Theory of Stochastic Signals/Cumulative Distribution Function - Revision history

Maintenance script: Add German interlanguage link

2026-03-16T12:28:53Z

Add German interlanguage link

← Older revision		Revision as of 14:28, 16 March 2026
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	*If we interchange integration and summation in this equation,  and consider that an integration over the Dirac delta function yields the step function,  we obtain:		*If we interchange integration and summation in this equation,  and consider that an integration over the Dirac delta function yields the step function,  we obtain:
	:$$F_{x}(r)=\sum\limits_{\mu= \rm 1}^{\it M}p_\mu\cdot \gamma_0 (r-x_\mu),\hspace{0.4cm}{\rm with} \hspace{0.4cm}\gamma_0(x)=\lim_{\epsilon\hspace{0.05cm}\to \hspace{0.05cm} 0}\int_{-\infty}^{x+\varepsilon}\delta (u)\,{\rm d} u = \left\{ \begin{array}{*{2}{c}} 0 \hspace{0.4cm} {\rm if}\hspace{0.1cm} x< 0,\\ 1 \hspace{0.4cm} {\rm if}\hspace{0.1cm}x\ge 0. \\ \end{array} \right.$$		:$$F_{x}(r)=\sum\limits_{\mu= \rm 1}^{\it M}p_\mu\cdot \gamma_0 (r-x_\mu),\hspace{0.4cm}{\rm with} \hspace{0.4cm}\gamma_0(x)=\lim_{\epsilon\hspace{0.05cm}\to \hspace{0.05cm} 0}\int_{-\infty}^{x+\varepsilon}\delta (u)\,{\rm d} u = \left\{ \begin{array}{*{2}{c}} 0 \hspace{0.4cm} {\rm if}\hspace{0.1cm} x< 0,\\ 1 \hspace{0.4cm} {\rm if}\hspace{0.1cm}x\ge 0. \\ \end{array} \right.$$
	::The function  $γ_0(x)$  differs from the  [[Signal_Representation/~~Fourier_Transform_Theorems~~#Assignment_Theorem\|»unit step function«]]  $γ(x)$  often used in systems theory in that at the jump point  $x = 0$  the right-hand side limit  $1$nbsp; is valid  $($instead of the mean value  $0.5$  between left– and right–hand side limits$)$.		::The function  $γ_0(x)$  differs from the  [[Signal_Representation/The_Fourier_Transform_Theorems#Assignment_Theorem\|»unit step function«]]  $γ(x)$  often used in systems theory in that at the jump point  $x = 0$  the right-hand side limit  $1$nbsp; is valid  $($instead of the mean value  $0.5$  between left– and right–hand side limits$)$.
	*With the above CDF definition,  the following probability equation holds for value-continuous and value-discrete random variables equally,  and of course also for  mixed random variables  with discrete and continuous parts:		*With the above CDF definition,  the following probability equation holds for value-continuous and value-discrete random variables equally,  and of course also for  mixed random variables  with discrete and continuous parts:
	:$${\rm Pr}(x_{\rm u}<x \le x_{\rm o})=F_x(x_{\rm o})-F_x(x_{\rm u}).$$		:$${\rm Pr}(x_{\rm u}<x \le x_{\rm o})=F_x(x_{\rm o})-F_x(x_{\rm u}).$$
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	{{Display}}		{{Display}}
			[[de:Stochastische_Signaltheorie/Verteilungsfunktion]]

Guenter at 15:47, 19 February 2024

2024-02-19T15:47:32Z

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	*The step-shaped CDF  $F_{q}(r)$  sketched on the right thus has six points of discontinuity,  where in each case the right-hand side limit is valid.}}		*The step-shaped CDF  $F_{q}(r)$  sketched on the right thus has six points of discontinuity,  where in each case the right-hand side limit is valid.}}


	»   The topic of this chapter is illustrated with examples in the  (German language)  learning video  <br>         [[Zusammenhang_zwischen_WDF_und_VTF_(Lernvideo)\|»Zusammenhang zwischen WDF und VTF»]]  $\Rightarrow$ »Relationship between PDF and CDF«.		»   The topic of this chapter is illustrated with examples in the  (German language)  learning video  <br>         [[Zusammenhang_zwischen_WDF_und_VTF_(Lernvideo)\|»Zusammenhang zwischen WDF und VTF»]]  $\Rightarrow$ »Relationship between PDF and CDF«.

Guenter at 15:46, 19 February 2024

2024-02-19T15:46:55Z

← Older revision		Revision as of 17:46, 19 February 2024
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	* Thus,  only six Dirac delta functions are visible in the PDF.  The two missing Diracs at  $q = 0$  and  $q =1$  are only indicated by dots.		* Thus,  only six Dirac delta functions are visible in the PDF.  The two missing Diracs at  $q = 0$  and  $q =1$  are only indicated by dots.

	*The step-shaped CDF  $F_{q}(r)$  sketched on the right thus has six points of discontinuity,  where in each case the right-hand side limit is valid.}}.		*The step-shaped CDF  $F_{q}(r)$  sketched on the right thus has six points of discontinuity,  where in each case the right-hand side limit is valid.}}

	»   The topic of this chapter is illustrated with examples in the  (German language)  learning video  <br>         [[Zusammenhang_zwischen_WDF_und_VTF_(Lernvideo)\|»Zusammenhang zwischen WDF und VTF»]]  $\Rightarrow$ »Relationship between PDF and CDF«.		»   The topic of this chapter is illustrated with examples in the  (German language)  learning video  <br>         [[Zusammenhang_zwischen_WDF_und_VTF_(Lernvideo)\|»Zusammenhang zwischen WDF und VTF»]]  $\Rightarrow$ »Relationship between PDF and CDF«.

Guenter at 17:57, 14 February 2024

2024-02-14T17:57:52Z

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Guenter at 16:53, 14 February 2024

2024-02-14T16:53:48Z

← Older revision		Revision as of 18:53, 14 February 2024
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	==CDF for value-continuous random variables==		==CDF for value-continuous random variables==
	<br>		<br>
	The equations given in the last section apply only to value-continuous random variables and will be illustrated here by an example.  In the next section it will be shown that for  [[Theory_of_Stochastic_Signals/Cumulative_Distribution_Function#~~CDF_for_discrete~~-~~valued_random_variables~~\|»value-discrete random variables«]]  the equations must be modified somewhat.		The equations given in the last section apply only to value-continuous random variables and will be illustrated here by an example.  In the next section it will be shown that for  [[Theory_of_Stochastic_Signals/Cumulative_Distribution_Function#CDF_for_value-discrete_random_variables\|»value-discrete random variables«]]  the equations must be modified somewhat.

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Guenter at 16:52, 14 February 2024

2024-02-14T16:52:50Z

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	==Relationship between PDF and CDF==		==Relationship between PDF and CDF==
	<br>		<br>
	To describe random variables,  in addition to the  [[Theory_of_Stochastic_Signals/Probability_Density_Function\|~~$\text{~~probability density function}$]]  $\rm (PDF)$,  we ~~also~~ use the  cumulative distribution function  $\rm (CDF)$  which is defined as follows:		To describe random variables,  in addition to the  [[Theory_of_Stochastic_Signals/Probability_Density_Function\|»probability density function»]]  $\rm (PDF)$,  we use the  »cumulative distribution function«  $\rm (CDF)$  which is defined as follows:

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	For a continuous random variable,  the following statements are possible regarding the CDF:		For a value-continuous random variable,  the following statements are possible regarding the CDF:
	*The CDF is computable from the probability density function  $f_{x}(x)$  by integration.  It holds:		*The CDF is computable from the probability density function  $f_{x}(x)$  by integration.  It holds:
	:$$F_{x}(r) = \int_{-\infty}^{r}f_x(x)\,{\rm d}x.$$		:$$F_{x}(r) = \int_{-\infty}^{r}f_x(x)\,{\rm d}x.$$
	*Since the PDF is never negative,  $F_{x}(r)$  increases at least weakly monotonically,  and always lies between the following limits:		*Since the PDF is never negative,  $F_{x}(r)$  increases at least weakly monotonically,  and the function always lies between the following limits:
	:$$F_{x}(r → \hspace{0.05cm} - \hspace{0.05cm} ∞) = 0, \hspace{0.5cm}F_{x}(r → +∞) = 1.$$		:$$F_{x}(r → \hspace{0.05cm} - \hspace{0.05cm} ∞) = 0, \hspace{0.5cm}F_{x}(r → +∞) = 1.$$
	*Inversely,  the probability density function can be determined from the CDF by differentiation:		*Inversely,  the probability density function can be determined from the CDF by differentiation:
	:$$f_{x}(x)=\frac{{\rm d} F_{x}(r)}{{\rm d} r}\Bigg \|_{\hspace{0.1cm}r=x}.$$		:$$f_{x}(x)=\frac{{\rm d} F_{x}(r)}{{\rm d} r}\Bigg \|_{\hspace{0.1cm}r=x}.$$
	:The addition  "$r = x$"  makes it clear that in our nomenclature the PDF argument is the random variable itself, while the CDF argument specifies any real variable  $r$ .		:The addition  »$r = x$«  makes it clear that in our nomenclature the PDF argument is the random variable  $x$  itself, while the CDF argument specifies any real variable  $r$ .


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	$\text{Notes on nomenclature:}$		$\text{Notes on nomenclature:}$  If in the definitions of  $\rm PDF$  and  $\rm CDF$  we had distinguished
			*between the random variable  $X$

	~~If in the definitions of  $\rm PDF$  and  $\rm CDF$  we had distinguished~~
	*between the random variable  $X$
	*and the realizations  $x ∈ X$    ⇒   $f_{X}(x), F_{X}(x)$,		*and the realizations  $x ∈ X$    ⇒   $f_{X}(x), F_{X}(x)$,

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	:$$F_{X}(x) = {\rm Pr}(X \le x) = \int_{-\infty}^{x}f_{x}(\xi)\,{\rm d}\xi.$$		:$$F_{X}(x) = {\rm Pr}(X \le x) = \int_{-\infty}^{x}f_{x}(\xi)\,{\rm d}\xi.$$

	Unfortunately,  at the beginning of our  $\rm LNTwww$ project  (2001)  we decided to use our nomenclature for quite legitimate reasons,  which now  (2017)  cannot be changed,  also with regard to the realized learning videos.		Unfortunately,  at the beginning of our  $\rm LNTwww$ project  $(2001)$  we decided to use our nomenclature for quite legitimate reasons,  which now  $(2017)$  cannot be changed,  also with regard to the realized learning videos.   '''So we stick with  $f_{x}(x)$  instead of  $f_{X}(x)$  as well as  $F_{x}(r)$  instead of  $F_{X}(x).$}}

	'''So we stick with  $f_{x}(x)$  instead of  $f_{X}(x)$  as well as  $F_{x}(r)$  instead of  $F_{X}(x).$}}

	==CDF for continuous~~-valued~~ random variables==		==CDF for value-continuous random variables==
	<br>		<br>
	The equations given in the last section apply only to continuous~~-valued~~ random variables and will be illustrated here by an example.  In the next section it will be shown that for  [[Theory_of_Stochastic_Signals/Cumulative_Distribution_Function#CDF_for_discrete-valued_random_variables\|~~$\text{~~discrete~~-valued~~ random variables}$]]  the equations must be modified somewhat.		The equations given in the last section apply only to value-continuous random variables and will be illustrated here by an example.  In the next section it will be shown that for  [[Theory_of_Stochastic_Signals/Cumulative_Distribution_Function#CDF_for_discrete-valued_random_variables\|»value-discrete random variables«]]  the equations must be modified somewhat.

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	==CDF for discrete~~-valued~~ random variables==		==CDF for value-discrete random variables==
	<br>		<br>
	For the CDF calculation of a discrete valued random variable  $x$  from its PDF,  a more general equation must always be assumed.  Here, with the auxiliary variable  $\varepsilon > 0$:		For the CDF calculation of a discrete valued random variable  $x$  from its PDF,  a more general equation must always be assumed.  Here, with the auxiliary variable  $\varepsilon > 0$:

Hwang at 19:54, 13 December 2022

2022-12-13T19:54:40Z

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	==Exercises for the chapter==		==Exercises for the chapter==
	<br>		<br>
	[[Aufgaben:Exercise_3.2:_CDF_for_Exercise_3.1\|Exercise 3.2: CDF for ~~Exercise_3~~.1]]		[[Aufgaben:Exercise_3.2:_CDF_for_Exercise_3.1\|Exercise 3.2: CDF for Exercise 3.1]]

	[[Aufgaben:Exercise_3.2Z:_Relationship_between_PDF_and_CDF\|Exercise 3.2Z: Relationship between PDF and CDF]]		[[Aufgaben:Exercise_3.2Z:_Relationship_between_PDF_and_CDF\|Exercise 3.2Z: Relationship between PDF and CDF]]

Hwang at 19:54, 13 December 2022

2022-12-13T19:54:03Z

← Older revision		Revision as of 21:54, 13 December 2022
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	*Since in the original image the gray levels  $x ≈0$  ("very deep black")  or  $x ≈1$  ("almost pure white")  are largely missing,  $p_1 ≈ p_8 ≈ 0$ result.		*Since in the original image the gray levels  $x ≈0$  ("very deep black")  or  $x ≈1$  ("almost pure white")  are largely missing,  $p_1 ≈ p_8 ≈ 0$ result.
	* Thus,  only six Dirac delta functions are visible in the PDF.  The two missing Diracs at  $q = 0$  and  $q =1$  are only indicated by dots.		* Thus,  only six Dirac delta functions are visible in the PDF.  The two missing Diracs at  $q = 0$  and  $q =1$  are only indicated by dots.
	*The step-shaped CDF  $F_{q}(r)$  sketched on the right thus has six points of discontinuity,  where in each case the right-hand side limit is valid.}}		*The step-shaped CDF  $F_{q}(r)$  sketched on the right thus has six points of discontinuity,  where in each case the right-hand side limit is valid.}}.

	The topic of this chapter is illustrated with examples in the  (German language)  learning video  <br>      [[Zusammenhang_zwischen_WDF_und_VTF_(Lernvideo)\|"Zusammenhang zwischen WDF und VTF"]]  $\Rightarrow$ "Relationship between PDF and CDF".		The topic of this chapter is illustrated with examples in the  (German language)  learning video  <br>      [[Zusammenhang_zwischen_WDF_und_VTF_(Lernvideo)\|"Zusammenhang zwischen WDF und VTF"]]  $\Rightarrow$ "Relationship between PDF and CDF".

Hwang at 19:53, 13 December 2022

2022-12-13T19:53:47Z

← Older revision		Revision as of 21:53, 13 December 2022
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	*Since in the original image the gray levels  $x ≈0$  ("very deep black")  or  $x ≈1$  ("almost pure white")  are largely missing,  $p_1 ≈ p_8 ≈ 0$ result.		*Since in the original image the gray levels  $x ≈0$  ("very deep black")  or  $x ≈1$  ("almost pure white")  are largely missing,  $p_1 ≈ p_8 ≈ 0$ result.
	* Thus,  only six Dirac delta functions are visible in the PDF.  The two missing Diracs at  $q = 0$  and  $q =1$  are only indicated by dots.		* Thus,  only six Dirac delta functions are visible in the PDF.  The two missing Diracs at  $q = 0$  and  $q =1$  are only indicated by dots.
	*The step-shaped CDF  $F_{q}(r)$  sketched on the right thus has six points of discontinuity,  where in each case the right-hand side limit is valid.}}.		*The step-shaped CDF  $F_{q}(r)$  sketched on the right thus has six points of discontinuity,  where in each case the right-hand side limit is valid.}}

	The topic of this chapter is illustrated with examples in the  (German language)  learning video  <br>      [[Zusammenhang_zwischen_WDF_und_VTF_(Lernvideo)\|"Zusammenhang zwischen WDF und VTF"]]  $\Rightarrow$ "Relationship between PDF and CDF".		The topic of this chapter is illustrated with examples in the  (German language)  learning video  <br>      [[Zusammenhang_zwischen_WDF_und_VTF_(Lernvideo)\|"Zusammenhang zwischen WDF und VTF"]]  $\Rightarrow$ "Relationship between PDF and CDF".

Hwang at 19:51, 13 December 2022

2022-12-13T19:51:19Z

← Older revision		Revision as of 21:51, 13 December 2022
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	==CDF for discrete-valued random variables==		==CDF for discrete-valued random variables==
	<br>		<br>
	For the CDF calculation of a discrete valued random variable  $x$  from its PDF,  a more general equation must always be assumed  Here, with the auxiliary variable  $\varepsilon > 0$:		For the CDF calculation of a discrete valued random variable  $x$  from its PDF,  a more general equation must always be assumed.  Here, with the auxiliary variable  $\varepsilon > 0$:
	:$$F_{x}(r)=\lim_{\varepsilon\hspace{0.05cm}\to \hspace{0.05cm}0}\int_{-\infty}^{r+\varepsilon}f_x(x)\,{\rm d}x.$$		:$$F_{x}(r)=\lim_{\varepsilon\hspace{0.05cm}\to \hspace{0.05cm}0}\int_{-\infty}^{r+\varepsilon}f_x(x)\,{\rm d}x.$$