm} \underline{=\rm 1}.$$
@@ Line 77: / Line 76: @@
 :$$ m_{k}=(1-p)\cdot 0^{ k} + p\cdot 1^{k}= p.$$
-*From this follows with&nbsp; $p = 1/4$:
+*From this follows with&nbsp; $p = 1/4$&nbsp; and&nbsp; $k=2$:
 :$$m_{b}= m_{2b}= p, \hspace{0.5cm} \sigma_{\it b}=\sqrt{p\cdot (1- p)}\hspace{0.15cm} \underline{=\rm 0.433} .$$
@@ Line 86: / Line 84: @@
 :$$ m_{2\it c}= {1}/{4}\cdot(-1)^2+{1}/{2}\cdot 0^2+{1}/{4}\cdot (1)^2={1}/{2} \hspace{0.5cm}$$
 :$$\Rightarrow \hspace{0.5cm}\sigma_{\it c}=\rm \sqrt{1/2}\hspace{0.15cm} \underline{=0.707}.$$
@@ Line 93: / Line 90: @@
-'''(5)'''&nbsp; Analogous to the subtask&nbsp; '''(4)'''&nbsp; we obtain for the root mean square:
+'''(5)'''&nbsp; Analogous to the subtask&nbsp; '''(4)'''&nbsp; we obtain for the second order moment:
 :$$m_{2d}= {\rm E}\big[( a-2b+c)^{\rm 2}\big] =  {\rm E}\big[a^{\rm 2}\big]\hspace{0.1cm}+\hspace{0.1cm}4\hspace{0.05cm}\cdot\hspace{0.05cm} {\rm E}\big[ b^{\rm 2}\big]\hspace{0.1cm}+\hspace{0.1cm} {\rm E}\big[c^{\rm 2}\big]\hspace{0.1cm}  -  \hspace{0.1cm}4\hspace{0.05cm}\cdot\hspace{0.05cm} {\rm E}\big[a\hspace{0.05cm}\cdot \hspace{0.05cm}b\big]\hspace{0.1cm}+\hspace{0.1cm} 2\hspace{0.05cm}\cdot\hspace{0.05cm}{\rm E}\big[ a\hspace{0.05cm}\cdot \hspace{0.05cm}c\big]\hspace{0.1cm}-\hspace{0.1cm} 4\hspace{0.05cm}\cdot\hspace{0.05cm}{\rm E}\big[ b\hspace{0.05cm}\cdot \hspace{0.05cm}c\big].$$
 *But since&nbsp; $a$&nbsp; and&nbsp; $b$&nbsp; are statistically independent of each other,&nbsp; also holds:
@@ Line 102: / Line 98: @@
 *The same holds for the other mixed terms.&nbsp; Therefore, using&nbsp; $p = 0.25$, we obtain:
 :$$ m_{2 d}=m_{2 a}+4\cdot m_{ 2 b}+m_{ 2 c}=1+4\cdot p+0.5\hspace{0.15cm} \underline{=\rm 2.5}.$$

@@ Line 5: / Line 5: @@
 [[File:P_ID84__Sto_Z_2_2.png|right|frame|Different rectangular signals]]
-Let be given three discrete random variables&nbsp; $a$,&nbsp; $b$&nbsp; and&nbsp; $c$,&nbsp; which are defined as the current values of the represented signals.&nbsp; These have the following properties:
+Let be given three discrete random variables&nbsp; $a$,&nbsp; $b$&nbsp; and&nbsp; $c$,&nbsp; which are defined as the instantaneous  values of the represented signals.&nbsp; These have the following properties:
-*The random variable&nbsp; $a$&nbsp; can take the values&nbsp; $+1$&nbsp; and&nbsp; $-1$&nbsp; with equal probability.
+*The random variable&nbsp; $a$&nbsp; can take the two values&nbsp; $+1$&nbsp; and&nbsp; $-1$&nbsp; with equal probability.
-*The random variable&nbsp; $b$&nbsp; is also two-point distributed, but with&nbsp; ${\rm Pr}(b = 1) = p$ &nbsp;and&nbsp; ${\rm Pr}(b = 0) = 1 - p$.
+*The random variable&nbsp; $b$&nbsp; is also two-point distributed,&nbsp;  but with&nbsp; ${\rm Pr}(b = 1) = p$ &nbsp;and&nbsp; ${\rm Pr}(b = 0) = 1 - p$.
-*The probabilities of&nbsp; $c$&nbsp; be&nbsp; ${\rm Pr}(c = 0) = 1/2$ &nbsp;and&nbsp; ${\rm Pr}(c = +1) = Pr(c = -1) =1/4$.
+*The probabilities of the random variable&nbsp; $c$&nbsp; be&nbsp; ${\rm Pr}(c = 0) = 1/2$ &nbsp;and&nbsp; ${\rm Pr}(c = +1) = Pr(c = -1) =1/4$.
-*There are no statistical dependencies between the three random variables&nbsp; $a$,&nbsp; $b$&nbsp; and&nbsp; $c$&nbsp; .
+*There are no statistical dependencies between the three random variables&nbsp; $a$,&nbsp; $b$&nbsp; and&nbsp; $c$.
-*Another random variable&nbsp; $d=a$,&nbsp; $b$&nbsp; and&nbsp; $c$&nbsp; is formed from the random variables&nbsp; $d=a-2 b+c$&nbsp;.
+*Another random variable&nbsp; $d$&nbsp; is formed from the random variables&nbsp; $a$,&nbsp; $b$&nbsp; and&nbsp; $c$:
 The graph shows sections of these random variables.&nbsp; It can be seen that&nbsp; $d$&nbsp; can take all integer values between&nbsp; $-4$&nbsp; and&nbsp; $+2$&nbsp;.
@@ Line 24: / Line 24: @@
 Hints:
 *The exercise belongs to the chapter&nbsp; [[Theory_of_Stochastic_Signals/Momente_einer_diskreten_Zufallsgröße|Moments of a Discrete Random Variable]].
+*The topic of this chapter is illustrated with examples in the&nbsp;  (German language)&nbsp;  learning video<br> &nbsp; &nbsp; [[Momentenberechnung_bei_diskreten_Zufallsgrößen_(Lernvideo)|"Momentenberechnung bei diskreten Zufallsgrößen"]] &nbsp; &rArr; &nbsp; "Calculating Moments for Discrete-Valued Random Variables"
@@ Line 33: / Line 32: @@
 <quiz display=simple>
-{What is the dispersion (standard deviation) of the random variable $a$?
+{What is the rms value of the random variable&nbsp; $a$?
 |type="{}"}
 $\sigma_a \ = \ $ { 1 3% }
-{What is the dispersion of the random variable&nbsp; $b$?&nbsp; Set&nbsp; $p = 0.25$.
+{What is the rms value of the random variable&nbsp; $b$?&nbsp; Set&nbsp; $p = 0.25$.
 |type="{}"}
 $\sigma_b \ = \ $ { 0.433 3% }
-{What is the spread of the random variable $c$?
+{What is the rms value of the random variable&nbsp; $c$?
 |type="{}"}
 $\sigma_c \ = \ $ { 0.707 3% }

← Older revision		Revision as of 20:33, 3 December 2021
Line 1:		Line 1:

−	{{quiz-Header\|Buchseite=~~Stochastische Signaltheorie~~/~~Momente einer diskreten Zufallsgröße~~	+	{{quiz-Header\|Buchseite=Theory_of_Stochastic_Signals/Moments_of_a_Discrete_Random_Variable
	}}		}}

@@ Line 109: / Line 109: @@
 :$$\sigma_{\it d}^{\rm 2}=1.5+4\cdot p - 4 \cdot p^{\rm 2}=2.25 \hspace{0.5cm}\Rightarrow \hspace{0.5cm}  \sigma_{d}\hspace{0.15cm} \underline{=\rm 1.5}.$$
-*The maximum variance for&nbsp; $p = 0.50$&nbsp;adds up to&nbsp; $\sigma_{\it d}^{\rm 2}=2.50$.
+*The maximum variance for&nbsp; $p = 0.50$&nbsp;results in&nbsp; $\sigma_{\it d}^{\rm 2}=2.50$.
 {{ML-Fuß}}

@@ Line 67: / Line 67: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; Aufgrund der Symmetrie gilt:
+'''(1)'''&nbsp; Due to the symmetry holds:
 :$$\rm \it m_{\it a}=\rm 0; \hspace{0.5cm}\it m_{\rm 2\it a}=\rm 0.5\cdot (-1)^2 + 0.5\cdot (1)^2{ = 1}.$$
-*Daraus erh&auml;lt man mit dem Satz von Steiner:
+*From this one obtains with Steiner's theorem:
-:$$\it\sigma_a^{\rm 2} = \rm\sqrt{1-0^2}=1 \hspace{0.5cm}bzw. \hspace{0.5cm}\it\sigma_a\hspace{0.15cm} \underline{=\rm 1}.$$
+:$$\it\sigma_a^{\rm 2} = \rm\sqrt{1-0^2}=1 \hspace{0.5cm}or \hspace{0.5cm}\it\sigma_a\hspace{0.15cm} \underline{=\rm 1}.$$
-'''(2)'''&nbsp; Allgemein gilt f&uuml;r das Moment&nbsp; $k$&ndash;ter Ordnung:
+'''(2)'''&nbsp; In general, for the&nbsp; $k$&ndash;th order moment:
 :$$ m_{k}=(1-p)\cdot 0^{ k} + p\cdot 1^{k}= p.$$
-*Daraus folgt mit&nbsp; $p = 1/4$:
+*From this follows with&nbsp; $p = 1/4$:
 :$$m_{b}= m_{2b}= p, \hspace{0.5cm} \sigma_{\it b}=\sqrt{p\cdot (1- p)}\hspace{0.15cm} \underline{=\rm 0.433} .$$
-'''(3)'''&nbsp; F&uuml;r die Zufallsgr&ouml;&szlig;e&nbsp; $c$&nbsp; gilt:
+'''(3)'''&nbsp; For the random variable&nbsp; $c$&nbsp; holds:
-:$$m_{\it c} =  0\hspace{0.3cm} ({\rm symmetrisch\hspace{0.1cm}um\hspace{0.1cm}0)},$$
+:$$m_{\it c} = 0\hspace{0.3cm} ({\rm symmetric\hspace{0.1cm}um\hspace{0.1cm}0)},$$
 :$$ m_{2\it c}= {1}/{4}\cdot(-1)^2+{1}/{2}\cdot 0^2+{1}/{4}\cdot (1)^2={1}/{2} \hspace{0.5cm}$$
 :$$\Rightarrow \hspace{0.5cm}\sigma_{\it c}=\rm \sqrt{1/2}\hspace{0.15cm} \underline{=0.707}.$$
@@ Line 90: / Line 90: @@
-'''(4)'''&nbsp; Nach den allgemeinen Regeln f&uuml;r Erwartungswerte gilt mit&nbsp; $p = 0.25$:
+'''(4)'''&nbsp; According to the general rules for expected values, with&nbsp; $p = 0.25$:
 :$$m_{\it d} = {\rm E}\big[a-2 b+c\big]= {\rm E}\big[a\big] \hspace{0.1cm} -\hspace{0.1cm}\rm 2 \hspace{0.05cm}\cdot\hspace{0.05cm} {\rm E}\big[ b\big]\hspace{0.1cm}+\hspace{0.1cm} {\rm E}\big[ c\big] =  m_{ a}\hspace{0.1cm}-\hspace{0.1cm}2\hspace{0.05cm}\cdot\hspace{0.05cm} m_{\it b}\hspace{0.1cm}+\hspace{0.1cm} m_{\it c} =    0-2\hspace{0.05cm}\cdot\hspace{0.05cm} p + 0 \hspace{0.15cm} \underline{= -0.5}.$$
-*Da aber&nbsp; $a$&nbsp; und&nbsp; $b$&nbsp; statistisch voneinander unabh&auml;ngig sind,&nbsp; gilt auch:
+*But since&nbsp; $a$&nbsp; and&nbsp; $b$&nbsp; are statistically independent of each other,&nbsp; also holds:
-:$${\rm E}\big[a\cdot b\big] = {\rm E}\big[ a\big] \cdot {\rm E}\big[ b\big]=  m_{ a}\cdot  m_{ b} = 0, \hspace{0.2cm} {\rm da}\hspace{0.2cm}  m_{ a}=\rm 0.$$
+:$${\rm E}\big[a\cdot b\big] = {\rm E}\big[ a\big] \cdot {\rm E}\big[ b\big]= m_{ a}\cdot m_{ b} = 0, \hspace{0.2cm} {\rm da}\hspace{0.2cm} m_{ a}=\rm 0.$$
-*Gleiches gilt f&uuml;r die anderen gemischten Terme.&nbsp; Daher erh&auml;lt man mit&nbsp; $p = 0.25$:
+*The same holds for the other mixed terms.&nbsp; Therefore, using&nbsp; $p = 0.25$, we obtain:
-:$$ m_{2 d}=m_{2 a}+4\cdot m_{ 2 b}+m_{ 2 c}=1+4\cdot  p+0.5\hspace{0.15cm} \underline{=\rm 2.5}.$$
+:$$ m_{2 d}=m_{2 a}+4\cdot m_{ 2 b}+m_{ 2 c}=1+4\cdot p+0.5\hspace{0.15cm} \underline{=\rm 2.5}.$$
-'''(6)'''&nbsp; Für allgemeines&nbsp; $p$ &nbsp;bzw.&nbsp; f&uuml;r&nbsp; $p = 0.25$&nbsp; ergibt sich:
+'''(6)'''&nbsp; For general&nbsp; $p$ &nbsp;resp.&nbsp; for&nbsp; $p = 0.25$&nbsp; results:
 :$$\sigma_{\it d}^{\rm 2}=1.5+4\cdot p - 4 \cdot p^{\rm 2}=2.25 \hspace{0.5cm}\Rightarrow \hspace{0.5cm}  \sigma_{d}\hspace{0.15cm} \underline{=\rm 1.5}.$$
-*Die maximale Varianz erg&auml;be sich f&uuml;r&nbsp; $p = 0.50$&nbsp; &nbsp;zu&nbsp; $\sigma_{\it d}^{\rm 2}=2.50$.
+*The maximum variance for&nbsp; $p = 0.50$&nbsp;adds up to&nbsp; $\sigma_{\it d}^{\rm 2}=2.50$.
 {{ML-Fuß}}

← Older revision		Revision as of 20:25, 3 December 2021
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	*The exercise belongs to the chapter  [[Theory_of_Stochastic_Signals/Momente_einer_diskreten_Zufallsgröße\|Moments of a Discrete Random Variable]].		*The exercise belongs to the chapter  [[Theory_of_Stochastic_Signals/Momente_einer_diskreten_Zufallsgröße\|Moments of a Discrete Random Variable]].

−	**The topic of this chapter is illustrated with examples in the  (German language)  learning video<br> [[Momentenberechnung_bei_diskreten_Zufallsgrößen_(Lernvideo)\|Momentenberechnung bei diskreten Zufallsgrößen]]  $\Rightarrow$   Calculating Moments for Discrete-Valued Random Variables	+	*The topic of this chapter is illustrated with examples in the  (German language)  learning video<br> [[Momentenberechnung_bei_diskreten_Zufallsgrößen_(Lernvideo)\|Momentenberechnung bei diskreten Zufallsgrößen]]  $\Rightarrow$   Calculating Moments for Discrete-Valued Random Variables

← Older revision	Revision as of 20:20, 3 December 2021
Line 16:	Line 16:


	+	[[File:P_ID84__Sto_Z_2_2.png\|right\|frame\|Discrete rectangular signals]]
	+	Let be given three discrete random variables  $a$,  $b$  and  $c$,  which are defined as the current values of the represented signals.  These have the following properties:
	+
	+	*The random variable  $a$  can take the values  $+1$  and  $-1$  with equal probability.
	+	*The random variable  $b$  is also two-point distributed, but with  ${\rm Pr}(b = 1) = p$  and  ${\rm Pr}(b = 0) = 1 - p$.
	+	*The probabilities of  $c$  be  ${\rm Pr}(c = 0) = 1/2$  and  ${\rm Pr}(c = +1) = Pr(c = -1) =1/4$.
	+	*There are no statistical dependencies between the three random variables  $a$,  $b$  and  $c$  .
	+	*Another random variable  $d=a$,  $b$  and  $c$  is formed from the random variables  $d=a-2 b+c$ .
	+
	+
	+	The graph shows sections of these four random variables.  It can be seen that  $d$  can take all integer values between  $-4$  and  $+2$ .