Theory of Stochastic Signals/Exponentially Distributed Random Variables - Revision history

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	#  This forms from the uniformly distributed random variable  $u$  with PDF  $f_{u}(u)$  a one-sided exponentially distributed random variable  $x$  with PDF  $f_{x}(x)$:		#  This forms from the uniformly distributed random variable  $u$  with PDF  $f_{u}(u)$  a one-sided exponentially distributed random variable  $x$  with PDF  $f_{x}(x)$:

	::$$f_{u}(u)= \left\{ \begin{array}{*{2}{c} } 1 & \rm if\hspace{0.3cm} 0 < {\it u} < 1,\\ 0.5 & \rm if\hspace{0.3cm} {\it u} = 0, {\it u} = 1,\\ 0 & \rm else, \\ \end{array} \right. \hspace{0.5cm}\Rightarrow \hspace{0.5cm}		::$$f_{u}(u)= \left\{ \begin{array}{{2}{c} } 1 & \rm if\hspace{0.3cm} 0 < {\it u} < 1,\\ 0.5 & \rm if\hspace{0.3cm} {\it u} = 0, {\it u} = 1,\\ 0 & \rm else, \\ \end{array} \right. \hspace{0.5cm}\Rightarrow \hspace{0.5cm}f_{x}(x)= \left\{ \begin{array}{{2}{c} } \lambda\cdot\rm e^{\it -\lambda\hspace{0.03cm} \cdot \hspace{0.03cm} x} & \rm if\hspace{0.3cm} {\it x} > 0,\\ \lambda/2 & \rm if\hspace{0.3cm} {\it x} = 0 ,\\ 0 & \rm else\hspace{0.3cm} {\it x} < 0. \\ \end{array} \right.$$}}
	f_{x}(x)= \left\{ \begin{array}{*{2}{c} } \lambda\cdot\rm e^{\it -\lambda\hspace{0.03cm} \cdot \hspace{0.03cm} x} & \rm if\hspace{0.3cm} {\it x} > 0,\\ \lambda/2 & \rm if\hspace{0.3cm} {\it x} = 0 ,\\ 0 & \rm else\hspace{0.3cm} {\it x} < 0. \\ \end{array} \right.$$}}

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	{{Display}}		{{Display}}
			[[de:Stochastische_Signaltheorie/Exponentialverteilte_Zufallsgrößen]]

Hwang at 20:22, 20 December 2022

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	* The  $k$–th order moments  $m_k$  of the Laplace distribution agree with those of the exponential distribution for even  $k$.		* The  $k$–th order moments  $m_k$  of the Laplace distribution agree with those of the exponential distribution for even  $k$.
	* For odd  $k$,  the  (symmetric)  Laplace distribution always yields  $m_k= 0$.		* For odd  $k$,  the  (symmetric)  Laplace distribution always yields  $m_k= 0$.
	*For generation the Laplace distribution,  one uses a between  $±1$  uniformly distributed random variable  $v$  $($where  $v = 0$  must be excluded$)$  and the transformation characteristic curve		*For generation of the Laplace distribution,  one uses a between  $±1$  uniformly distributed random variable  $v$  $($where  $v = 0$  must be excluded$)$  and the transformation characteristic curve
	:$$x=\frac{{\rm sign}(v)}{\lambda}\cdot \rm ln(\it v \rm ).$$		:$$x=\frac{{\rm sign}(v)}{\lambda}\cdot \rm ln(\it v \rm ).$$

Hwang at 20:19, 20 December 2022

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	::$$x = g_1(u)= \frac{1}{\lambda} \cdot {\rm ln} \left(\frac{1}{1 - u} \right) .$$		::$$x = g_1(u)= \frac{1}{\lambda} \cdot {\rm ln} \left(\frac{1}{1 - u} \right) .$$

	*In a computer implementation,  however,  ensure that the critical value  $1$  is excluded for the uniformly distributed input variable  $u$~~ ~~		*In a computer implementation,  however,  ensure that the critical value  $1$  is excluded for the uniformly distributed input variable  $u$.
	*This,  however,  has (almost) no effect on the final result. }}		*This,  however,  has (almost) no effect on the final result. }}

Hwang at 20:12, 20 December 2022

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	*However,  the above equation is only valid under the condition that the derivative  $g\hspace{0.03cm}'(u) \ne 0$.		*However,  the above equation is only valid under the condition that the derivative  $g\hspace{0.03cm}'(u) \ne 0$.
	*For a characteristic with horizontal sections  $(g\hspace{0.05cm}'(u) = 0)$:  Additional Dirac delta functions appear in the PDF if the input variable has components in these ranges.		*For a characteristic with horizontal sections  $(g\hspace{0.05cm}'(u) = 0)$:  Additional Dirac delta functions appear in the PDF if the input variable has components in these ranges.
	*The weights of these Dirac functions are equal to the probabilities that the input variable lies in these ranges.		*The weights of these Dirac delta functions are equal to the probabilities that the input variable lies in these ranges.

Hwang at 20:10, 20 December 2022

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	{{BlaueBox\|TEXT=		{{BlaueBox\|TEXT=
	$\text{Procedure:}$  If a continuous valued random variable  $u$  possesses the PDF  $f_{u}(u)$,  then the probability density function of the random variable transformed at the nonlinear characteristic  $x = g(u)$  holds:		$\text{Procedure:}$  If a continuous-valued random variable  $u$  possesses the PDF  $f_{u}(u)$,  then the probability density function of the random variable transformed at the nonlinear characteristic  $x = g(u)$  holds:
	:$$f_{x}(x)=\frac{f_u(u)}{\mid g\hspace{0.05cm}'(u)\mid}\Bigg \vert_{\hspace{0.1cm} u=h(x)}.$$		:$$f_{x}(x)=\frac{f_u(u)}{\mid g\hspace{0.05cm}'(u)\mid}\Bigg \vert_{\hspace{0.1cm} u=h(x)}.$$

Hwang at 20:03, 20 December 2022

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	The left sketch shows the  "probability density function"  $\rm (PDF)$  of such an exponentially distributed random variable  $x$.  To be emphasized:		The left sketch shows the  "probability density function"  $\rm (PDF)$  of such an exponentially distributed random variable  $x$.  To be emphasized:
	#The larger the distribution parameter  $λ$  ~~is,~~  the steeper the decay ~~occurs~~.		#The larger the distribution parameter  $λ$,    the steeper the decay.
	#By definition  $f_{x}(0) = λ/2$,  i.e. the mean of left-hand limit  $(0)$  and right-hand limit  $(\lambda)$.		#By definition  $f_{x}(0) = λ/2$,  i.e. the mean of left-hand limit  $(0)$  and right-hand limit  $(\lambda)$.

Hwang at 20:02, 20 December 2022

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	[[File: P_ID72__Sto_T_3_6_S1_neu.png \|right\|frame\| PDF and CDF of an exponentially distributed random variable]]		[[File: P_ID72__Sto_T_3_6_S1_neu.png \|right\|frame\| PDF and CDF of an exponentially distributed random variable]]

	The left sketch shows the  "probability density function"  $\rm (PDF)$  of such an exponentially distributed random variable  $x$.  ~~Highlight~~:		The left sketch shows the  "probability density function"  $\rm (PDF)$  of such an exponentially distributed random variable  $x$.  To be emphasized:
	#The larger the distribution parameter  $λ$  is,  the steeper the decay occurs.		#The larger the distribution parameter  $λ$  is,  the steeper the decay occurs.
	#By definition  $f_{x}(0) = λ/2$,  i.e. the mean of left-hand limit  $(0)$  and right-hand limit  $(\lambda)$.		#By definition  $f_{x}(0) = λ/2$,  i.e. the mean of left-hand limit  $(0)$  and right-hand limit  $(\lambda)$.

Hwang at 20:01, 20 December 2022

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	{{BlaueBox\|TEXT=		{{BlaueBox\|TEXT=
	$\text{Definition:}$		$\text{Definition:}$
	A continuous random variable  $x$  is called  (one-sided)  '''exponentially distributed'''  if it can take only non–negative values and the probability density function has the following shape for  $x>0$:		A continuous random variable  $x$  is called  (one-sided)  »'''exponentially distributed'''«  if it can take only non–negative values and the probability density function has the following shape for  $x>0$:
	:$$f_x(x)=\it \lambda\cdot\rm e^{\it -\lambda \hspace{0.05cm}\cdot \hspace{0.03cm} x}.$$}}		:$$f_x(x)=\it \lambda\cdot\rm e^{\it -\lambda \hspace{0.05cm}\cdot \hspace{0.03cm} x}.$$}}

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	$\text{Example 1:}$  The exponential distribution has great importance for reliability studies,  and the term  "lifetime distribution"  is also commonly used in this context.		$\text{Example 1:}$  The exponential distribution has great importance for reliability studies,  and the term  "lifetime distribution"  is also commonly used in this context.
	#In these applications,  the random variable is often the time  $t$  that elapses before a component fails.		#In these applications,  the random variable is often the time  $t$  that elapses before a component fails.
	#Furthermore,  it should be noted that the exponential distribution is closely related to the  [[Theory_of_Stochastic_Signals/Poisson_Distribution\|Poisson distribution]]. }}		#Furthermore,  it should be noted that the exponential distribution is closely related to the  [[Theory_of_Stochastic_Signals/Poisson_Distribution\|$\text{Poisson distribution}$]]. }}

	==Transformation of random variables==		==Transformation of random variables==
	<br>		<br>
	To generate such an exponentially distributed random variable on a digital computer,  you can use e.g. a  '''nonlinear transformation'''.  The underlying principle is first stated here in general terms.		To generate such an exponentially distributed random variable on a digital computer,  you can use e.g. a  »'''nonlinear transformation'''«.  The underlying principle is first stated here in general terms.

	{{BlaueBox\|TEXT=		{{BlaueBox\|TEXT=
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	:$$x=g_1(u) =\frac{1}{\lambda}\cdot \rm ln \ (\frac{1}{1-\it u}).$$		:$$x=g_1(u) =\frac{1}{\lambda}\cdot \rm ln \ (\frac{1}{1-\it u}).$$

	*It can be shown that by this characteristic  $x=g_1(u)$  a one-sided exponentially distributed random variable  $x$  with the following PDF arises  <br>(derivation see [[Theory_of_Stochastic_Signals/Exponentially_Distributed_Random_Variables#Derivation_of_the_corresponding_transformation_characteristic\|next section]]):		*It can be shown that by this characteristic  $x=g_1(u)$  a one-sided exponentially distributed random variable  $x$  with the following PDF arises  <br>(derivation see [[Theory_of_Stochastic_Signals/Exponentially_Distributed_Random_Variables#Derivation_of_the_corresponding_transformation_characteristic\|"next section"]]):
	:$$f_{x}(x)=\lambda\cdot\rm e^{\it -\lambda \hspace{0.05cm}\cdot \hspace{0.03cm} x}\hspace{0.2cm}{\rm for}\hspace{0.2cm} {\it x}>0.$$		:$$f_{x}(x)=\lambda\cdot\rm e^{\it -\lambda \hspace{0.05cm}\cdot \hspace{0.03cm} x}\hspace{0.2cm}{\rm for}\hspace{0.2cm} {\it x}>0.$$
	*Note:		*Note:
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	==Two-sided exponential distribution - Laplace distribution==		==Two-sided exponential distribution - Laplace distribution==
	<br>		<br>
	Closely related to the exponential distribution is the  [https://en.wikipedia.org/wiki/Laplace_distribution Laplace distribution]  with the probability density function		Closely related to the exponential distribution is the  [https://en.wikipedia.org/wiki/Laplace_distribution $\text{Laplace distribution}$]  with the probability density function
	:$$f_{x}(x)=\frac{\lambda}{2}\cdot\rm e^{\it -\lambda \hspace{0.05cm} \cdot \hspace{0.05cm} \| x\|}.$$		:$$f_{x}(x)=\frac{\lambda}{2}\cdot\rm e^{\it -\lambda \hspace{0.05cm} \cdot \hspace{0.05cm} \| x\|}.$$

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	Further notes:		Further notes:
	*From the  [[Aufgaben:Exercise_3.8:_Amplification_and_Limitation\| Exercise 3.8]]  one can see further properties of the Laplace distribution.		*From the  [[Aufgaben:Exercise_3.8:_Amplification_and_Limitation\|"Exercise 3.8"]]  one can see further properties of the Laplace distribution.
	*With the HTML 5/JavaScript applet  [[Applets:PDF,_CDF_and_Moments_of_Special_Distributions\|"PDF, CDF and Moments of Special Distributions"]]  you can display the characteristics of the exponential and the Laplace distribution.		*With the HTML 5/JavaScript applet  [[Applets:PDF,_CDF_and_Moments_of_Special_Distributions\|"PDF, CDF and Moments of Special Distributions"]]  you can display the characteristics of the exponential and the Laplace distribution.
	*In the  (German language)  learning video  [[Wahrscheinlichkeit_und_WDF_(Lernvideo)\|"Wahrscheinlichkeit und WDF"]]   $\Rightarrow$   "Probability and PDF",  it is shown which meaning the Laplace distribution has for the description of speech and music signals.		*In the  (German language)  learning video  [[Wahrscheinlichkeit_und_WDF_(Lernvideo)\|"Wahrscheinlichkeit und WDF"]]   $\Rightarrow$   "Probability and PDF",  it is shown which meaning the Laplace distribution has for the description of speech and music signals.

Hwang at 23:17, 12 November 2022

2022-11-12T23:17:02Z

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	:$$x=g_1(u) =\frac{1}{\lambda}\cdot \rm ln \ (\frac{1}{1-\it u}).$$		:$$x=g_1(u) =\frac{1}{\lambda}\cdot \rm ln \ (\frac{1}{1-\it u}).$$

	*It can be shown that by this characteristic  $x=g_1(u)$  a one-sided exponentially distributed random variable  $x$  with the following PDF arises  <br>(derivation see [[Theory_of_Stochastic_Signals/Exponentially_Distributed_Random_Variables#Derivation_of_the_corresponding_transformation_characteristic\|next ~~page~~]]):		*It can be shown that by this characteristic  $x=g_1(u)$  a one-sided exponentially distributed random variable  $x$  with the following PDF arises  <br>(derivation see [[Theory_of_Stochastic_Signals/Exponentially_Distributed_Random_Variables#Derivation_of_the_corresponding_transformation_characteristic\|next section]]):
	:$$f_{x}(x)=\lambda\cdot\rm e^{\it -\lambda \hspace{0.05cm}\cdot \hspace{0.03cm} x}\hspace{0.2cm}{\rm for}\hspace{0.2cm} {\it x}>0.$$		:$$f_{x}(x)=\lambda\cdot\rm e^{\it -\lambda \hspace{0.05cm}\cdot \hspace{0.03cm} x}\hspace{0.2cm}{\rm for}\hspace{0.2cm} {\it x}>0.$$
	*Note:		*Note:
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	{{BlaueBox\|TEXT=		{{BlaueBox\|TEXT=
	$\text{Task:}$		$\text{Task:}$
	#  Now the transformation characteristic  $x = g_1(u)= g(u)$  already used on the last ~~page~~ is derived.		#  Now the transformation characteristic  $x = g_1(u)= g(u)$  already used in the last section is derived.
	#  This forms from the uniformly distributed random variable  $u$  with PDF  $f_{u}(u)$  a one-sided exponentially distributed random variable  $x$  with PDF  $f_{x}(x)$:		#  This forms from the uniformly distributed random variable  $u$  with PDF  $f_{u}(u)$  a one-sided exponentially distributed random variable  $x$  with PDF  $f_{x}(x)$: