Aufgaben:Exercise 3.2: CDF for Exercise 3.1 - Revision history

Guenter at 14:18, 4 January 2022

2022-01-04T14:18:25Z

Guenter at 13:58, 4 January 2022

2022-01-04T13:58:25Z

Guenter at 13:57, 4 January 2022

2022-01-04T13:57:50Z

Guenter: Guenter moved page Aufgaben:Exercise 3.2: cos²-CDF and CDF with Step Functions to Aufgaben:Exercise 3.2: CDF for Exercise 3.1

2022-01-04T13:41:14Z

Guenter moved page Exercise 3.2: cos²-CDF and CDF with Step Functions to Exercise 3.2: CDF for Exercise 3.1

Guenter at 13:40, 4 January 2022

2022-01-04T13:40:46Z

Noah at 19:59, 26 December 2021

2021-12-26T19:59:25Z

Noah at 13:23, 26 December 2021

2021-12-26T13:23:55Z

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Guenter at 10:04, 20 December 2021

2021-12-20T10:04:14Z

Guenter: Guenter moved page Aufgaben:Exercise 3.2: cos² CDF and CDF with Step Functions to Aufgaben:Exercise 3.2: cos²-CDF and CDF with Step Functions

2021-12-20T09:15:48Z

Guenter moved page Exercise 3.2: cos² CDF and CDF with Step Functions to Exercise 3.2: cos²-CDF and CDF with Step Functions

Guenter: Guenter moved page Aufgaben:Aufgabe 3.2: cos²- und Dirac-VTF to Aufgaben:Exercise 3.2: cos² CDF and CDF with Step Functions

2021-12-20T09:14:38Z

Guenter moved page Aufgabe 3.2: cos²- und Dirac-VTF to Exercise 3.2: cos² CDF and CDF with Step Functions

@@ Line 75: / Line 75: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; Since&nbsp; $x$&nbsp; is a continuous random variable and limited to the range&nbsp; $|\hspace{0.05cm}x\hspace{0.05cm}< 2|$&nbsp; , <u>all three given statements</u> are correct.
+'''(1)'''&nbsp; Since the continuous valued random variable&nbsp; $x$&nbsp; is limited to the range&nbsp; $|\hspace{0.05cm}x\hspace{0.05cm}< 2|$,&nbsp;  <u>all three given statements</u>&nbsp; are correct.
-'''(2)'''&nbsp; Only <u>statements 2 and 3</u> are correct here:
+'''(2)'''&nbsp; Only&nbsp; <u>statements 2 and 3</u>&nbsp; are correct here:
-*For a discrete random variable, the distribution function increases only weakly monotonically.
+*For a discrete valued random variable,&nbsp; the cumulative distribution function increases only weakly monotonically.
-* That means: Except for unit steps, there are only horizontal sections of the CDF.
+* That means:&nbsp; Except for unit steps,&nbsp; there are only horizontal sections of the CDF.
-*Since at the unit step points the right-hand side limit value is valid, $F_y(-2) = 0.1$, i.e. not equal to zero.
+*Since at the unit step points the right-hand side limit value &nbsp; &rArr; &nbsp; $F_y(-2) = 0.1$,&nbsp; i.e. not equal to zero.
@@ Line 88: / Line 88: @@
 '''(3)'''&nbsp; The CDF&nbsp; $F_x(r)$&nbsp; is calculated as the integral from&nbsp; $-\infty$&nbsp; to&nbsp; $r$&nbsp; over the PDF&nbsp; $f_x(x)$.
-Due to symmetry, herefore can be written in the range&nbsp; $0 \le r \le +2$&nbsp; :
+*Due to symmetry,&nbsp; herefore can be written in the range&nbsp; $0 \le r \le +2$:
 :$$F_{x} (r) =\frac{1}{2} + \int_{0}^{r} f_x(x)\;{\rm d}x = \frac{1}{2} + \int_{0}^{ r} {1}/{2}\cdot \cos^2 ({\pi}/{4}\cdot x)\;{\rm d}x.$$
-In the same way as for the subtask&nbsp; '''(7)'''&nbsp; of Exercise 3.1, we thus obtain:
+*In the same way as for the subtask&nbsp; '''(7)'''&nbsp; of Exercise 3.1,&nbsp; we thus obtain:
 :$$F_{x} (r) =\frac{1}{2} + \frac{ r}{ 4} + \frac{1}{2 \pi}   \cdot \sin({\pi}/{2}\cdot r),$$
 :$$F_{x} (r=0) =\rm \frac{1}{2} + \rm \frac{1}{2 \pi}   \cdot\rm sin(\rm 0)\hspace{0.15cm}{= 0.500},$$
@@ Line 98: / Line 98: @@
-'''(4)'''&nbsp; Because of the point symmetry around&nbsp; $r=0$&nbsp; &nbsp;resp.&nbsp; $F_{x} (0) = 1/2$&nbsp; and because of&nbsp; $\sin(-x) = -\sin(x)$&nbsp; this formula holds in the whole domain, as the following control calculation shows:
+'''(4)'''&nbsp; Because of the point symmetry around&nbsp; $r=0$&nbsp; &nbsp;resp.&nbsp; $F_{x} (0) = 1/2$&nbsp; and because of&nbsp; $\sin(-x) = -\sin(x)$&nbsp; this formula holds in the whole domain,&nbsp; as the following control calculation shows:
 :$$F_{x} (r=-2) =\rm \frac{1}{2} - \frac{\rm1}{\rm 2} - \rm \frac{1}{2 \pi}   \cdot\rm sin(\pi)=0,$$
 :$$F_{x} (r=-1) =\rm \frac{1}{2} - \frac{\rm1}{\rm 4} - \rm \frac{1}{2 \pi}   \cdot\rm sin({\pi}/{2})\hspace{0.15cm}\underline{= 0.091}.$$
-'''(5)'''&nbsp; For the probability that&nbsp; $x$&nbsp; lies between&nbsp; $-1$&nbsp; and&nbsp; $+1$&nbsp; holds:
+'''(5)'''&nbsp; For the probability that&nbsp; $x$&nbsp; lies between&nbsp; $-1$&nbsp; and&nbsp; $+1$:
 :$${\rm Pr}(|\hspace{0.05cm}x\hspace{0.05cm}|< 1)= F_{x}(+1) - F_{ x}(-1)= 0.909-0.091\hspace{0.15cm}\underline{= 0.818}.$$
-*This result agrees exactly with the result of the subtask&nbsp; '''(7)'''&nbsp; of Exercise 3.1 &uuml;which was obtained by direct integration &uuml;over the WDF.
+*This result agrees exactly with the result of the subtask&nbsp; '''(7)'''&nbsp; of Exercise 3.1&nbsp; which was obtained by direct integration &uuml;over the WDF.
-'''(6)'''&nbsp; The VTF of the discrete random size&ouml;&aerospace;e&nbsp; $y$&nbsp; at the location&nbsp; $y =0$&nbsp; is the sum of the probabilities of&nbsp; $-2$,&nbsp; $-1$&nbsp; and&nbsp; $0$,&nbsp; so holds
 :$$F_y(r = 0)\hspace{0.15cm}\underline{= 0.7}.$$
 {{ML-Fuß}}

@@ Line 21: / Line 21: @@
 Hints:
-*The exercise belongs to the chapter&nbsp; [[Theory_of_Stochastic_Signals/Cumulative_Distribution_Function|cumulative distribution function]].
+*The exercise belongs to the chapter&nbsp; [[Theory_of_Stochastic_Signals/Cumulative_Distribution_Function|Cumulative Distribution Function]].
 *The topic of this chapter is illustrated with examples in the&nbsp; (German language)&nbsp; learning video <br> &nbsp; &nbsp; [[Zusammenhang_zwischen_WDF_und_VTF_(Lernvideo)|"Zusammenhang zwischen WDF und VTF"]] &nbsp; $\Rightarrow$ &nbsp; "Relationship between PDF and CDF".
 *Given the following equation:

@@ Line 3: / Line 3: @@
 }}
-[[File:P_ID114__Sto_A_3_2.png|right|frame|"Cosine&ndash;Square"&nbsp; CDF (top),<br> "Dirac"&nbsp; CDF (bottom)]]
+[[File:P_ID114__Sto_A_3_2.png|right|frame|Given cumulative distribution function&nbsp; $\rm (CDF)$]]
 The same conditions apply as for&nbsp; [[Aufgaben:Exercise_3.1:_cos²-PDF_and_PDF_with_Dirac_Functions|Exercise 3.1]].
-*The PDF of the continuous valued random variable is identically zero in the ranges&nbsp; $|x| > 2$&nbsp; and in the range&nbsp; $-2 \le x \le +2$&nbsp; holds:
+*The PDF of the continuous valued random variable is identically zero in the ranges&nbsp; $|x| > 2$.&nbsp;
 :$$f_x(x)={1}/{2}\cdot \cos^2({\pi}/{4}\cdot x).$$
-*Also, the discrete random variable&nbsp; $y$&nbsp; is limited to the range&nbsp; $\pm 2$&nbsp; Here, the following probabilities apply:
+*The discrete valued random variable&nbsp; $y$&nbsp; is limited too to the range&nbsp; $\pm 2$&nbsp; Here,&nbsp; the following probabilities apply:
 :$${\rm \Pr}(y=0)=0.4,$$
 :$${\rm \Pr}(y=+1)={\rm \Pr}(y=-1)=0.2,$$
@@ Line 21: / Line 22: @@
 Hints:
 *The exercise belongs to the chapter&nbsp; [[Theory_of_Stochastic_Signals/Cumulative_Distribution_Function|cumulative distribution function]].
 *Given the following equation:
 :$$\int \cos^{\rm 2}( ax)\, {\rm d}x=\frac{x}{2}+\frac{1}{4 a}\cdot \sin(2 ax).$$
@@ Line 31: / Line 32: @@
 <quiz display=simple>
-{Which of the following statements are true for the distribution function&nbsp; $F_x(r)$&nbsp; of continuous valued random variable&nbsp; $x$&nbsp;?
+{Which of the following statements are true for the cumulative distribution function&nbsp; $F_x(r)$&nbsp; of the continuous valued random variable&nbsp; $x$&nbsp;?
 |type="[]"}
-+ The CDF is equal for all values&nbsp; $r \le -2$&nbsp; $F_x(r) \equiv 0$.
++ The CDF is equal for all values&nbsp; $r \le -2$ &nbsp; &rArr; &nbsp; $F_x(r) \equiv 0$.
-+ The CDF is equal for all values&nbsp; $r \ge +2$&nbsp; $F_x(r) \equiv 1$.
++ The CDF is equal for all values&nbsp; $r \ge +2$ &nbsp; &rArr; &nbsp; $F_x(r) \equiv 1$.
-+ Der Verlauf von&nbsp; $F_x(r)$&nbsp; ist monoton steigend.
++ The curve of&nbsp; $F_x(r)$&nbsp; is monotonically increasing.
-{Which of the following statements are true for the distribution function&nbsp; $F_y(r)$&nbsp; of the discrete value random variable&nbsp; $y$&nbsp;?
+{Which of the following statements are true for the cumulative distribution function&nbsp; $F_y(r)$&nbsp; of the discrete valued random variable&nbsp; $y$&nbsp;?
 |type="[]"}
-- The CDF is equal for all values&nbsp; $r \le -2$&nbsp; $F_y(r) \equiv 0$.
+- The CDF is equal for all values&nbsp; $r \le -2$ &nbsp; &rArr; &nbsp; $F_y(r) \equiv 0$.
-+ The CDF is equal for all values&nbsp; $r \ge +2$&nbsp; $F_y(r) \equiv 1$.
++ The CDF is equal for all values&nbsp; $r \ge +2$ &nbsp; &rArr; &nbsp; $F_y(r) \equiv 1$.
 + The curve of&nbsp; $F_y(r)$&nbsp; is monotonically increasing.
-{Calculate the CDF&nbsp; $F_x(r)$.&nbsp; Restrict yourself here to the range&nbsp; $0 \le r \le +2$. <br>What value results for&nbsp; $r = +1$?
+{Calculate the CDF&nbsp; $F_x(r)$.&nbsp; Restrict yourself to the range&nbsp; $0 \le r \le +2$.&nbsp; What value results for&nbsp; $r = +1$?
 |type="{}"}
 $F_x(r=+1) \ = \ $ { 0.909 3% }
-{What is the relationship between&nbsp; $F_x(r)$&nbsp; and&nbsp; $F_x(-r)$?&nbsp; Enter the CDF value&nbsp; $F_x(r=-1)$&nbsp; .
+{What is the relationship between&nbsp; $F_x(r)$&nbsp; and&nbsp; $F_x(-r)$?&nbsp; Enter the CDF value&nbsp; $F_x(r=-1)$.
 |type="{}"}
 $F_x(r=-1) \ = \ $ { 0.091 3% }
@@ Line 57: / Line 58: @@
-{Calculate the probability that&nbsp; $x$&nbsp; is smaller in absolute value than&nbsp; $1$&nbsp;. <br>Compare the result with the result of the subtask&nbsp; '''(7)'''&nbsp; of task 3.1.
+{Calculate the probability that&nbsp; $|\hspace{0.05cm}x\hspace{0.05cm}|$&nbsp; is smaller  than&nbsp; $1$.&nbsp; Compare the result with the result of subtask&nbsp; '''(7)'''&nbsp; of Exercise 3.1.
 |type="{}"}
 ${\rm Pr}(|\hspace{0.05cm}x\hspace{0.05cm}| < 1) \ = \ $ { 0.818 3% }
@@ Line 63: / Line 64: @@
-{What value is obtained for the CDF of the discrete random variable&nbsp; $y$&nbsp; at the location&nbsp; $r = 0$?
+{What value is obtained for the CDF of the discrete valued random variable&nbsp; $y$&nbsp; at the location&nbsp; $r = 0$?
 |type="{}"}
 $F_y(r = 0)\ = \ $ { 0.7 3% }

@@ Line 3: / Line 3: @@
 }}
-[[File:P_ID114__Sto_A_3_2.png|right|frame|Cosine&ndash;Square&ndash;CDF (top),<br> Dirac&ndash;CDF (bottom)]]
+[[File:P_ID114__Sto_A_3_2.png|right|frame|"Cosine&ndash;Square"&nbsp; CDF (top),<br> "Dirac"&nbsp; CDF (bottom)]]
 The same conditions apply as for&nbsp; [[Aufgaben:Exercise_3.1:_cos²-PDF_and_PDF_with_Dirac_Functions|Exercise 3.1]].
 *The PDF of the continuous valued random variable is identically zero in the ranges&nbsp; $|x| > 2$&nbsp; and in the range&nbsp; $-2 \le x \le +2$&nbsp; holds:

@@ Line 20: / Line 20: @@
 Hints:
-*The task belongs to the chapter&nbsp; [[Theory_of_Stochastic_Signals/Cumulative_Distribution_Function|cumulative distribution function]].
+*The exercise belongs to the chapter&nbsp; [[Theory_of_Stochastic_Signals/Cumulative_Distribution_Function|cumulative distribution function]].
 *Given the following equation:
 :$$\int \cos^{\rm 2}( ax)\, {\rm d}x=\frac{x}{2}+\frac{1}{4 a}\cdot \sin(2 ax).$$
-The topic of this chapter is illustrated with examples in the (German language) learning video&nbsp; [[Zusammenhang_zwischen_WDF_und_VTF_(Lernvideo)|Zusammenhang zwischen WDF und VTF]]&nbsp; $\Rightarrow$ relationship between PDF and CDF.
+*The topic of this chapter is illustrated with examples in the (German language) learning video&nbsp; [[Zusammenhang_zwischen_WDF_und_VTF_(Lernvideo)|Zusammenhang zwischen WDF und VTF]]&nbsp; $\Rightarrow$ relationship between PDF and CDF.

← Older revision		Revision as of 10:04, 20 December 2021
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−	[[Category:Theory of Stochastic Signals: Exercises\|^3.2 ~~Verteilungsfunktion~~^]]	+	[[Category:Theory of Stochastic Signals: Exercises\|^3.2 Cumulative Distribution Function^]]