Theory of Stochastic Signals/Probability Density Function - Revision history

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2022-02-03T13:24:25Z

@@ Line 92: / Line 92: @@
 ==PDF definition for discrete random variables==
-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.
+For reasons of a uniform representation of all random variables&nbsp; $($both value-discrete and value-continuous$)$,&nbsp; it is convenient to define the probability density function also for value-discrete random variables.
 {{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 value-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|$\text{Dirac delta functions}$]] &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|&raquo;Dirac delta functions&laquo;]] &nbsp; &rArr; &nbsp; &raquo;distributions&laquo;:
 :$$f_{x}(x)=\sum_{\mu=1}^{M}p_\mu\cdot {\rm \delta}( x-x_\mu).$$
@@ Line 104: / Line 104: @@
-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"]]
+Here is another note to help classify the different descriptive quantities for value-discrete and value-continuous random variables: &nbsp; Probability and probability density function are related in a similar way as in the book&nbsp; [[Signal_Representation|&raquo;Signal Representation&laquo;]]
 *a discrete spectral component of a harmonic oscillation ⇒ line spectrum,&nbsp; and
 *a continuous spectrum of an energy-limited&nbsp; (pulse-shaped)&nbsp; signal.
 [[File:P_ID40__Sto_T_3_1_S3_NEU.png|right|frame|Signal and PDF of a ternary signal]]
-{{GraueBox|TEXT=
+$\text{Example 2:}$&nbsp; Below you can see a section
-$\text{Example 2:}$&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; $(\pm 1 \ \rm V)$.
-Thus,&nbsp; the corresponding&nbsp; PDF&nbsp; (values from top to bottom)&nbsp; is:
+Thus,&nbsp; the corresponding&nbsp; PDF&nbsp; $($values from top to bottom$)$:
-:$$f_{x}(x) = 0.25 \cdot \delta(x - {\rm 1 V})+ 0.5\cdot \delta(x) + 0.25\cdot \delta (x + 1\rm V).$$}}
+:$$f_{x}(x) = 0.25 \cdot \delta(x - {\rm 1 V})+ 0.5\cdot \delta(x) + 0.25\cdot \delta (x + 1\rm V).$$
-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)|&raquo;Wahrscheinlichkeit und WDF&laquo;]] &nbsp; &rArr; &nbsp; &raquo;Probability and probability density function&laquo;}}
 ==Numerical determination of the PDF==
 <br>
-You can see here a scheme for the numerical determination of the probability density function:&nbsp;
+You can see here a scheme for the numerical determination of the probability density function.&nbsp; 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:
-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: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; 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( $i_{\rm act}$) 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 $i_{\rm act}$.
+#After $N$ iterations, 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; $N \cdot Δx$.
+#The actual PDF values are obtained if,&nbsp; at the end,&nbsp; all field elements&nbsp; PDF$[i]$&nbsp; with&nbsp; $0 ≤ i ≤ I-1$&nbsp; are still divided by&nbsp; $N \cdot Δx$.
 <br clear=all>
 {{GraueBox|TEXT=
@@ Line 141: / Line 143: @@
 From the drawn green arrows in the graph above,&nbsp;  one can see:
 *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; $i_{\rm act} = 102$.
+* Here&nbsp; &raquo;(int)&laquo;&nbsp; means an integer conversion after float division &nbsp; ⇒ &nbsp; $i_{\rm act} = 102$.
 *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; $x_{\rm act} = 0.09$. }}

@@ Line 10: / Line 10: @@
 <br>
 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.
@@ Line 29: / Line 30: @@
-&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.
+&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&nbsp; $4$&nbsp; and&nbsp; $5$.
 ==Properties of continuous random variables==
 In the second chapter it was shown that the amplitude distribution of a discrete random variable is completely determined by its&nbsp; $M$&nbsp; occurrence probabilities,&nbsp; where the level number&nbsp; $M$&nbsp; usually has a finite value.
 {{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; In the following,&nbsp; we will often use the short form&nbsp; &raquo;continuous random variable&laquo;.}}
 Further it shall hold:
-*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.
+#We&nbsp; (mostly)&nbsp; denote value-continuous random variables with&nbsp; $x$&nbsp; in contrast to the value-discrete random variables,&nbsp; which are denoted with&nbsp; $z$&nbsp; as before.
-*No statement is made here about a possible time discretization,&nbsp; 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; value-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.
+#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=
 $\text{Example 1:}$
 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&nbsp; &raquo;probability density function&raquo; &nbsp; $\rm (PDF)$&nbsp; shown on the right,&nbsp; it can be seen that instantaneous values around the mean&nbsp; $m_1$&nbsp; occur most frequently for this example signal.
+*From the&nbsp; &raquo;probability density function&raquo; &nbsp; $\rm (PDF)$&nbsp; shown on the right,&nbsp; it can be seen that instantaneous values around the mean&nbsp; $m_1$&nbsp; occur most frequently for any exemplary signal.
-*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 often also referred to as&nbsp; &raquo;'''white noise'''&laquo;.}}
 ==Definition of the probability density function==
-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; $\rm (PDF)$.
+<br>
 {{BlaueBox|TEXT=
 $\text{Definition:}$ &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; $Δx$:
+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; $Δx$:
 :$$f_x(x=x_\mu) = \lim_{\rm \Delta \it x \hspace{0.05cm}\to \hspace{0.05cm}\rm 0}\frac{\rm Pr \{\it x_\mu-\rm \Delta \it x/\rm 2 \le \it x \le x_\mu \rm +\rm \Delta \it x/\rm 2\} }{\rm \Delta \it  x}.$$}}
@@ Line 67: / Line 70: @@
 This extremely important descriptive variable has the following properties:
-*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.
+*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 value-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.
-*For the probability that the random variable lies in the range between&nbsp; $x_{\rm u}$&nbsp; and&nbsp; $x_{\rm o}$:
+*The probability that the random variable lies in the range between&nbsp; $x_{\rm u}$&nbsp; and&nbsp; $x_{\rm o}$:
 :$${\rm Pr}(x_{\rm u} \le  x \le x_{\rm o})= \int_{x_{\rm u} }^{x_{\rm o} }f_{x}(x) \,{\rm d}x.$$
@@ Line 76: / Line 78: @@
 :$$\int_{-\infty}^{+\infty} f_{x}(x) \,{\rm d}x = \rm 1.$$
-*The corresponding equation for discrete-value,&nbsp; $M$-level random variables states that the sum over the&nbsp; $M$&nbsp; occurrence probabilities gives the value&nbsp; $1$.
+*The corresponding equation for value-discrete,&nbsp; $M$-level random variables states that the sum over the&nbsp; $M$&nbsp; occurrence probabilities gives the value&nbsp; $1$.
 {{BlaueBox|TEXT=
 $\text{Note on nomenclature:}$&nbsp;
-Thus, the above definition equation is
+In the literature,&nbsp; a distinction is often made between the random variable&nbsp; $X$&nbsp; and its realizations&nbsp; $x ∈ X$.&nbsp; Thus,&nbsp; the above definition equation is
 :$$f_{X}(X=x) = \lim_{ {\rm \Delta} x \hspace{0.05cm}\to \hspace{0.05cm} 0}\frac{ {\rm Pr} \{ x - {\rm \Delta} x/2 \le X \le x +{\rm \Delta} x/ 2\} }{ {\rm \Delta} x}.$$
-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.
+We have largely dispensed with this more precise nomenclature in our&nbsp; $\rm LNTwww$&nbsp; so as not to use up two letters for one quantity.
-*Lowercase letters&nbsp; $($as&nbsp; $x)$&nbsp; often denote signals and uppercase letters&nbsp; $($as&nbsp; $X)$&nbsp; the associated spectra in our case.
+#Lowercase letters&nbsp; $($as&nbsp; $x)$&nbsp; often denote signals and uppercase letters&nbsp; $($as&nbsp; $X)$&nbsp; the associated spectra in our case.
-*Nevertheless,&nbsp; today (2017)&nbsp; we have to honestly admit that the 2001 decision was not entirely happy.}}
+#Nevertheless,&nbsp; today (2017)&nbsp; we have to honestly admit that the 2001 decision was not entirely happy.}}
 ==PDF definition for discrete random variables==
@@ Line 125: / Line 126: @@
 ==Numerical determination of the PDF==
 You can see here a scheme for the numerical determination of the probability density function:&nbsp;
@@ Line 143: / Line 145: @@
 ==Exercises for the chapter==
+<br>
 [[Aufgaben:Exercise_3.1:_cos²-PDF_and_PDF_with_Dirac_Functions|Exercise 3.1: Cosine-square PDF and PDF with Dirac Functions]]

@@ Line 42: / Line 42: @@
 *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,&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 '''KORREKTUR: bindings or ties?''' between the individual samples&nbsp; $x_ν$,&nbsp; or at least leave them out of consideration.
+*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.

@@ Line 42: / Line 42: @@
 *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,&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.
+*Further,&nbsp; we assume for this chapter that there are no statistical bindings '''KORREKTUR: bindings or ties?''' between the individual samples&nbsp; $x_ν$,&nbsp; or at least leave them out of consideration.

@@ Line 95: / 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|$\text{Dirac delta functions}$]] &nbsp; &rArr; &nbsp; "distributions":

← Older revision		Revision as of 19:10, 13 December 2022
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	{{BlaueBox\|TEXT=		{{BlaueBox\|TEXT=
−	$\text{Definition:}$  By a  »'''~~»~~value-continuous random variable~~«~~'''«  is meant a random variable whose possible numerical values are uncountable   ⇒   $M \to \infty$.     In the following,  we will often use the short form "continuous random variable".}}	+	$\text{Definition:}$  By a  »'''value-continuous random variable'''«  is meant a random variable whose possible numerical values are uncountable   ⇒   $M \to \infty$.     In the following,  we will often use the short form "continuous random variable".}}

@@ Line 9: / Line 9: @@
 == # 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 29: / 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 36: / 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;'''&raquo;value-continuous random variable&laquo;'''&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 97: / Line 97: @@
 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 103: / 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 122: / 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 144: / 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]]