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

Maintenance script: Add German interlanguage link

2026-03-16T12:28:55Z

Add German interlanguage link

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	{{Display}}		{{Display}}
			[[de:Stochastische_Signaltheorie/Wahrscheinlichkeitsdichtefunktion_(WDF)]]

Hwang at 09:32, 12 April 2023

2023-04-12T09:32:13Z

Noah at 10:30, 13 January 2022

2022-01-13T10:30:21Z

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	\|Untermenü=Continuous Random Variables		\|Untermenü=Continuous Random Variables
	\|Vorherige Seite=Generation of Discrete Random Variables		\|Vorherige Seite=Generation of Discrete Random Variables
	\|Nächste Seite=~~Verteilungsfunktion (VTF)~~		\|Nächste Seite=Cumulative Distribution Function
	}}		}}

Guenter at 16:14, 31 December 2021

2021-12-31T16:14:00Z

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

Guenter at 16:10, 31 December 2021

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	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,  we recommend the following  (German language)  learning video:
	~~#####~~
			:[[Wahrscheinlichkeit_und_WDF_(Lernvideo)\|Wahrscheinlichkeit und WDF]]   ⇒   "Probability and probability density function"

	==Numerical determination of the PDF==		==Numerical determination of the PDF==

Guenter at 16:04, 31 December 2021

2021-12-31T16:04:47Z

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	Applying the definition equation of the last section 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&nbsp $Δx → 0$.		Applying the definition equation of the last section 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&nbsp $Δx → 0$.

	Thus,  the PDF results in a sum of  [[Signal_Representation/~~General_Description/Gleichsignal_~~-~~_Grenzfall_eines_periodischen_Signals~~#~~Diracfunktion_im_Frequenzbereich~~\|Dirac delta functions]]   ⇒   "distributions":		Thus,  the PDF results in a sum of  [[Signal_Representation/Direct_Current_Signal_-_Limit_Case_of_a_Periodic_Signal#Dirac_.28delta.29_function_in_frequency_domain\|Dirac delta functions]]   ⇒   "distributions":
	:$$f_{x}(x)=\sum_{\mu=1}^{M}p_\mu\cdot {\rm \delta}( x-x_\mu).$$		:$$f_{x}(x)=\sum_{\mu=1}^{M}p_\mu\cdot {\rm \delta}( x-x_\mu).$$

Guenter at 16:00, 31 December 2021

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	*Since there are no statistical dependencies between the samples $x_ν$,  such a signal is also referred to as  »white noise«.		*Since there are no statistical dependencies between the samples $x_ν$,  such a signal is also referred to as  »white noise«.}}
	~~{{end~~}}

	==Definition of the probability density function==		==Definition of the probability density function==

Guenter at 09:54, 31 December 2021

2021-12-31T09:54:05Z

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	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  $x$  at hand has negligible values outside the range from  $x_{\rm min} = -4.02$  to  $x_{\rm max} = +4.02$,  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  $x$-values into  $I$  intervals of equal width  $Δx$  and define a field  $\text{PDF}[0 : I-1]$.  In the sketch  $I = 201$  and accordingly  $Δx = 0.04$  is chosen.		#Divide the range of  $x$-values into  $I$  intervals of equal width  $Δx$  and define a field  $\text{PDF}[0 : I-1]$.  In the sketch  $I = 201$  and accordingly  $Δx = 0.04$  is chosen.
	#The random variable  $x$  is now called  $N$  times in succession,  each time checking to which interval  $i_{\rm act}$  the current random variable  $x_{\rm act}$ belongs: <br>     $i_{\rm act} = ({\rm int})((x + x_{\rm max})/Δx).$		#The random variable  $x$  is now called  $N$  times in succession,  each time checking to which interval  $i_{\rm act}$  the current random variable  $x_{\rm act}$ belongs: <br>     $i_{\rm act} = ({\rm int})((x + x_{\rm max})/Δx).$
	#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  $1$.
	#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,  at the end,  all field elements  $\text{PDF}[i]$  with  $0 ≤ i ≤ I-1$  are still divided by  $N - Δx$.
			<br clear=all>
	{~~{Beispiel}~~}		{{GraueBox\|TEXT=
	From the drawn green arrows in the graph above, one can see:		$\text{Example 3:}$
	*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   ⇒   $i_{\rm ~~akt~~} = 102$.		From the drawn green arrows in the graph above,  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  $x_{\rm act} = 0.07$  leads to the result  $i_{\rm act} =$ (int) ((0.07 + 4.02)/0.04) = (int) $102.25$.
			* Here  "(int)"  means an integer conversion after float division   ⇒   $i_{\rm act} = 102$.
	~~{{end~~}}		*The same interval  $i_{\rm act} = 102$  results for  $0.06 < x_{\rm act} < 0.10$,  so for example also for  $x_{\rm act} = 0.09$. }}

	==Exercises for the chapter==		==Exercises for the chapter==

Guenter at 17:23, 30 December 2021

2021-12-30T17:23:37Z

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	==PDF definition for discrete random variables==		==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  (both discrete-value and continuous-value),  it is convenient to define the probability density function also for discrete random variables.
	~~$$f_{x}(x)=\sum_{\mu=1}^{M}p_\mu\cdot {\rm \delta}( x-x_\mu).$$~~

	~~The weights~~ of ~~these Dirac functions are equal~~ to the ~~probabilities~~ $~~p_\mu = {\rm Pr}(x =~~ x_\mu$).		{{BlaueBox\|TEXT=
			$\text{Definition:}$
			Applying the definition equation of the last section 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&nbsp $Δx → 0$.

	~~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,  the PDF results in a sum of  [[Signal_Representation/General_Description/Gleichsignal_-_Grenzfall_eines_periodischen_Signals#Diracfunktion_im_Frequenzbereich\|Dirac delta functions]]   ⇒   "distributions":
	*a discrete spectral component of a harmonic oscillation ⇒ line spectrum, and		:$$f_{x}(x)=\sum_{\mu=1}^{M}p_\mu\cdot {\rm \delta}( x-x_\mu).$$
	*a continuous spectrum of an energy-limited (~~pulse~~-~~shaped~~) ~~signal~~.

			The weights of these Dirac delta functions are equal to the probabilities  $p_\mu = {\rm Pr}(x = x_\mu$).}}

	~~{{Beispiel}}~~
	~~Below is a section of a rectangular signal with three possible values, where the signal value $0 \ \rm V$ occurs twice as often as the outer signal values ($\pm 1 \ \rm V$).~~

	[[~~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:   Probability and probability density function are related in a similar way as in the book  [[Signal Representation]]
			*a discrete spectral component of a harmonic oscillation ⇒ line spectrum,  and
			*a continuous spectrum of an energy-limited  (pulse-shaped)  signal.

	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=
			$\text{Example 2:}$  Below is a section
			*of a rectangular signal with three possible values,
			*where the signal value  $0 \ \rm V$&nbsp occurs twice as often as the outer signal values  $(\pm 1 \ \rm V)$.



			Thus,  the corresponding  PDF  (values from top to bottom)  is:
			:$$f_{x}(x) = 0.25 \cdot \delta(x - {\rm 1 V})+ 0.5\cdot \delta(x) + 0.25\cdot \delta (x + 1\rm V).$$}}


	For a more in-depth look at the topic covered here, we recommend the following educational video:		For a more in-depth look at the topic covered here, we recommend the following educational video:
			#####

	==Numerical determination of the PDF==		==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:

	~~[[File~~:~~P_ID175__Sto_T_3_1_S4_ganzneu.png \| For numerical determination of the WDF]]~~

	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  $x$  at hand has negligible values outside the range from  $x_{\rm min} = -4.02$  to  $x_{\rm max} = +4.02$,  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  $x$-values into  $I$  intervals of equal width  $Δx$  and define a field  $\text{PDF}[0 : I-1]$.  In the sketch  $I = 201$  and accordingly  $Δx = 0.04$  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  $x$  is now called  $N$  times in succession,  each time checking to which interval  $i_{\rm act}$  the current random variable  $x_{\rm act}$ belongs: <br>     $i_{\rm act} = ({\rm int})((x + x_{\rm max})/Δx).$
	*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 $i_{\rm akt}$.
			#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 $N - Δx$.

Guenter at 16:26, 30 December 2021

2021-12-30T16:26:31Z

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	$\text{Definition:}$		$\text{Definition:}$
	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  $Δx$:		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  $Δx$:
	:$$f_x(x=x_\mu) = \lim_{\rm \Delta \it x \hspace{0.05cm}\to \hspace{0.05cm}\rm 0}\frac{\rm Pr \{\it x_\mu-\rm \Delta \it x/\rm 2 \le \it x \le x_\mu \rm +\rm \Delta \it x/\rm 2\} }{\rm \Delta \it x}.$$		:$$f_x(x=x_\mu) = \lim_{\rm \Delta \it x \hspace{0.05cm}\to \hspace{0.05cm}\rm 0}\frac{\rm Pr \{\it x_\mu-\rm \Delta \it x/\rm 2 \le \it x \le x_\mu \rm +\rm \Delta \it x/\rm 2\} }{\rm \Delta \it x}.$$}}
	~~{{end~~}}


	This extremely important descriptive variable has the following properties:		This extremely important descriptive variable has the following properties:

	*Although from the time course in  [[~~Stochastic_Signal_theory~~/~~Probability_density_function~~#~~Properties_of_continuous_random_gr.C3.B6.C3. 9Fen~~\|$\text{Example 1}$]]  it can be seen  that the most frequent signal components lie at  $x = m_1$  and the ~~WDF~~ has its largest value here,  for a continuous random variable the probability  ${\rm Pr}(x = m_1)$,  that the instantaneous value is exactly equal to the mean  $m_1$  is~~, ~~ identically zero.		*Although from the time course in  [[Theory_of_Stochastic_Signals/Probability_Density_Function_(PDF)#Properties_of_continuous_random_variables\|$\text{Example 1}$]]  it can be seen  that the most frequent signal components lie at  $x = m_1$  and the PDF has its largest value here,  for a continuous random variable the probability  ${\rm Pr}(x = m_1)$,  that the instantaneous value is exactly equal to the mean  $m_1$,  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  $x_{\rm u}$  and  $x_{\rm o}$:
	*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,  this yields for the area under the PDF with the boundary transitions  $x_{\rm u} → \hspace{0.05cm} - \hspace{0.05cm} ∞$  and  $x_{\rm o} → +∞:$
	:$$\int_{-\infty}^{+\infty} f_{x}(x) \,{\rm d}x = \rm 1.$$		:$$\int_{-\infty}^{+\infty} f_{x}(x) \,{\rm d}x = \rm 1.$$
	*The corresponding equation for discrete-value, $M$-level random variables states that the sum over the $M$ occurrence probabilities gives the value $1$.

			*The corresponding equation for discrete-value,  $M$-level random variables states that the sum over the  $M$  occurrence probabilities gives the value  $1$.


			{{BlaueBox\|TEXT=
			$\text{Note on nomenclature:}$
			In the literature,  a distinction is often made between the random variable  $X$  and its realizations  $x ∈ X$.

	~~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  $\rm LNTwww$  so as not to use up two letters for one quantity.
			*Lowercase letters  $($as  $x)$  often denote signals and uppercase letters  $($as  $X)$  the associated spectra in our case.
			*Nevertheless,  today (2017)  we have to honestly admit that the 2001 decision was not entirely happy.}}

	==PDF definition for discrete random variables==		==PDF definition for discrete random variables==

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	{{Header		{{Header
	\|Untermenü=Continuous Random Variables		\|Untermenü=Continuous Random Variables
	\|Vorherige Seite=Generation of Discrete Random Variables		\|Vorherige Seite=Generation of Discrete Random Variables
	\|Nächste Seite=Cumulative Distribution Function		\|Nächste Seite=Cumulative Distribution Function
	}}		}}

	== # OVERVIEW OF THE THIRD MAIN CHAPTER # ==		== # OVERVIEW OF THE THIRD MAIN CHAPTER # ==
	<br>		<br>
	We consider here  '''continuous random variables''',  i.e.,  random variables which can assume infinitely many different values,  at least in certain ranges of real numbers.		We consider here  »'''continuous random variables'''«,  i.e.,  random variables which can assume infinitely many different values,  at least in certain ranges of real numbers.
	*Their applications in information and communication technology are manifold.		*Their applications in information and communication technology are manifold.
	*They are used,  among other things,  for the simulation of noise signals and for the description of fading effects.		*They are used,  among other things,  for the simulation of noise signals and for the description of fading effects.


	We restrict ourselves at first to the statistical description of the  '''amplitude distribution'''.  In detail,  the following are treated:		We restrict ourselves at first to the statistical description of the  »'''amplitude distribution'''«.  In detail,  the following are treated:

	*The relationship between  »probability density function«  $\rm (PDF)$  and  »cumulative distribution function«  $\rm (CDF)$;		*The relationship between  »probability density function«  $\rm (PDF)$  and  »cumulative distribution function«  $\rm (CDF)$;
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	'''Inner statistical dependencies'''  of the underlying processes  '''are not considered here'''.  For this,  we refer to the following main chapters 4 and 5.		»'''Inner statistical dependencies'''«  of the underlying processes  '''are not considered here'''.  For this,  we refer to the following main chapters 4 and 5.


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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".}}


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	{{BlaueBox\|TEXT=		{{BlaueBox\|TEXT=
	$\text{Definition:}$		$\text{Definition:}$
	Applying the definition equation of the last section 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&nbsp $Δx → 0$.		Applying the definition equation of the last section 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/Direct_Current_Signal_-_Limit_Case_of_a_Periodic_Signal#Dirac_.28delta.29_function_in_frequency_domain\|Dirac delta functions]]   ⇒   "distributions":		Thus,  the PDF results in a sum of  [[Signal_Representation/Direct_Current_Signal_-_Limit_Case_of_a_Periodic_Signal#Dirac_.28delta.29_function_in_frequency_domain\|$\text{Dirac delta functions}$]]   ⇒   "distributions":
	:$$f_{x}(x)=\sum_{\mu=1}^{M}p_\mu\cdot {\rm \delta}( x-x_\mu).$$		:$$f_{x}(x)=\sum_{\mu=1}^{M}p_\mu\cdot {\rm \delta}( x-x_\mu).$$

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	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]]		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\|"Signal Representation"]]
	*a discrete spectral component of a harmonic oscillation ⇒ line spectrum,  and		*a discrete spectral component of a harmonic oscillation ⇒ line spectrum,  and
	*a continuous spectrum of an energy-limited  (pulse-shaped)  signal.		*a continuous spectrum of an energy-limited  (pulse-shaped)  signal.
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	$\text{Example 2:}$  Below is a section		$\text{Example 2:}$  Below is a section
	*of a rectangular signal with three possible values,		*of a rectangular signal with three possible values,
	*where the signal value  $0 \ \rm V$&nbsp occurs twice as often as the outer signal values  $(\pm 1 \ \rm V)$.		*where the signal value  $0 \ \rm V$  occurs twice as often as the outer signal values  $(\pm 1 \ \rm V)$.


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	For a more in-depth look at the topic covered here,  we recommend the following  (German language)  learning video:		For a more in-depth look at the topic covered here,  we recommend the following  (German language)  learning video:

	:[[Wahrscheinlichkeit_und_WDF_(Lernvideo)\|Wahrscheinlichkeit und WDF]]   ⇒   "Probability and probability density function"		:[[Wahrscheinlichkeit_und_WDF_(Lernvideo)\|"Wahrscheinlichkeit und WDF"]]   ⇒   "Probability and probability density function"

	==Numerical determination of the PDF==		==Numerical determination of the PDF==
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	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  $x$  at hand has negligible values outside the range from  $x_{\rm min} = -4.02$  to  $x_{\rm max} = +4.02$,  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  $x$-values into  $I$  intervals of equal width  $Δx$  and define a field  $\text{PDF}[0 : I-1]$.  In the sketch  $I = 201$  and accordingly  $Δx = 0.04$  is chosen.		#Divide the range of  $x$-values into  $I$  intervals of equal width  $Δx$  and define a field  $\text{PDF}[0 : I-1]$.  In the sketch  $I = 201$  and accordingly  $Δx = 0.04$  is chosen.
	#The random variable  $x$  is now called  $N$  times in succession,  each time checking to which interval  $i_{\rm act}$  the current random variable  $x_{\rm act}$ belongs: <br>     $i_{\rm act} = ({\rm int})((x + x_{\rm max})/Δx).$		#The random variable  $x$  is now called  $N$  times in succession,  each time checking to which interval  $i_{\rm act}$  the current random variable  $x_{\rm act}$ belongs: <br>     $i_{\rm act} = ({\rm int})((x + x_{\rm max})/Δx).$
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	==Exercises for the chapter==		==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]]		[[Aufgaben:Exercise_3.1Z:_Triangular_PDF\|Exercise 3.1Z: Triangular PDF]]