Theory of Stochastic Signals/Probability Density Function (PDF) - Revision history

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@@ Line 1: / Line 1: @@
 {{Header
 |Untermenü=Continuous Random Variables
 |Vorherige Seite=Generation of Discrete Random Variables
 |Nächste Seite=Cumulative Distribution Function
 }}
 == # OVERVIEW OF THE THIRD MAIN CHAPTER # ==
 <br>
-We consider here&nbsp; '''continuous random variables''',&nbsp; i.e.,&nbsp; random variables which can assume infinitely many different values,&nbsp; at least in certain ranges of real numbers.&nbsp;
+We consider here&nbsp; &raquo;'''continuous random variables'''&laquo;,&nbsp; i.e.,&nbsp; random variables which can assume infinitely many different values,&nbsp; at least in certain ranges of real numbers.&nbsp;
 *Their applications in information and communication technology are manifold.
 *They are used,&nbsp; among other things,&nbsp; for the simulation of noise signals and for the description of fading effects.
-We restrict ourselves at first to the statistical description of the&nbsp; '''amplitude distribution'''.&nbsp; In detail,&nbsp; the following are treated:
+We restrict ourselves at first to the statistical description of the&nbsp; &raquo;'''amplitude distribution'''&laquo;.&nbsp; In detail,&nbsp; the following are treated:
 *The relationship between&nbsp; &raquo;probability density function&laquo;&nbsp; $\rm (PDF)$&nbsp; and&nbsp; &raquo;cumulative distribution function&laquo;&nbsp; $\rm (CDF)$;
@@ Line 28: / Line 29: @@
-'''Inner statistical dependencies'''&nbsp; of the underlying processes&nbsp; '''are not considered here'''.&nbsp; For this,&nbsp; we refer to the following main chapters 4 and 5.
+&raquo;'''Inner statistical dependencies'''&laquo;&nbsp; of the underlying processes&nbsp; '''are not considered here'''.&nbsp; For this,&nbsp; we refer to the following main chapters 4 and 5.
@@ Line 35: / Line 36: @@
 {{BlaueBox|TEXT=
-$\text{Definition:}$&nbsp; By a&nbsp; '''&raquo;value-continuous random variable&laquo;'''&nbsp; is meant a random variable whose possible numerical values are uncountable &nbsp; &rArr; &nbsp; $M \to \infty$. &nbsp; &nbsp; In the following,&nbsp; we will often use the short form "continuous random variable".}}
+$\text{Definition:}$&nbsp; By a&nbsp; &raquo;'''value-continuous random variable'''&laquo;&nbsp; is meant a random variable whose possible numerical values are uncountable &nbsp; &rArr; &nbsp; $M \to \infty$. &nbsp; &nbsp; In the following,&nbsp; we will often use the short form "continuous random variable".}}
@@ Line 94: / Line 95: @@
 {{BlaueBox|TEXT=
 $\text{Definition:}$ &nbsp;
-Applying the definition equation of the last section to discrete random variables,&nbsp; the PDF takes infinitely large values at some points&nbsp; $x_\mu$&nbsp; due to the nonvanishingly small probability value and the limit transition&nbsp $Δx → 0$.
+Applying the definition equation of the last section to discrete random variables,&nbsp; the PDF takes infinitely large values at some points&nbsp; $x_\mu$&nbsp; due to the nonvanishingly small probability value and the limit transition&nbsp; $Δx → 0$.
-Thus,&nbsp; the PDF results in a sum of&nbsp; [[Signal_Representation/Direct_Current_Signal_-_Limit_Case_of_a_Periodic_Signal#Dirac_.28delta.29_function_in_frequency_domain|Dirac delta functions]] &nbsp; &rArr; &nbsp; "distributions":
+Thus,&nbsp; the PDF results in a sum of&nbsp; [[Signal_Representation/Direct_Current_Signal_-_Limit_Case_of_a_Periodic_Signal#Dirac_.28delta.29_function_in_frequency_domain|$\text{Dirac delta functions}$]] &nbsp; &rArr; &nbsp; "distributions":
 :$$f_{x}(x)=\sum_{\mu=1}^{M}p_\mu\cdot {\rm \delta}( x-x_\mu).$$
@@ Line 102: / Line 103: @@
-Here is another note to help classify the different descriptive quantities for discrete and continuous random variables: &nbsp; Probability and probability density function are related in a similar way as in the book&nbsp; [[Signal Representation]]
+Here is another note to help classify the different descriptive quantities for discrete and continuous random variables: &nbsp; Probability and probability density function are related in a similar way as in the book&nbsp; [[Signal_Representation|"Signal Representation"]]
 *a discrete spectral component of a harmonic oscillation ⇒ line spectrum,&nbsp; and
 *a continuous spectrum of an energy-limited&nbsp; (pulse-shaped)&nbsp; signal.
@@ Line 111: / Line 112: @@
 $\text{Example 2:}$&nbsp; Below is a section
 *of a rectangular signal with three possible values,
-*where the signal value&nbsp; $0 \ \rm V$&nbsp occurs twice as often as the outer signal values&nbsp; $(\pm 1 \ \rm V)$.
+*where the signal value&nbsp; $0 \ \rm V$&nbsp; occurs twice as often as the outer signal values&nbsp; $(\pm 1 \ \rm V)$.
@@ Line 121: / Line 122: @@
 For a more in-depth look at the topic covered here,&nbsp; we recommend the following&nbsp; (German language)&nbsp; learning video:
-:[[Wahrscheinlichkeit_und_WDF_(Lernvideo)|Wahrscheinlichkeit und WDF]] &nbsp; &rArr; &nbsp; "Probability and probability density function"
+:[[Wahrscheinlichkeit_und_WDF_(Lernvideo)|"Wahrscheinlichkeit und WDF"]] &nbsp; &rArr; &nbsp; "Probability and probability density function"
 ==Numerical determination of the PDF==
@@ Line 127: / Line 128: @@
 Assuming that the random variable&nbsp; $x$&nbsp; at hand has negligible values outside the range from&nbsp; $x_{\rm min} = -4.02$&nbsp; to&nbsp; $x_{\rm max} = +4.02$,&nbsp; proceed as follows:
-[[File:P_ID175__Sto_T_3_1_S4_ganzneu.png |right|frame| For numerical determination of the PDF '''Korrektur''']]
+[[File:EN_Sto_T_3_1_S4_neu.png |right|frame| For numerical determination of the PDF]]
 #Divide the range of&nbsp; $x$-values into&nbsp; $I$&nbsp; intervals of equal width&nbsp; $Δx$&nbsp; and define a field&nbsp; $\text{PDF}[0 : I-1]$.&nbsp; In the sketch&nbsp; $I = 201$&nbsp; and accordingly&nbsp; $Δx = 0.04$&nbsp; is chosen.
 #The random variable&nbsp; $x$&nbsp; is now called&nbsp; $N$&nbsp; times in succession,&nbsp; each time checking to which interval&nbsp; $i_{\rm act}$&nbsp; the current random variable&nbsp; $x_{\rm act}$ belongs: <br> &nbsp; &nbsp; $i_{\rm act} = ({\rm int})((x + x_{\rm max})/Δx).$
@@ Line 143: / Line 144: @@
 ==Exercises for the chapter==
-[[Aufgaben:Exercise_3.1:_cos²-PDF_and_PDF_with_Dirac_Functions|Exercise 3.1: cos²-PDF and PDF with Dirac Functions]]
+[[Aufgaben:Exercise_3.1:_cos²-PDF_and_PDF_with_Dirac_Functions|Exercise 3.1: Cosine-square PDF and PDF with Dirac Functions]]
 [[Aufgaben:Exercise_3.1Z:_Triangular_PDF|Exercise 3.1Z: Triangular PDF]]

@@ Line 3: / Line 3: @@
 |Untermenü=Continuous Random Variables
 |Vorherige Seite=Generation of Discrete Random Variables
-|Nächste Seite=Verteilungsfunktion (VTF)
+|Nächste Seite=Cumulative Distribution Function
 }}

@@ Line 130: / Line 130: @@
 #Divide the range of&nbsp; $x$-values into&nbsp; $I$&nbsp; intervals of equal width&nbsp; $Δx$&nbsp; and define a field&nbsp; $\text{PDF}[0 : I-1]$.&nbsp; In the sketch&nbsp; $I = 201$&nbsp; and accordingly&nbsp; $Δx = 0.04$&nbsp; is chosen.
 #The random variable&nbsp; $x$&nbsp; is now called&nbsp; $N$&nbsp; times in succession,&nbsp; each time checking to which interval&nbsp; $i_{\rm act}$&nbsp; the current random variable&nbsp; $x_{\rm act}$ belongs: <br> &nbsp; &nbsp; $i_{\rm act} = ({\rm int})((x + x_{\rm max})/Δx).$
-#The corresponding field element PDF( $i_{\rm akt}$) is then incremented by&nbsp; $1$.&nbsp;
+#The corresponding field element PDF( $i_{\rm act}$) is then incremented by&nbsp; $1$.&nbsp;
 #After $N$ iterations, $\text{PDF}[i_{\rm act}]$ then contains the number of random numbers belonging to the interval $i_{\rm act}$.
-#The actual PDF values are obtained if,&nbsp; at the end,&nbsp; all field elements&nbsp; $\text{PDF}[i]$&nbsp; with&nbsp; $0 ≤ i ≤ I-1$&nbsp; are still divided by&nbsp; $N - Δx$.
+#The actual PDF values are obtained if,&nbsp; at the end,&nbsp; all field elements&nbsp; $\text{PDF}[i]$&nbsp; with&nbsp; $0 ≤ i ≤ I-1$&nbsp; are still divided by&nbsp; $N \cdot Δx$.
 <br clear=all>
 {{GraueBox|TEXT=

@@ Line 119: / Line 119: @@
-For a more in-depth look at the topic covered here, we recommend the following educational video:
+For a more in-depth look at the topic covered here,&nbsp; we recommend the following&nbsp; (German language)&nbsp; learning video:
-#####
 ==Numerical determination of the PDF==

@@ Line 96: / Line 96: @@
 Applying the definition equation of the last section to discrete random variables,&nbsp; the PDF takes infinitely large values at some points&nbsp; $x_\mu$&nbsp; due to the nonvanishingly small probability value and the limit transition&nbsp $Δx → 0$.
-Thus,&nbsp; the PDF results in a sum of&nbsp; [[Signal_Representation/General_Description/Gleichsignal_-_Grenzfall_eines_periodischen_Signals#Diracfunktion_im_Frequenzbereich|Dirac delta functions]] &nbsp; &rArr; &nbsp; "distributions":
+Thus,&nbsp; the PDF results in a sum of&nbsp; [[Signal_Representation/Direct_Current_Signal_-_Limit_Case_of_a_Periodic_Signal#Dirac_.28delta.29_function_in_frequency_domain|Dirac delta functions]] &nbsp; &rArr; &nbsp; "distributions":
 :$$f_{x}(x)=\sum_{\mu=1}^{M}p_\mu\cdot {\rm \delta}( x-x_\mu).$$

@@ Line 52: / Line 52: @@
-*Since there are no statistical dependencies between the samples $x_ν$,&nbsp; such a signal is also referred to as&nbsp; &raquo;white noise&laquo;.
+*Since there are no statistical dependencies between the samples $x_ν$,&nbsp; such a signal is also referred to as&nbsp; &raquo;white noise&laquo;.}}
-{{end}}
 ==Definition of the probability density function==

@@ Line 126: / Line 126: @@
 Assuming that the random variable&nbsp; $x$&nbsp; at hand has negligible values outside the range from&nbsp; $x_{\rm min} = -4.02$&nbsp; to&nbsp; $x_{\rm max} = +4.02$,&nbsp; proceed as follows:
-[[File:P_ID175__Sto_T_3_1_S4_ganzneu.png |right|frame| For numerical determination of the PDF]]
+[[File:P_ID175__Sto_T_3_1_S4_ganzneu.png |right|frame| For numerical determination of the PDF '''Korrektur''']]
 #Divide the range of&nbsp; $x$-values into&nbsp; $I$&nbsp; intervals of equal width&nbsp; $Δx$&nbsp; and define a field&nbsp; $\text{PDF}[0 : I-1]$.&nbsp; In the sketch&nbsp; $I = 201$&nbsp; and accordingly&nbsp; $Δx = 0.04$&nbsp; is chosen.
 #The random variable&nbsp; $x$&nbsp; is now called&nbsp; $N$&nbsp; times in succession,&nbsp; each time checking to which interval&nbsp; $i_{\rm act}$&nbsp; the current random variable&nbsp; $x_{\rm act}$ belongs: <br> &nbsp; &nbsp; $i_{\rm act} = ({\rm int})((x + x_{\rm max})/Δx).$
-#The corresponding field element WDF( $i_{\rm akt}$) is then incremented by $1$. After $N$ iterations, $\text{WDF}[i_{\rm akt}]$ then contains the number of random numbers belonging to the interval $i_{\rm akt}$.
+#The corresponding field element PDF( $i_{\rm akt}$) is then incremented by&nbsp; $1$.&nbsp;
-#The actual WDF values are obtained if, at the end, all field elements $\text{WDF}[i]$ with $0 ≤ i ≤ I-1$ are still divided by $N - Δx$.
+#After $N$ iterations, $\text{PDF}[i_{\rm act}]$ then contains the number of random numbers belonging to the interval $i_{\rm act}$.
+#The actual PDF values are obtained if,&nbsp; at the end,&nbsp; all field elements&nbsp; $\text{PDF}[i]$&nbsp; with&nbsp; $0 ≤ i ≤ I-1$&nbsp; are still divided by&nbsp; $N - Δx$.
+<br clear=all>
-{{Beispiel}}
+{{GraueBox|TEXT=
-From the drawn green arrows in the graph above, one can see:
+$\text{Example 3:}$&nbsp;
-*The value $x_{\rm akt} = 0.07$ leads to the result $i_{\rm akt} =$ (int) ((0.07 + 4.02)/0.04) = (int) $102.25$. Here (int) means an integer conversion after float division &nbsp; ⇒ &nbsp; $i_{\rm akt} = 102$.
+From the drawn green arrows in the graph above,&nbsp;  one can see:
-*The same interval $i_{\rm akt} = 102$ results for $0.06 < x_{\rm akt} < 0.10$, so for example also for $x_{\rm akt} = 0.09$.
+*The value&nbsp; $x_{\rm act} = 0.07$&nbsp; leads to the result&nbsp; $i_{\rm act} =$ (int) ((0.07 + 4.02)/0.04) = (int) $102.25$.
+* Here&nbsp; "(int)"&nbsp; means an integer conversion after float division &nbsp; ⇒ &nbsp; $i_{\rm act} = 102$.
-{{end}}
+*The same interval&nbsp; $i_{\rm act} = 102$&nbsp; results for&nbsp; $0.06 < x_{\rm act} < 0.10$,&nbsp; so for example also for&nbsp; $x_{\rm act} = 0.09$. }}
 ==Exercises for the chapter==

@@ Line 90: / Line 90: @@
 ==PDF definition for discrete random variables==
-For reasons of a uniform representation of all random variables (both discrete-value and continuous-value), it is convenient to define the probability density function also for discrete random variables. Applying the definition equation of the last subchapter to discrete random variables, the PDF takes infinitely large values at some points $x_\mu$ due to the nonvanishingly small probability value and the limit transition $Δx → 0$. Thus, the PDF results in a sum of [[Signal_Representation/General_Description/Gleichsignal_-_Grenzfall_eines_periodischen_Signals#Diracfunktion_im_Frequenzbereich|Dirac functions]]  (resp. ''distributions''):
+For reasons of a uniform representation of all random variables&nbsp; (both discrete-value and continuous-value),&nbsp; it is convenient to define the probability density function also for discrete random variables.
-The weights of these Dirac functions are equal to the probabilities $p_\mu = {\rm Pr}(x = x_\mu$).
+{{BlaueBox|TEXT=
-Here is another note to help classify the different descriptive quantities for discrete and continuous random variables: Probability and probability density function are related in a similar way as in the book [[Signal Representation]]
+Thus,&nbsp; the PDF results in a sum of&nbsp; [[Signal_Representation/General_Description/Gleichsignal_-_Grenzfall_eines_periodischen_Signals#Diracfunktion_im_Frequenzbereich|Dirac delta functions]] &nbsp; &rArr; &nbsp; "distributions":
-*a discrete spectral component of a harmonic oscillation ⇒ line spectrum, and
+:$$f_{x}(x)=\sum_{\mu=1}^{M}p_\mu\cdot {\rm \delta}( x-x_\mu).$$
-[[File:P_ID40__Sto_T_3_1_S3_NEU.png|Signal and PDF of a digital signal]]
+Here is another note to help classify the different descriptive quantities for discrete and continuous random variables: &nbsp; Probability and probability density function are related in a similar way as in the book&nbsp; [[Signal Representation]]
 Thus, the corresponding PDF (values from top to bottom) is:
-$$f_{x}(x) = 0.25 \cdot \delta(x-{\rm 1 \rm V}) + 0.5\cdot \delta(x) + 0.25\cdot \delta (x + 1\rm V).$$
+[[File:P_ID40__Sto_T_3_1_S3_NEU.png|right|frame|Signal and PDF of a ternary signal]]
-{{end}}
+{{GraueBox|TEXT=
 For a more in-depth look at the topic covered here, we recommend the following educational video:
 ==Numerical determination of the PDF==
-You can see here a scheme for the numerical determination of the probability density function.
+You can see here a scheme for the numerical determination of the probability density function:&nbsp;
-Assuming that the random variable $x$ at hand has negligible values outside the range from $x_{\rm min} = -4.02$ to $x_{\rm max} = +4.02$, proceed as follows:
+Assuming that the random variable&nbsp; $x$&nbsp; at hand has negligible values outside the range from&nbsp; $x_{\rm min} = -4.02$&nbsp; to&nbsp; $x_{\rm max} = +4.02$,&nbsp; proceed as follows:
-*Divide the range of values of $x$ into $I$ intervals of equal width $Δx$ and define a field $\text{WDF}[0 : I-1]$. In the above sketch $I = 201$ and accordingly $Δx = 0.04$ is chosen.
+[[File:P_ID175__Sto_T_3_1_S4_ganzneu.png |right|frame| For numerical determination of the PDF]]
-*The random variable $x$ is now called $N$ times in succession, each time checking to which interval $i_{\rm akt}$ the current random variable $x_{\rm akt}$ belongs: $i_{\rm akt} = ({\rm int})((x + x_{\rm max})/Δx)$.
+#Divide the range of&nbsp; $x$-values into&nbsp; $I$&nbsp; intervals of equal width&nbsp; $Δx$&nbsp; and define a field&nbsp; $\text{PDF}[0 : I-1]$.&nbsp; In the sketch&nbsp; $I = 201$&nbsp; and accordingly&nbsp; $Δx = 0.04$&nbsp; is chosen.
-*The corresponding field element WDF( $i_{\rm akt}$) is then incremented by $1$. After $N$ iterations, $\text{WDF}[i_{\rm akt}]$ then contains the number of random numbers belonging to the interval $i_{\rm akt}$.
+#The random variable&nbsp; $x$&nbsp; is now called&nbsp; $N$&nbsp; times in succession,&nbsp; each time checking to which interval&nbsp; $i_{\rm act}$&nbsp; the current random variable&nbsp; $x_{\rm act}$ belongs: <br> &nbsp; &nbsp; $i_{\rm act} = ({\rm int})((x + x_{\rm max})/Δx).$
-*The actual WDF values are obtained if, at the end, all field elements $\text{WDF}[i]$ with $0 ≤ i ≤ I-1$ are still divided by $N - Δx$.
+#The corresponding field element WDF( $i_{\rm akt}$) is then incremented by $1$. After $N$ iterations, $\text{WDF}[i_{\rm akt}]$ then contains the number of random numbers belonging to the interval $i_{\rm akt}$.

@@ Line 61: / Line 61: @@
 $\text{Definition:}$ &nbsp;
 The value of the&nbsp; &raquo;'''probability density function'''&laquo;&nbsp; $f_{x}(x)$&nbsp; at location&nbsp; $x_\mu$&nbsp; is equal to the probability that the instantaneous value of the random variable&nbsp; $x$&nbsp; lies in an&nbsp; (infinitesimally small)&nbsp; interval of width&nbsp; $Δx$&nbsp; around&nbsp; $x_\mu$,&nbsp; divided by&nbsp; $Δx$:
-:$$f_x(x=x_\mu) = \lim_{\rm \Delta \it x \hspace{0.05cm}\to \hspace{0.05cm}\rm 0}\frac{\rm Pr \{\it x_\mu-\rm \Delta \it x/\rm 2 \le \it x \le x_\mu \rm +\rm \Delta \it x/\rm 2\} }{\rm \Delta \it  x}.$$
+:$$f_x(x=x_\mu) = \lim_{\rm \Delta \it x \hspace{0.05cm}\to \hspace{0.05cm}\rm 0}\frac{\rm Pr \{\it x_\mu-\rm \Delta \it x/\rm 2 \le \it x \le x_\mu \rm +\rm \Delta \it x/\rm 2\} }{\rm \Delta \it  x}.$$}}
 This extremely important descriptive variable has the following properties:
-*Although from the time course in&nbsp; [[Stochastic_Signal_theory/Probability_density_function#Properties_of_continuous_random_gr.C3.B6.C3. 9Fen|$\text{Example 1}$]]&nbsp; it can be seen&nbsp; that the most frequent signal components lie at&nbsp; $x = m_1$&nbsp; and the WDF has its largest value here,&nbsp; for a continuous random variable the probability&nbsp; ${\rm Pr}(x = m_1)$,&nbsp; that the instantaneous value is exactly equal to the mean&nbsp; $m_1$&nbsp; is,&nbsp; identically zero.
+*Although from the time course in&nbsp; [[Theory_of_Stochastic_Signals/Probability_Density_Function_(PDF)#Properties_of_continuous_random_variables|$\text{Example 1}$]]&nbsp; it can be seen&nbsp; that the most frequent signal components lie at&nbsp; $x = m_1$&nbsp; and the PDF has its largest value here,&nbsp; for a continuous random variable the probability&nbsp; ${\rm Pr}(x = m_1)$,&nbsp; that the instantaneous value is exactly equal to the mean&nbsp; $m_1$,&nbsp; is identically zero.
-Translated with www.DeepL.com/Translator (free version)
-*Although it can be seen from the time course on the last subchapter that the most frequent signal components are at $x = m_1$ and the probability density function has its largest value here, the probability ${\rm Pr}(x = m_1$) that the instantaneous value is exactly equal to the mean $m_1$ is identical $0$.
+*For the probability that the random variable lies in the range between&nbsp; $x_{\rm u}$&nbsp; and&nbsp; $x_{\rm o}$:
-*For the probability that the random variable lies in the range between $x_u$ and $x_o$:
+:$${\rm Pr}(x_{\rm u} \le  x \le x_{\rm o})= \int_{x_{\rm u} }^{x_{\rm o} }f_{x}(x) \,{\rm d}x.$$
 :$${\rm Pr}(x_{\rm u} \le x \le x_{\rm o}) = \int_{x_{\rm u}}^{x_{\rm o}} f_{x}(x) \,{\rm d}x.$$
-*As an important normalization property, this yields for the area under the PDF with the boundary transitions $x_{\rm u} → \hspace{0.05cm} - \hspace{0.05cm} ∞$ and $x_{\rm o} → +∞:$
+*As an important normalization property,&nbsp; this yields for the area under the PDF with the boundary transitions&nbsp; $x_{\rm u} → \hspace{0.05cm} - \hspace{0.05cm} ∞$&nbsp; and&nbsp; $x_{\rm o} → +∞:$
 :$$\int_{-\infty}^{+\infty} f_{x}(x) \,{\rm d}x = \rm 1.$$
-Note on nomenclature: In the literature, we usually distinguish between the random variable $X$ and its realizations $x ∈ X$. Thus, the above definition equation is
+Thus, the above definition equation is
-$$f_{X}(X=x) = \lim_{{\rm \Delta} x \hspace{0.05cm}\to \hspace{0.05cm} 0}\frac{{\rm Pr} \{ x-{\rm \Delta} x/2 \le X \le x +{\rm \Delta} x/ 2\}}{{\rm \Delta} x}.$$
+:$$f_{X}(X=x) = \lim_{ {\rm \Delta} x \hspace{0.05cm}\to \hspace{0.05cm} 0}\frac{ {\rm Pr} \{ x - {\rm \Delta} x/2 \le X \le x +{\rm \Delta} x/ 2\} }{ {\rm \Delta} x}.$$
-We have largely dispensed with this more precise nomenclature in our learning tutorial so as not to use up two letters for one variable. Lowercase letters (like $x$) often denote signals and uppercase letters (like $X$) denote the associated spectra in our case. Nevertheless, today (2017) we honestly have to admit that the decision of 2001 was not quite fortunate.
+We have largely dispensed with this more precise nomenclature in our learning tutorial&nbsp; $\rm LNTwww$&nbsp; so as not to use up two letters for one quantity.
 ==PDF definition for discrete random variables==

@@ Line 35: / Line 35: @@
 {{BlaueBox|TEXT=
-$\text{Definition:}$&nbsp; By a&nbsp; '''continuous random variable'''&nbsp; is meant a random variable whose possible numerical values are uncountable &nbsp; &rArr; &nbsp; $M \to \infty$.}}
+$\text{Definition:}$&nbsp; By a&nbsp; '''&raquo;value-continuous random variable&laquo;'''&nbsp; is meant a random variable whose possible numerical values are uncountable &nbsp; &rArr; &nbsp; $M \to \infty$. &nbsp; &nbsp; In the following,&nbsp; we will often use the short form "continuous random variable".}}
 Further it shall hold:
-*In the following we denote continuous random variables (mostly) with&nbsp; $x$&nbsp; in contrast to the discrete random variables,&nbsp; which are denoted with&nbsp; $z$&nbsp; as before.
+*In the following we denote continuous random variables&nbsp; (mostly)&nbsp; with&nbsp; $x$&nbsp; in contrast to the discrete random variables,&nbsp; which are denoted with&nbsp; $z$&nbsp; as before.
-*No statement is made here about a possible time discretization, i.e.,&nbsp; continuous random variables can be discrete in time.
+*No statement is made here about a possible time discretization,&nbsp; i.e.,&nbsp; continuous random variables can be discrete in time.
 *Further,&nbsp; we assume for this chapter that there are no statistical bindings between the individual samples&nbsp; $x_ν$,&nbsp; or at least leave them out of consideration.
 {{GraueBox|TEXT=
-$\text{Example}$
+$\text{Example 1:}$
-The following image shows a section of a stochastic noise signal $x(t)$ whose instantaneous value can be taken as a continuous random variable $x$.
+The graphic shows a section of a stochastic noise signal&nbsp; $x(t)$&nbsp; whose instantaneous value can be taken as a continuous random variable&nbsp; $x$.
-From the ''probability density function'' (PDF) shown on the right, it can be seen that instantaneous values around the mean $m_1$ occur most frequently for this example signal. Since there are no statistical bindings between the samples $x_ν$, such a signal is also referred to as '''white noise.'''
+*Since there are no statistical dependencies between the samples $x_ν$,&nbsp; such a signal is also referred to as&nbsp; &raquo;white noise&laquo;.
 {{end}}
 ==Definition of the probability density function==
-For a continuous random variable, the probabilities that it takes on quite specific values are equal to (wie übersetzt man 'identisch 0' besser?) $0$. Therefore, to describe a continuous random variable, we must always refer to the ''probability density function'' - abbreviated PDF.
+For a continuous random variable,&nbsp; the probabilities that it takes on quite specific values are zero.&nbsp; Therefore,&nbsp; to describe a continuous random variable,&nbsp; we must always refer to the&nbsp; "probability density function"&nbsp; $\rm (PDF)$.
-{{Definition}}
+{{BlaueBox|TEXT=
-The value of the '''probability density function''' $f_{x}(x)$ at location $x_\mu$ is equal to the probability that the instantaneous value of the random variable $x$ lies in an (infinitesimally small) interval of width $Δx$ around $x_\mu$ divided by $Δx$:
+$\text{Definition:}$ &nbsp;
-$$f_x(x=x_\mu) = \lim_{\rm \Delta \it x \hspace{0.05cm}\to \hspace{0.05cm}\rm 0}\frac{\rm Pr \{\it x_\mu-\rm \Delta \it x/\rm 2 \le \it x \le x_\mu \rm +\rm \Delta \it x/\rm 2\}}{\rm \Delta \it x}.$$
+The value of the&nbsp; &raquo;'''probability density function'''&laquo;&nbsp; $f_{x}(x)$&nbsp; at location&nbsp; $x_\mu$&nbsp; is equal to the probability that the instantaneous value of the random variable&nbsp; $x$&nbsp; lies in an&nbsp; (infinitesimally small)&nbsp; interval of width&nbsp; $Δx$&nbsp; around&nbsp; $x_\mu$,&nbsp; divided by&nbsp; $Δx$:
 {{end}}
 This extremely important descriptive variable has the following properties:
 *Although it can be seen from the time course on the last subchapter that the most frequent signal components are at $x = m_1$ and the probability density function has its largest value here, the probability ${\rm Pr}(x = m_1$) that the instantaneous value is exactly equal to the mean $m_1$ is identical $0$.
 *For the probability that the random variable lies in the range between $x_u$ and $x_o$: