Theory of Stochastic Signals/Further Distributions - Revision history

Guenter at 09:03, 22 December 2022

2022-12-22T09:03:47Z

Hwang at 20:46, 20 December 2022

2022-12-20T20:46:36Z

Hwang at 20:27, 20 December 2022

2022-12-20T20:27:14Z

Hwang at 04:32, 18 September 2022

2022-09-18T04:32:31Z

Bene: Text replacement - "rms value" to "standard deviation"

2022-02-15T15:45:53Z

Text replacement - "rms value" to "standard deviation"

Guenter at 16:00, 8 February 2022

2022-02-08T16:00:00Z

Guenter at 13:04, 21 January 2022

2022-01-21T13:04:51Z

Guenter at 13:04, 21 January 2022

2022-01-21T13:04:03Z

Guenter at 13:03, 21 January 2022

2022-01-21T13:03:18Z

Guenter at 13:00, 21 January 2022

2022-01-21T13:00:56Z

@@ Line 8: / Line 8: @@
 <br>
 {{BlaueBox|TEXT=
-$\text{Definition:}$&nbsp; A continuous random variable&nbsp; $x$&nbsp; is called&nbsp; &raquo;'''Rayleigh distributed'''&laquo;&nbsp; if it cannot take negative values and the probability density function&nbsp; $\rm (PDF)$&nbsp; for&nbsp; $x \ge 0$&nbsp;  has the following shape with the distribution parameter&nbsp; $λ$:
+$\text{Definition:}$&nbsp; A continuous-valued random variable&nbsp; $x$&nbsp; is called&nbsp; &raquo;'''Rayleigh distributed'''&laquo;&nbsp; if it cannot take negative values and the probability density function&nbsp; $\rm (PDF)$&nbsp; for&nbsp; $x \ge 0$&nbsp;  has the following shape with the distribution parameter&nbsp; $λ$:
 :$$f_{x}(x)=\frac{x}{\lambda^2}\cdot {\rm e}^{-x^2 / ( 2 \hspace{0.05cm}\cdot \hspace{0.05cm}\lambda^2) } .$$}}
@@ Line 58: / Line 58: @@
 {{BlaueBox|TEXT=
-$\text{Definition:}$&nbsp; A continuous random variable&nbsp; $x$&nbsp; is called&nbsp; &raquo;'''Rice distributed'''&laquo;&nbsp;  if it cannot take negative values and the probability density function&nbsp;  $\rm (PDF)$&nbsp; for&nbsp; $x > 0$&nbsp; has the following shape:
+$\text{Definition:}$&nbsp; A continuous-valued random variable&nbsp; $x$&nbsp; is called&nbsp; &raquo;'''Rice distributed'''&laquo;&nbsp;  if it cannot take negative values and the probability density function&nbsp;  $\rm (PDF)$&nbsp; for&nbsp; $x > 0$&nbsp; has the following shape:
 :$$f_{\rm x}(x)=\frac{x}{\lambda^2}\cdot{\rm e}^{-({C^2+\it x^{\rm 2} })/ ({\rm 2 \it \lambda^{\rm 2} })}\cdot {\rm I_0}(\frac{\it x\cdot C}{\lambda^{\rm 2} }) \hspace{0.4cm}{\rm with} \hspace{0.4cm} {\rm I_0}(x) = \sum_{k=0}^{\infty}\frac{(x/2)^{2k} }{k! \cdot {\rm \gamma ({\it k}+1)} }.$$
 ${\rm I_0}( ... )$ denotes the&nbsp; [https://en.wikipedia.org/wiki/Bessel_function $\text{modified zero-order Bessel function}$].}}
@@ Line 93: / Line 93: @@
 <br>
 {{BlaueBox|TEXT=
-$\text{Definition:}$&nbsp; A continuous random variable&nbsp; $x$&nbsp; is called&nbsp; &raquo;'''Cauchy distributed'''&laquo;&nbsp; if the probability density function&nbsp; $\rm (PDF)$&nbsp; and the cumulative distribution function&nbsp; $\rm  (CDF)$&nbsp; with parameter&nbsp; $λ$&nbsp; have the following form:
+$\text{Definition:}$&nbsp; A continuous-valued random variable&nbsp; $x$&nbsp; is called&nbsp; &raquo;'''Cauchy distributed'''&laquo;&nbsp; if the probability density function&nbsp; $\rm (PDF)$&nbsp; and the cumulative distribution function&nbsp; $\rm  (CDF)$&nbsp; with parameter&nbsp; $λ$&nbsp; have the following form:
 :$$f_{x}(x)=\frac{1}{\pi}\cdot\frac{\lambda}{\lambda^2+x^2},$$
 :$$F_{x}(r)={\rm 1}/{2}+{\rm arctan}({r}/{\lambda}).$$

@@ Line 139: / Line 139: @@
 $\text{Example 4:}$&nbsp; We assume a Gaussian distributed and zero mean random variable&nbsp; $x$.
 *Thus,&nbsp; the probability that its absolute value&nbsp; $\vert x \vert $&nbsp; is greater than three times the rms&nbsp; $(3 \cdot σ_{x})$&nbsp; is easily computable.&nbsp; Result:&nbsp; ${\rm 2 - Q(3) ≈ 2.7 \cdot 10^{-3} }.$
-*Tschebyshev's inequality yields here as an upper bound the clearly too large value&nbsp; $1/9 ≈ 0.111$.
+*Chebyshev's inequality yields here as an upper bound the clearly too large value&nbsp; $1/9 ≈ 0.111$.
 *This Chebyshev bound would hold for any PDF form.}}

@@ Line 8: / Line 8: @@
 <br>
 {{BlaueBox|TEXT=
-$\text{Definition:}$&nbsp; A continuous random variable&nbsp; $x$&nbsp; is called&nbsp; '''Rayleigh distributed'''&nbsp; if it cannot take negative values and the probability density function&nbsp; $\rm (PDF)$&nbsp; for&nbsp; $x \ge 0$&nbsp;  has the following shape with the distribution parameter&nbsp; $λ$:
+$\text{Definition:}$&nbsp; A continuous random variable&nbsp; $x$&nbsp; is called&nbsp; &raquo;'''Rayleigh distributed'''&laquo;&nbsp; if it cannot take negative values and the probability density function&nbsp; $\rm (PDF)$&nbsp; for&nbsp; $x \ge 0$&nbsp;  has the following shape with the distribution parameter&nbsp; $λ$:
 :$$f_{x}(x)=\frac{x}{\lambda^2}\cdot {\rm e}^{-x^2 / ( 2 \hspace{0.05cm}\cdot \hspace{0.05cm}\lambda^2) } .$$}}
-The name goes back to the English physicist&nbsp; [https://en.wikipedia.org/wiki/John_William_Strutt,_3rd_Baron_Rayleigh John William Strutt]&nbsp; the&nbsp; "third Baron Rayleigh".&nbsp; In 1904 he received the Physics Nobel Prize.
+The name goes back to the English physicist&nbsp; [https://en.wikipedia.org/wiki/John_William_Strutt,_3rd_Baron_Rayleigh $\text{John William Strutt}$]&nbsp; the&nbsp; "third Baron Rayleigh".&nbsp; In 1904 he received the Physics Nobel Prize.
-*The Rayleigh distribution plays a central role in the description of time-varying channels.&nbsp; Such channels are described in the book&nbsp; [[Mobile_Communications/Probability_Density_of_Rayleigh_Fading|Mobile Communications]].
+*The Rayleigh distribution plays a central role in the description of time-varying channels.&nbsp; Such channels are described in the book&nbsp; [[Mobile_Communications/Probability_Density_of_Rayleigh_Fading|"Mobile Communications"]].
 *For example,&nbsp; "non-frequency selective fading"&nbsp; exhibits such a distribution when there is no&nbsp; "line-of-sight"&nbsp; between the base station and the mobile user.
@@ Line 58: / Line 58: @@
 {{BlaueBox|TEXT=
-$\text{Definition:}$&nbsp; A continuous random variable&nbsp; $x$&nbsp; is called&nbsp; '''Rice distributed'''&nbsp;  if it cannot take negative values and the probability density function&nbsp;  $\rm (PDF)$&nbsp; for&nbsp; $x > 0$&nbsp; has the following shape:
+$\text{Definition:}$&nbsp; A continuous random variable&nbsp; $x$&nbsp; is called&nbsp; &raquo;'''Rice distributed'''&laquo;&nbsp;  if it cannot take negative values and the probability density function&nbsp;  $\rm (PDF)$&nbsp; for&nbsp; $x > 0$&nbsp; has the following shape:
 :$$f_{\rm x}(x)=\frac{x}{\lambda^2}\cdot{\rm e}^{-({C^2+\it x^{\rm 2} })/ ({\rm 2 \it \lambda^{\rm 2} })}\cdot {\rm I_0}(\frac{\it x\cdot C}{\lambda^{\rm 2} }) \hspace{0.4cm}{\rm with} \hspace{0.4cm} {\rm I_0}(x) = \sum_{k=0}^{\infty}\frac{(x/2)^{2k} }{k! \cdot {\rm \gamma ({\it k}+1)} }.$$
-${\rm I_0}( ... )$ denotes the&nbsp; [https://en.wikipedia.org/wiki/Bessel_function "modified zero-order Bessel function"].}}
+${\rm I_0}( ... )$ denotes the&nbsp; [https://en.wikipedia.org/wiki/Bessel_function $\text{modified zero-order Bessel function}$].}}
-The name is due to the mathematician and logician&nbsp; [https://en.wikipedia.org/wiki/Henry_Gordon_Rice Henry Gordon Rice].&nbsp; He taught as a mathematics professor at the University of New Hampshire.
+The name is due to the mathematician and logician&nbsp; [https://en.wikipedia.org/wiki/Henry_Gordon_Rice $\text{Henry Gordon Rice}$].&nbsp; He taught as a mathematics professor at the University of New Hampshire.
 '''Characteristic properties of the Rice distribution''':
-*The additional parameter&nbsp; $C$&nbsp; compared to the Rayleigh distribution is a measure of the&nbsp; "strength"&nbsp; of the direct component.&nbsp; The larger the quotient&nbsp; $C/λ$,&nbsp;  the more the&nbsp; Rice channel approximates the Gaussian channel.&nbsp; For&nbsp; $C = 0$&nbsp; the Rice distribution transitions to the&nbsp; [[Theory_of_Stochastic_Signals/Further_Distributions#Rayleigh_PDF|Rayleigh distribution]].
+*The additional parameter&nbsp; $C$&nbsp; compared to the Rayleigh distribution is a measure of the&nbsp; "strength"&nbsp; of the direct component.&nbsp; The larger the quotient&nbsp; $C/λ$,&nbsp;  the more the&nbsp; Rice channel approximates the Gaussian channel.&nbsp; For&nbsp; $C = 0$&nbsp; the Rice distribution transitions to the&nbsp; [[Theory_of_Stochastic_Signals/Further_Distributions#Rayleigh_PDF|$\text{Rayleigh distribution}$]].
 *In the Rice distribution,&nbsp; the expression for the moment&nbsp; $m_k$&nbsp; is much more complicated and can only be specified using hypergeometric functions.&nbsp;
 *However,&nbsp; if&nbsp; $λ \ll C$,&nbsp; then&nbsp; $m_1 ≈ C$&nbsp; and&nbsp; $σ ≈ λ$&nbsp; holds.
-*Under these conditions,&nbsp; the Rice distribution can be approximated by a&nbsp; [[Theory_of_Stochastic_Signals/Gaussian_Distributed_Random_Variables|Gaussian distribution]]&nbsp; with mean&nbsp; $C$&nbsp; and rms&nbsp; $λ$.
+*Under these conditions,&nbsp; the Rice distribution can be approximated by a&nbsp; [[Theory_of_Stochastic_Signals/Gaussian_Distributed_Random_Variables|$\text{Gaussian distribution}$]]&nbsp; with mean&nbsp; $C$&nbsp; and rms&nbsp; $λ$.
 *To model a Rice distributed random variable&nbsp; $x$&nbsp; we use a similar model as for the Rayleigh distribution,&nbsp; except that now at least one of the two Gaussian distributed and statistically independent random variables&nbsp; $(u$&nbsp; and/or &nbsp;$v$&nbsp;)&nbsp; must have a non-zero mean.
 :$$x=\sqrt{u^2+v^2}\hspace{0.5cm}{\rm with}\hspace{0.5cm}|m_u| + |m_v| > 0 .$$
@@ Line 93: / Line 93: @@
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 {{BlaueBox|TEXT=
-$\text{Definition:}$&nbsp; A continuous random variable&nbsp; $x$&nbsp; is called&nbsp; '''Cauchy distributed'''&nbsp; if the probability density function&nbsp; $\rm (PDF)$&nbsp; and the cumulative distribution function&nbsp; $\rm  (CDF)$&nbsp; with parameter&nbsp; $λ$&nbsp; have the following form:
+$\text{Definition:}$&nbsp; A continuous random variable&nbsp; $x$&nbsp; is called&nbsp; &raquo;'''Cauchy distributed'''&laquo;&nbsp; if the probability density function&nbsp; $\rm (PDF)$&nbsp; and the cumulative distribution function&nbsp; $\rm  (CDF)$&nbsp; with parameter&nbsp; $λ$&nbsp; have the following form:
 :$$f_{x}(x)=\frac{1}{\pi}\cdot\frac{\lambda}{\lambda^2+x^2},$$
 :$$F_{x}(r)={\rm 1}/{2}+{\rm arctan}({r}/{\lambda}).$$
@@ Line 99: / Line 99: @@
-The name derives from the French mathematician&nbsp; [https://en.wikipedia.org/wiki/Augustin-Louis_Cauchy Augustin-Louis Cauchy],&nbsp; a pioneer of calculus who further developed the foundations established by&nbsp; [https://en.wikipedia.org/wiki/Gottfried_Wilhelm_Leibniz Gottfried Wilhelm Leibniz]&nbsp; and&nbsp; [https://en.wikipedia.org/wiki/Isaac_Newton Sir Isaac Newton]&nbsp; and formally proved fundamental propositions.&nbsp; In particular,&nbsp; many central theorems of&nbsp; "Function Theory"&nbsp; derive from Cauchy.
+The name derives from the French mathematician&nbsp; [https://en.wikipedia.org/wiki/Augustin-Louis_Cauchy $\text{Augustin-Louis Cauchy}$],&nbsp; a pioneer of calculus who further developed the foundations established by&nbsp; [https://en.wikipedia.org/wiki/Gottfried_Wilhelm_Leibniz $\text{Gottfried Wilhelm Leibniz}$]&nbsp; and&nbsp; [https://en.wikipedia.org/wiki/Isaac_Newton $\text{Sir Isaac Newton}$]&nbsp; and formally proved fundamental propositions.&nbsp; In particular,&nbsp; many central theorems of&nbsp; "Function Theory"&nbsp; derive from Cauchy.
 The Cauchy distribution has less practical significance for communications engineering,&nbsp; but is mathematically very interesting.&nbsp; It has the following properties in the symmetric form&nbsp; $($with mean&nbsp; $m_1 = 0)$:
@@ Line 105: / Line 105: @@
 *Thus,&nbsp; this distribution also has an infinitely large variance&nbsp; $\sigma^2 = m_2$ &nbsp; ⇒ &nbsp; "power" &nbsp; &rArr; &nbsp; it is obvious that no physical variable can be Cauchy distributed.
 *The quotient&nbsp; $u/v$&nbsp; of two independent Gaussian distributed zero mean variables&nbsp; $u$&nbsp; and&nbsp; $v$&nbsp; is Cauchy distributed with the distribution parameter&nbsp; $λ = σ_u/σ_v$&nbsp; .
-*A Cauchy distributed random variable&nbsp; $x$&nbsp; can be generated from a random variable&nbsp; $\pm1$&nbsp; uniformly distributed between&nbsp; $u$&nbsp; by the following&nbsp; [[Theory_of_Stochastic_Signals/Exponentially_Distributed_Random_Variables#Transformation_of_random_variables|nonlinear transformation]]:
+*A Cauchy distributed random variable&nbsp; $x$&nbsp; can be generated from a random variable&nbsp; $\pm1$&nbsp; uniformly distributed between&nbsp; $u$&nbsp; by the following&nbsp; [[Theory_of_Stochastic_Signals/Exponentially_Distributed_Random_Variables#Transformation_of_random_variables|$\text{nonlinear transformation}$]]:
 :$$x=\lambda \cdot {\tan}( {\pi}/{2}\cdot u).$$
 *Because of symmetry,&nbsp; for odd&nbsp; $k$&nbsp; all moments&nbsp; $m_k = 0$,&nbsp; assuming the&nbsp; "Cauchy Principal Value".
@@ Line 134: / Line 134: @@
 *If besides the mean&nbsp; $m_{x}$&nbsp; the standard deviation&nbsp; $σ_{x}$&nbsp; is known,&nbsp; but not the exact&nbsp; $\rm PDF$&nbsp; course&nbsp; $f_{x}(x)$,&nbsp; at least an upper bound can be given for this probability:
 :$${\rm Pr}(|x - m_{\rm x}|\ge\varepsilon)\le\frac{\sigma_{x}^{\rm 2}}{\varepsilon^{\rm 2}}. $$
-*This bound given by&nbsp; [https://en.wikipedia.org/wiki/Pafnuty_Chebyshev Pafnuti L. Chebyshev]&nbsp; - known as&nbsp; "Chebyshev's inequality"&nbsp; - is in general only a very rough approximation for the actual exceeding probability.&nbsp; It should therefore be applied only in the case of an unknown course of the PDF&nbsp; $f_{x}(x)$.
+*This bound given by&nbsp; [https://en.wikipedia.org/wiki/Pafnuty_Chebyshev $\text{Pafnuti L. Chebyshev}$]&nbsp; - known as&nbsp; "Chebyshev's inequality"&nbsp; - is in general only a very rough approximation for the actual exceeding probability.&nbsp; It should therefore be applied only in the case of an unknown course of the PDF&nbsp; $f_{x}(x)$.
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@@ Line 130: / Line 130: @@
 <br>
 [[File:EN_Sto_T_3_7_S4_neu.png |frame| Chebyshev's inequality | right]]
-Given a random variable&nbsp; $x$&nbsp; with known probability density function&nbsp; $f_{x}(x)$&nbsp; the probability that the random variable&nbsp; $x$&nbsp; deviates in amount by more than the value&nbsp; $ε$&nbsp; from its mean&nbsp; $m_{x}$&nbsp; can be calculated exactly according to the way generally described in this chapter.
+Given a random variable&nbsp; $x$&nbsp; with known probability density function&nbsp; $f_{x}(x)$&nbsp; the probability that the random variable&nbsp; $x$&nbsp; deviates in magnitude by more than the value&nbsp; $ε$&nbsp; from its mean&nbsp; $m_{x}$&nbsp; can be calculated exactly according to the way generally described in this chapter.
 *If besides the mean&nbsp; $m_{x}$&nbsp; the standard deviation&nbsp; $σ_{x}$&nbsp; is known,&nbsp; but not the exact&nbsp; $\rm PDF$&nbsp; course&nbsp; $f_{x}(x)$,&nbsp; at least an upper bound can be given for this probability:

@@ Line 24: / Line 24: @@
 :$$m_k=(2\cdot \lambda^{\rm 2})^{\it k/\rm 2}\cdot {\rm \Gamma}( 1+ {\it k}/{\rm 2}) \hspace{0.3cm}{\rm with }\hspace{0.3cm}{\rm \Gamma}(x)= \int_{0}^{\infty} t^{x-1} \cdot
 {\rm e}^{-t} \hspace{0.1cm}{\rm d}t.$$
-*From this,&nbsp; the mean&nbsp; $m_1$&nbsp; and the rms value&nbsp; $\sigma$&nbsp; can be calculated as follows:
+*From this,&nbsp; the mean&nbsp; $m_1$&nbsp; and the standard deviation&nbsp; $\sigma$&nbsp; can be calculated as follows:
 :$$m_1=\sqrt{2}\cdot \lambda\cdot {\rm \Gamma}(1.5) =
 \sqrt{2}\cdot \lambda\cdot {\sqrt{\pi}}/{2} =\lambda\cdot\sqrt{{\pi}/{2}},$$
@@ Line 132: / Line 132: @@
 Given a random variable&nbsp; $x$&nbsp; with known probability density function&nbsp; $f_{x}(x)$&nbsp; the probability that the random variable&nbsp; $x$&nbsp; deviates in amount by more than the value&nbsp; $ε$&nbsp; from its mean&nbsp; $m_{x}$&nbsp; can be calculated exactly according to the way generally described in this chapter.
-*If besides the mean&nbsp; $m_{x}$&nbsp; the rms value&nbsp; $σ_{x}$&nbsp; is known,&nbsp; but not the exact&nbsp; $\rm PDF$&nbsp; course&nbsp; $f_{x}(x)$,&nbsp; at least an upper bound can be given for this probability:
+*If besides the mean&nbsp; $m_{x}$&nbsp; the standard deviation&nbsp; $σ_{x}$&nbsp; is known,&nbsp; but not the exact&nbsp; $\rm PDF$&nbsp; course&nbsp; $f_{x}(x)$,&nbsp; at least an upper bound can be given for this probability:
 :$${\rm Pr}(|x - m_{\rm x}|\ge\varepsilon)\le\frac{\sigma_{x}^{\rm 2}}{\varepsilon^{\rm 2}}. $$
 *This bound given by&nbsp; [https://en.wikipedia.org/wiki/Pafnuty_Chebyshev Pafnuti L. Chebyshev]&nbsp; - known as&nbsp; "Chebyshev's inequality"&nbsp; - is in general only a very rough approximation for the actual exceeding probability.&nbsp; It should therefore be applied only in the case of an unknown course of the PDF&nbsp; $f_{x}(x)$.

@@ Line 129: / Line 129: @@
 ==Chebyshev's inequality==
 <br>
-[[File:EN_Sto_T_3_7_S4.png |frame| Chebyshev's inequality | right]]
+[[File:EN_Sto_T_3_7_S4_neu.png |frame| Chebyshev's inequality | right]]
 Given a random variable&nbsp; $x$&nbsp; with known probability density function&nbsp; $f_{x}(x)$&nbsp; the probability that the random variable&nbsp; $x$&nbsp; deviates in amount by more than the value&nbsp; $ε$&nbsp; from its mean&nbsp; $m_{x}$&nbsp; can be calculated exactly according to the way generally described in this chapter.

@@ Line 14: / Line 14: @@
 The name goes back to the English physicist&nbsp; [https://en.wikipedia.org/wiki/John_William_Strutt,_3rd_Baron_Rayleigh John William Strutt]&nbsp; the&nbsp; "third Baron Rayleigh".&nbsp; In 1904 he received the Physics Nobel Prize.
-*The Rayleigh distribution plays a central role in the description of time-varying channels.&nbsp; Such channels are described in the book&nbsp; [[Mobile_Communications/Probability_Density_of_Rayleigh_Fading]].
+*The Rayleigh distribution plays a central role in the description of time-varying channels.&nbsp; Such channels are described in the book&nbsp; [[Mobile_Communications/Probability_Density_of_Rayleigh_Fading|Probability Density of Rayleigh Fading]].
 *For example,&nbsp; "non-frequency selective fading"&nbsp; exhibits such a distribution when there is no&nbsp; "line-of-sight"&nbsp; between the base station and the mobile user.