Theory of Stochastic Signals/Cumulative Distribution Function - Revision history

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← Older revision		Revision as of 15:47, 19 February 2024
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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«.

@@ Line 94: / Line 94: @@
 * Thus,&nbsp; only six Dirac delta functions are visible in the PDF.&nbsp; The two missing Diracs at&nbsp; $q = 0$&nbsp; and&nbsp; $q =1$&nbsp; are only indicated by dots.
-*The step-shaped CDF&nbsp; $F_{q}(r)$&nbsp; sketched on the right thus has six points of discontinuity,&nbsp; where  in each case the right-hand side limit is valid.}}.
+*The step-shaped CDF&nbsp; $F_{q}(r)$&nbsp; sketched on the right thus has six points of discontinuity,&nbsp; where  in each case the right-hand side limit is valid.}}
 &raquo; &nbsp; The topic of this chapter is illustrated with examples in the&nbsp; (German language)&nbsp; learning video&nbsp; <br> &nbsp; &nbsp; &nbsp;&nbsp; &nbsp;[[Zusammenhang_zwischen_WDF_und_VTF_(Lernvideo)|&raquo;Zusammenhang zwischen WDF und VTF&raquo;]]&nbsp; $\Rightarrow$ &raquo;Relationship between PDF and CDF&laquo;.

@@ Line 41: / Line 41: @@
 {{GraueBox|TEXT=
-$\text{Example 1:}$&nbsp; The left image shows the photo&nbsp; "Lena",&nbsp; which is often used as a test template for image coding procedures.
+$\text{Example 1:}$&nbsp; The left image shows the photo&nbsp; &raquo;Lena&laquo;,&nbsp; which is often used as a test template for image coding procedures.
-[[File:P_ID617__Sto_T_3_2_S1b_neu.png |right|frame| PDF and CDF of a continuous-valued image]]
+[[File:P_ID617__Sto_T_3_2_S1b_neu.png |right|frame| PDF and CDF of a value-continuous image]]
-*If this image is divided into&nbsp; $256 × 256$&nbsp; (image) pixels,&nbsp;  and the brightness is determined for each pixel,&nbsp; a sequence&nbsp; $〈x_ν〉$&nbsp; of gray values is obtained whose length&nbsp; $N = 256^2 = 65536$.
+*If this image is divided into&nbsp; $256 × 256$&nbsp; pixels,&nbsp;  and the brightness is determined for each pixel,&nbsp; a sequence&nbsp; $〈x_ν〉$&nbsp; of gray values is obtained whose length&nbsp; $N = 256^2 = 65\hspace{0.06cm}536$.
-The middle diagram shows the PDF&nbsp; $f_{x}(x)$&nbsp; which is also often referred to in the literature as&nbsp; "gray value statistics".
-*It can be seen that in the original image some gray values are preferred and the two extreme values&nbsp; $x =0$&nbsp; ("deep black")&nbsp; or&nbsp; $x =1$&nbsp; ("pure white")&nbsp; occur very rarely.
+The middle diagram shows the PDF&nbsp; $f_{x}(x)$&nbsp; which is also often referred to in the literature as&nbsp; &raquo;gray value statistics&laquo;.
-*The distribution function&nbsp; $F_{x}(r)$&nbsp; of this continuous random variable is continuous and increases monotonically from&nbsp; $0$&nbsp; to&nbsp; $1$&nbsp; as the right figure shows.&nbsp;
+*In the original image some gray values are preferred and the two extreme values&nbsp; $x =0$&nbsp; ("deep black")&nbsp; or&nbsp; $x =1$&nbsp; ("pure white")&nbsp; occur very rarely.
 *For&nbsp; $r \approx 0$&nbsp; and&nbsp; $r \approx 1$&nbsp; the CDF is horizontal due to the lack of PDF components.
-Note: &nbsp; Strictly speaking,&nbsp; for an image that can be displayed on a computer:&nbsp; In contrast to an "analog" photograph:
+$\text{Note:}$  &nbsp; Strictly speaking,&nbsp; for an image that can be displayed on a computer&nbsp; $($in contrast to an analog photograph$)$:
-# &nbsp; '''The gray value is always a discrete value random variable'''.&nbsp;
+# &nbsp; The gray value is always a discrete in value.&nbsp;
-# &nbsp; However,&nbsp; with large resolution of the color information&nbsp; ("color depth"),&nbsp; this random variable can be approximated to be continuous in value. }}
+# &nbsp; However,&nbsp; with large resolution of the color information&nbsp; $($&raquo;color depth&laquo;$)$,&nbsp; this random variable can be approximated to be continuous in value. }}
-The topic of this chapter is illustrated with examples in the&nbsp; (German language)&nbsp; learning video&nbsp; <br> &nbsp; &nbsp; &nbsp;[[Zusammenhang_zwischen_WDF_und_VTF_(Lernvideo)|"Zusammenhang zwischen WDF und VTF"]]&nbsp; $\Rightarrow$ "Relationship between PDF and CDF".
+&raquo; &nbsp; The topic of this chapter is illustrated with examples in the&nbsp; (German language)&nbsp; learning video&nbsp; <br> &nbsp; &nbsp; &nbsp;&nbsp; &nbsp;[[Zusammenhang_zwischen_WDF_und_VTF_(Lernvideo)|&raquo;Zusammenhang zwischen WDF und VTF&raquo;]]&nbsp; $\Rightarrow$ &raquo;Relationship between PDF and CDF&laquo;.
 ==CDF for value-discrete random variables==
 <br>
-For the CDF calculation of a discrete valued random variable&nbsp; $x$&nbsp; from its PDF,&nbsp; a more general equation must always be assumed.&nbsp; Here, with the auxiliary variable&nbsp; $\varepsilon > 0$:
+For the CDF calculation of a value-discrete random variable&nbsp; $x$&nbsp; from its PDF,&nbsp; a more general equation must always be assumed.&nbsp; Here,&nbsp; with the auxiliary variable&nbsp; $\varepsilon > 0$:
 :$$F_{x}(r)=\lim_{\varepsilon\hspace{0.05cm}\to \hspace{0.05cm}0}\int_{-\infty}^{r+\varepsilon}f_x(x)\,{\rm d}x.$$
-*The CDF calculation is here required  by boundary value formation due to the&nbsp; "less than/equal"&nbsp; sign in the&nbsp; [[Theory_of_Stochastic_Signals/Cumulative_Distribution_Function#Relationship_between_PDF_and_CDF|$\text{general definition}$]].
+*Due to the&nbsp; &raquo;less than/equal&raquo;&nbsp; sign in the&nbsp; [[Theory_of_Stochastic_Signals/Cumulative_Distribution_Function#Relationship_between_PDF_and_CDF|&raquo;general definition&laquo;]], a limit value must be formed for the CDF calculation.&nbsp; If we also take into account that,&nbsp; for a value-discrete random variable,&nbsp; the PDF consists of a sum of weighted&nbsp; [[Signal_Representation/Direct_Current_Signal_-_Limit_Case_of_a_Periodic_Signal#Dirac_.28delta.29_function_in_frequency_domain|&raquo;Dirac delta functions&laquo;]],&nbsp; we obtain:
 :$$F_{x}(r)=\lim_{\varepsilon\hspace{0.05cm}\to \hspace{0.05cm} 0}\int_{-\infty}^{r+\varepsilon}\sum\limits_{\mu= 1}^{ M}p_\mu\cdot \delta(x-x_\mu)\,{\rm d}x.$$
-*If we interchange integration and summation in this equation,&nbsp; and consider that integration over the Dirac delta function yields the step function,&nbsp; we obtain:
+*If we interchange integration and summation in this equation,&nbsp; and consider that an integration over the Dirac delta function yields the step function,&nbsp; we obtain:
 :$$F_{x}(r)=\sum\limits_{\mu= \rm 1}^{\it M}p_\mu\cdot \gamma_0 (r-x_\mu),\hspace{0.4cm}{\rm with} \hspace{0.4cm}\gamma_0(x)=\lim_{\epsilon\hspace{0.05cm}\to \hspace{0.05cm} 0}\int_{-\infty}^{x+\varepsilon}\delta (u)\,{\rm d} u = \left\{ \begin{array}{*{2}{c}}  0 \hspace{0.4cm}  {\rm if}\hspace{0.1cm} x< 0,\\ 1  \hspace{0.4cm} {\rm if}\hspace{0.1cm}x\ge 0. \\ \end{array} \right.$$
+::The function&nbsp; $γ_0(x)$&nbsp; differs from the&nbsp; [[Signal_Representation/Fourier_Transform_Theorems#Assignment_Theorem|&raquo;unit step function&laquo;]]&nbsp; $γ(x)$&nbsp; often used in systems theory in that at the jump point&nbsp; $x = 0$&nbsp; the right-hand side limit&nbsp; $1$nbsp; is valid&nbsp; $($instead of the mean value&nbsp; $0.5$&nbsp; between left&ndash; and right&ndash;hand side limits$)$.
-It should be noted that:
+*With the above CDF definition,&nbsp; the following probability equation holds for value-continuous and value-discrete random variables equally,&nbsp; and of course also for&nbsp; mixed random variables&nbsp; with discrete and continuous parts:
 :$${\rm Pr}(x_{\rm u}<x \le x_{\rm o})=F_x(x_{\rm o})-F_x(x_{\rm u}).$$
-*For purely continuous random variables,&nbsp; the&nbsp; "less than"&nbsp; sign and the&nbsp; "less than/equal to"&nbsp; sign could be substituted for each other here.
+*For purely value-continuous random variables,&nbsp; the&nbsp; &raquo;less than&laquo;&nbsp; sign and the&nbsp; &raquo;less than/equal to&laquo;&nbsp; sign could be substituted for each other here.
 :$${\rm Pr}(x_{\rm u}<x \le x_{\rm o}) ={\rm Pr}(x_{\rm u}\le x \le x_{\rm o}) ={\rm Pr}(x_{\rm u}\le x < x_{\rm o}) ={\rm Pr}(x_{\rm u}<x < x_{\rm o}).$$
 {{GraueBox|TEXT=
-$\text{Example 2:}$&nbsp; If the gray value of the&nbsp; [[Theory_of_Stochastic_Signals/Cumulative_Distribution_Function#CDF_for_continuous-valued_random_variables|$\text{original Lena photo}$]]&nbsp; is quantized by eight levels,&nbsp; so that each pixel can be represented by three bits and transmitted digitally,&nbsp; the discrete random variable&nbsp; $q$&nbsp; is obtained. &nbsp; However, due to the quantization,&nbsp; a part of the image information is lost,&nbsp; which is reflected in the quantized image by clearly recognizable&nbsp; "contours".
+$\text{Example 2:}$&nbsp; If the gray value of the&nbsp; [[Theory_of_Stochastic_Signals/Cumulative_Distribution_Function#CDF_for_continuous-valued_random_variables|&raquo;original Lena photo&laquo;]]&nbsp; is quantized by eight levels,&nbsp; so that each pixel can be represented by three bits and transmitted digitally,&nbsp; the discrete random variable&nbsp; $q$&nbsp; is obtained. &nbsp; However, due to the quantization,&nbsp; a part of the image information is lost,&nbsp; which is reflected in the quantized image by clearly recognizable&nbsp; &raquo;contours&laquo;.
-[[File:P_ID74__Sto_T_3_2_S2b_neu.png |right|frame| PDF and CDF of a discrete valued image]]
+[[File:P_ID74__Sto_T_3_2_S2b_neu.png |right|frame| PDF and CDF of a value-discrete image]]
 *The associated PDF&nbsp; $f_{q}(q)$&nbsp; is composed of&nbsp; $M = 8$&nbsp; Dirac delta functions, where,&nbsp; in the quantization chosen here,&nbsp; the possible gray levels are assigned the values&nbsp; $q_\mu = (\mu - 1)/7$&nbsp; with&nbsp; $\mu = 1, 2,$ ... , $8$.
 *The weights of the Dirac delta functions can be calculated from the PDF&nbsp; $f_{x}(x)$&nbsp; of the original image.&nbsp; One obtains
 :$$p_\mu={\rm Pr}(q = q_\mu ) = {\rm Pr}(\frac{2\mu-\rm 3}{14}< {x} \le\frac{2\it \mu- \rm 1}{14}) $$
 :$$\Rightarrow \hspace{0.3cm} p_\mu={\rm Pr}(q = q_\mu ) = \int_{(2\it \mu- \rm 3)/14}^{(2\mu-1)/14}\it f_{x}{\rm (}x{\rm )}\,{\rm d}x.$$
-*For the undefined areas&nbsp; $(x<0$, &nbsp;  $x>1)$&nbsp; is to be set&nbsp; $f_{x}(x) = 0$&nbsp;.
+*For the undefined areas&nbsp; $(x<0$, &nbsp;  $x>1)$&nbsp; is to be set&nbsp; $f_{x}(x) = 0$.&nbsp; Since in the original image the gray levels&nbsp; $x ≈0$&nbsp; $($&raquo;very deep black&laquo;$)$&nbsp; or&nbsp; $x ≈1$&nbsp; $($&raquo;almost pure white&laquo;$)$&nbsp; are largely missing,&nbsp; $p_1 ≈ p_8 ≈ 0$ result.
-*Since in the original image the gray levels&nbsp; $x ≈0$&nbsp; ("very deep black") &nbsp;or&nbsp; $x ≈1$&nbsp; ("almost pure white")&nbsp; are largely missing,&nbsp; $p_1 ≈ p_8 ≈ 0$ result.
 * Thus,&nbsp; only six Dirac delta functions are visible in the PDF.&nbsp; The two missing Diracs at&nbsp; $q = 0$&nbsp; and&nbsp; $q =1$&nbsp; are only indicated by dots.
 *The step-shaped CDF&nbsp; $F_{q}(r)$&nbsp; sketched on the right thus has six points of discontinuity,&nbsp; where  in each case the right-hand side limit is valid.}}.
-The topic of this chapter is illustrated with examples in the&nbsp; (German language)&nbsp; learning video&nbsp; <br> &nbsp; &nbsp; &nbsp;[[Zusammenhang_zwischen_WDF_und_VTF_(Lernvideo)|"Zusammenhang zwischen WDF und VTF"]]&nbsp; $\Rightarrow$ "Relationship between PDF and CDF".
+&raquo; &nbsp; The topic of this chapter is illustrated with examples in the&nbsp; (German language)&nbsp; learning video&nbsp; <br> &nbsp; &nbsp; &nbsp;&nbsp; &nbsp;[[Zusammenhang_zwischen_WDF_und_VTF_(Lernvideo)|&raquo;Zusammenhang zwischen WDF und VTF&raquo;]]&nbsp; $\Rightarrow$ &raquo;Relationship between PDF and CDF&laquo;.

@@ Line 38: / Line 38: @@
 ==CDF for value-continuous random variables==
 <br>
-The equations given in the last section apply only to value-continuous random variables and will be illustrated here by an example.&nbsp; In the next section it will be shown that for&nbsp; [[Theory_of_Stochastic_Signals/Cumulative_Distribution_Function#CDF_for_discrete-valued_random_variables|&raquo;value-discrete random variables&laquo;]]&nbsp; 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.&nbsp; In the next section it will be shown that for&nbsp; [[Theory_of_Stochastic_Signals/Cumulative_Distribution_Function#CDF_for_value-discrete_random_variables|&raquo;value-discrete random variables&laquo;]]&nbsp; the equations must be modified somewhat.
 {{GraueBox|TEXT=

@@ Line 7: / Line 7: @@
 ==Relationship between PDF and CDF==
 <br>
-To describe random variables,&nbsp; in addition to the&nbsp; [[Theory_of_Stochastic_Signals/Probability_Density_Function|$\text{probability density function}$]]&nbsp; $\rm (PDF)$,&nbsp; we also use the&nbsp; cumulative distribution function&nbsp; $\rm (CDF)$&nbsp; which is defined as follows:
+To describe random variables,&nbsp; in addition to the&nbsp; [[Theory_of_Stochastic_Signals/Probability_Density_Function|&raquo;probability density function&raquo;]]&nbsp; $\rm (PDF)$,&nbsp; we use the&nbsp; &raquo;cumulative distribution function&laquo;&nbsp; $\rm (CDF)$&nbsp; which is defined as follows:
 {{BlaueBox|TEXT=
@@ Line 14: / Line 14: @@
-For a continuous random variable,&nbsp; the following statements are possible regarding the CDF:
+For a value-continuous random variable,&nbsp; the following statements are possible regarding the CDF:
 *The CDF is computable from the probability density function&nbsp; $f_{x}(x)$&nbsp; by integration.&nbsp; It holds:
 :$$F_{x}(r) = \int_{-\infty}^{r}f_x(x)\,{\rm d}x.$$
-*Since the PDF is never negative,&nbsp; $F_{x}(r)$&nbsp; increases at least weakly monotonically,&nbsp; and always lies between the following limits:
+*Since the PDF is never negative,&nbsp; $F_{x}(r)$&nbsp; increases at least weakly monotonically,&nbsp; and the function always lies between the following limits:
 :$$F_{x}(r → \hspace{0.05cm} - \hspace{0.05cm} ∞) = 0, \hspace{0.5cm}F_{x}(r → +∞) = 1.$$
 *Inversely,&nbsp; the probability density function can be determined from the CDF by differentiation:
 :$$f_{x}(x)=\frac{{\rm d} F_{x}(r)}{{\rm d} r}\Bigg |_{\hspace{0.1cm}r=x}.$$
-:The addition&nbsp; "$r = x$"&nbsp; makes it clear that in our nomenclature the PDF argument is the random variable itself, while the CDF argument specifies any real variable&nbsp; $r$&nbsp;.
+:The addition&nbsp; &raquo;$r = x$&laquo;&nbsp; makes it clear that in our nomenclature the PDF argument is the random variable&nbsp; $x$&nbsp; itself, while the CDF argument specifies any real variable&nbsp; $r$&nbsp;.
 {{BlaueBox|TEXT=
-$\text{Notes on nomenclature:}$&nbsp;
+$\text{Notes on nomenclature:}$&nbsp; If in the definitions of&nbsp; $\rm PDF$&nbsp; and&nbsp; $\rm CDF$&nbsp; we had distinguished
 *and the realizations&nbsp; $x ∈ X$&nbsp; &nbsp; ⇒ &nbsp; $f_{X}(x), F_{X}(x)$,
@@ Line 34: / Line 34: @@
 :$$F_{X}(x) = {\rm Pr}(X \le x) = \int_{-\infty}^{x}f_{x}(\xi)\,{\rm d}\xi.$$
-Unfortunately,&nbsp; at the beginning of our&nbsp; $\rm LNTwww$ project&nbsp; (2001)&nbsp; we decided to use our nomenclature for quite legitimate reasons,&nbsp; which now&nbsp; (2017)&nbsp; cannot be changed,&nbsp; also with regard to the realized learning videos.
+Unfortunately,&nbsp; at the beginning of our&nbsp; $\rm LNTwww$ project&nbsp; $(2001)$&nbsp; we decided to use our nomenclature for quite legitimate reasons,&nbsp; which now&nbsp; $(2017)$&nbsp; cannot be changed,&nbsp; also with regard to the realized learning videos. &nbsp; '''So we stick with&nbsp; $f_{x}(x)$&nbsp; instead of&nbsp; $f_{X}(x)$&nbsp; as well as&nbsp; $F_{x}(r)$&nbsp; instead of&nbsp; $F_{X}(x).$}}
-==CDF for continuous-valued random variables==
+==CDF for value-continuous random variables==
 <br>
-The equations given in the last section apply only to continuous-valued random variables and will be illustrated here by an example.&nbsp; In the next section it will be shown that for&nbsp; [[Theory_of_Stochastic_Signals/Cumulative_Distribution_Function#CDF_for_discrete-valued_random_variables|$\text{discrete-valued random variables}$]]&nbsp; 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.&nbsp; In the next section it will be shown that for&nbsp; [[Theory_of_Stochastic_Signals/Cumulative_Distribution_Function#CDF_for_discrete-valued_random_variables|&raquo;value-discrete random variables&laquo;]]&nbsp; the equations must be modified somewhat.
 {{GraueBox|TEXT=
@@ Line 64: / Line 62: @@
-==CDF for discrete-valued random variables==
+==CDF for value-discrete random variables==
 <br>
 For the CDF calculation of a discrete valued random variable&nbsp; $x$&nbsp; from its PDF,&nbsp; a more general equation must always be assumed.&nbsp; Here, with the auxiliary variable&nbsp; $\varepsilon > 0$:

@@ Line 101: / Line 101: @@
 ==Exercises for the chapter==
 <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]]