Aufgaben:Exercise 3.2Z: Relationship between PDF and CDF - Revision history

Guenter at 14:39, 4 January 2022

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Guenter at 14:27, 4 January 2022

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Guenter at 14:25, 4 January 2022

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Guenter at 14:24, 4 January 2022

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Noah at 19:59, 26 December 2021

2021-12-26T19:59:13Z

Noah at 19:58, 26 December 2021

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Guenter at 10:05, 20 December 2021

2021-12-20T10:05:09Z

Guenter: Guenter moved page Aufgaben:Aufgabe 3.2Z: Zusammenhang zwischen WDF und VTF to Aufgaben:Exercise 3.2Z: Relationship between PDF and CDF

2021-12-20T09:16:54Z

Guenter moved page Aufgabe 3.2Z: Zusammenhang zwischen WDF und VTF to Exercise 3.2Z: Relationship between PDF and CDF

Javier: Text replacement - "”" to """

2021-05-28T14:31:16Z

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Javier: Text replacement - "„" to """

2021-05-28T14:27:32Z

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 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; The <u>statements 1, 3 and 4</u> are always correct:
+'''(1)'''&nbsp; The&nbsp; <u>statements 1, 3 and 4</u>&nbsp; are always correct:
-*A horizontal intercept in the VTF indicates that the random size has no values in that region.
+*A horizontal intercept in the CDF indicates that the random variable has no values in that region.
-*In contrast, a vertical intercept in the VTF indicates a Dirac function in the WDF&nbsp; $($at the same location&nbsp; $x_0)$&nbsp;.
+*In contrast,&nbsp; a vertical intercept in the CDF indicates a Dirac delta function in the PDF&nbsp; $($at the same location&nbsp; $x_0)$.
-*This means that the random size takes the value&nbsp; $x_0$&nbsp; very frequently, namely with finite probability.
+*This means that the random variable takes the value&nbsp; $x_0$&nbsp; very frequently,&nbsp; namely with finite probability.
-*All other values occur exactly with probability&nbsp; $0$&nbsp;.
+*All other values occur exactly with probability&nbsp; $0$.
-*If, however&nbsp; $x$&nbsp; is limited to the range from&nbsp; $x_{\rm min}$&nbsp; to&nbsp; $x_{\rm max}$&nbsp; then&nbsp; $F_x(r) = 0$ &nbsp;f&uuml;r&nbsp; $r < x_{\rm min}$&nbsp; and&nbsp; $F_x(r) = 1$ &nbsp;f&uuml;r&nbsp; $r > x_{\rm max}$.
+*If,&nbsp; however&nbsp; $x$&nbsp; is limited to the range from&nbsp; $x_{\rm min}$&nbsp; to&nbsp; $x_{\rm max}$&nbsp; then&nbsp; $F_x(r) = 0$&nbsp; &nbsp;for&nbsp; $r < x_{\rm min}$&nbsp; and&nbsp; $F_x(r) = 1$ &nbsp;for&nbsp; $r > x_{\rm max}$.
-*In this special case, the second statement would also be true.
+*In this special case,&nbsp; the second statement would also be true.
-'''(2)'''&nbsp; The sought probability can be calculated from the difference of the VTF&ndash;values at the boundaries:
+'''(2)'''&nbsp; The sought probability can be calculated from the difference of the CDF&nbsp; values at the boundaries:
 :$${\rm Pr}( x> 0)= F_x(\infty)- F_x(\rm 0)
 \hspace{0.15cm}\underline{=\rm 0.25}.$$
@@ Line 84: / Line 84: @@
 \hspace{0.15cm}{\approx0.092}. $$
-*For reasons of symmetry ${\rm Pr}(x<- 0.5)$&nbsp; is just as large. From this follows:
+*For reasons of symmetry:&nbsp; ${\rm Pr}(x<- 0.5)$&nbsp; is just as large.&nbsp; From this follows:
-$${\rm Pr}( |\hspace{0.05cm} x\hspace{0.05cm}| >\rm 0.5) \hspace{0.15cm}\underline{= \rm 0.184}.$$
+:$${\rm Pr}( |\hspace{0.05cm} x\hspace{0.05cm}| >\rm 0.5) \hspace{0.15cm}\underline{= \rm 0.184}.$$
 [[File: P_ID116__Sto_Z_3_2_c.png|right|frame|PDF of Laplace distribution]]
 '''(4)'''&nbsp; The PDF is obtained from the corresponding CDF by differentiating the two areas.
-*The result is a two-sided exponential function as well as a Dirac function at&nbsp; $x = 0$&nbsp;:
+*The result is a two-sided exponential function as well as a Dirac delta function at&nbsp; $x = 0$:
 :$$f_x(x)=\rm 0.5\cdot \rm e^{-2\cdot |\hspace{0.05cm}\it x\hspace{0.05cm}|} + \rm 0.5\cdot\delta(\it x).$$
 *The numerical value we are looking for is&nbsp; $f_x(x = 1)\hspace{0.15cm}\underline{= \rm 0.0677}$.
+'''(5)'''&nbsp; In the range around&nbsp; $1$&nbsp; describes&nbsp; $x$&nbsp; a continuous valued random variable.
 *The probability that&nbsp; $x$&nbsp; has exactly the value&nbsp; $1$&nbsp; is therefore&nbsp; ${\rm Pr}(x = 1)\hspace{0.15cm}\underline{= \rm 0}.$
 *The PDF of a speech signal is often described by a two-sided exponential function.
-*The Dirac function at&nbsp; $x = 0$&nbsp; mainly takes into account speech pauses &ndash; here in&nbsp; $50\%$&nbsp; all times.
+*The Dirac delta function at&nbsp; $x = 0$&nbsp; mainly takes into account speech pauses &ndash; here in&nbsp; $50\%$&nbsp; of all times.
 {{ML-Fuß}}

@@ Line 30: / Line 30: @@
 |type="[]"}
 + The CDF increases from&nbsp; $0$&nbsp; to&nbsp; $1$&nbsp; at least weakly monotonically.
-- The&nbsp; $F_x(r)$ values&nbsp; $0$&nbsp; and&nbsp; $1$&nbsp; are possible for finite&nbsp; $r$values.
+- The&nbsp; $F_x(r)$&nbsp; values&nbsp; $0$&nbsp; and&nbsp; $1$&nbsp; are possible for finite&nbsp; $r$&nbsp; values.
 +A horizontal section indicates that in this range the random size has no proportions.
 +Vertical sections are possible.

@@ Line 17: / Line 17: @@
 Hints:
 *The exercise belongs to the chapter&nbsp; [[Theory_of_Stochastic_Signals/Cumulative_Distribution_Function|Cumulative Distribution Function]].
-*Reference is made to the chapter&nbsp; [[Theory_of_Stochastic_Signals/Probability_Density_Function]].
+*Reference is made to the chapter&nbsp; [[Theory_of_Stochastic_Signals/Probability_Density_Function|Probability Density 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".

@@ Line 3: / Line 3: @@
 }}
-[[File:P_ID117__Sto_Z_3_2.png|right|frame|Cumulative distribution function&nbsp; $ F_x(r)$]]
+[[File:P_ID117__Sto_Z_3_2.png|right|frame|Given CDF&nbsp; $ F_x(r)$]]
-Given is the random variable&nbsp; $x$&nbsp; with the distribution function.
+Given is the random variable&nbsp; $x$&nbsp; with the cumulative distribution function&nbsp; $\rm (CDF)$.
 :$$ F_x(r)=\left\{\begin{array}{*{4}{c}} 0.25\cdot {\rm e}^{2\it r}  &\rm for\hspace{0.1cm}\it r<\rm 0, \\ 1-0.25\cdot {\rm e}^{-2\it r} & \rm for\hspace{0.1cm}\it r\ge\rm 0.  \\\end{array}\right.$$
@@ Line 16: / Line 16: @@
 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]].
 *Reference is made to the chapter&nbsp; [[Theory_of_Stochastic_Signals/Probability_Density_Function]].
-*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&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".
@@ Line 27: / Line 27: @@
 <quiz display=simple>
-{What properties of a CDF hold when the random variable has no limits?
+{What properties of the CDF hold when the random variable has no limits?
 |type="[]"}
 + The CDF increases from&nbsp; $0$&nbsp; to&nbsp; $1$&nbsp; at least weakly monotonically.
-- The&nbsp; $F_x(r)$&ndash;values&nbsp; $0$&nbsp; and&nbsp; $1$&nbsp; are possible f&uuml;r finite&nbsp; $r$&ndash;values.
+- The&nbsp; $F_x(r)$ values&nbsp; $0$&nbsp; and&nbsp; $1$&nbsp; are possible for finite&nbsp; $r$values.
 +A horizontal section indicates that in this range the random size has no proportions.
 +Vertical sections are possible.

@@ Line 3: / Line 3: @@
 }}
-[[File:P_ID117__Sto_Z_3_2.png|right|frame|cumulative distribution function&nbsp; $ F_x(r)$]]
+[[File:P_ID117__Sto_Z_3_2.png|right|frame|Cumulative distribution function&nbsp; $ F_x(r)$]]
 Given is the random variable&nbsp; $x$&nbsp; with the distribution function.
 :$$ F_x(r)=\left\{\begin{array}{*{4}{c}} 0.25\cdot {\rm e}^{2\it r}  &\rm for\hspace{0.1cm}\it r<\rm 0, \\ 1-0.25\cdot {\rm e}^{-2\it r} & \rm for\hspace{0.1cm}\it r\ge\rm 0.  \\\end{array}\right.$$

← Older revision		Revision as of 10:05, 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^]]

← Older revision		Revision as of 14:31, 28 May 2021
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−	<i>Hinweis:</i>   Die zweiseitige Exponentialverteilung nennt man auch "Laplaceverteilung~~”~~.	+	<i>Hinweis:</i>   Die zweiseitige Exponentialverteilung nennt man auch "Laplaceverteilung".

← Older revision		Revision as of 14:27, 28 May 2021
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−	<i>Hinweis:</i>   Die zweiseitige Exponentialverteilung nennt man auch ~~&bdquo;~~Laplaceverteilung”.	+	<i>Hinweis:</i>   Die zweiseitige Exponentialverteilung nennt man auch "Laplaceverteilung”.