Applets:PDF, CDF and Moments of Special Distributions - Revision history

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← Older revision		Revision as of 15:49, 16 March 2026
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	'''(3)'''    »''''''«		'''(3)'''    »''''''«
	*Mean and standard deviation have the following values for the uniform distribution:		*Mean and standard deviation have the following values for the uniform distribution:
	:$$m_X = \frac{\it x_ {\rm max} \rm + \it x_{\rm min}}{2},\hspace{0.5cm}		:$$m_X = \frac{\it x_ {\rm max} \rm + \it x_{\rm min}}{2},\hspace{0.5cm}\sigma_X^2 = \frac{(\it x_{\rm max} - \it x_{\rm min}\rm )^2}{12}.$$
	\sigma_X^2 = \frac{(\it x_{\rm max} - \it x_{\rm min}\rm )^2}{12}.$$
	*For symmetric PDF   ⇒   $x_{\rm min} = -x_{\rm max}$  the mean value  $m_X = 0$  and the variance  $σ_X^2 = x_{\rm max}^2/3.$		*For symmetric PDF   ⇒   $x_{\rm min} = -x_{\rm max}$  the mean value  $m_X = 0$  and the variance  $σ_X^2 = x_{\rm max}^2/3.$
	*Because of the symmetry around the mean  $m_X$  the Charlier skewness  $S_X = 0$.		*Because of the symmetry around the mean  $m_X$  the Charlier skewness  $S_X = 0$.
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	*Probability density function:		*Probability density function:
	:$$f_X(x) =		:$$f_X(x) =\left\{ \begin{array}{c} x/\lambda_X^2 \cdot {\rm e}^{- x^2/(2 \hspace{0.05cm}\cdot\hspace{0.05cm} \lambda_X^2)} \\0 \end{array} \right.\hspace{0.15cm}\begin{array}{*{1}c} {\rm for}\hspace{0.1cm} x\hspace{-0.05cm} \ge \hspace{-0.05cm}0,\\ {\rm for}\hspace{0.1cm} x \hspace{-0.05cm}<\hspace{-0.05cm} 0. \\ \end{array}.$$
	\left\{ \begin{array}{c} x/\lambda_X^2 \cdot {\rm e}^{- x^2/(2 \hspace{0.05cm}\cdot\hspace{0.05cm} \lambda_X^2)} \\
	0 \end{array} \right.\hspace{0.15cm}
	\begin{array}{*{1}c} {\rm for}\hspace{0.1cm} x\hspace{-0.05cm} \ge \hspace{-0.05cm}0,
	\\ {\rm for}\hspace{0.1cm} x \hspace{-0.05cm}<\hspace{-0.05cm} 0. \\ \end{array}.$$
	*Application:     Modeling of the cellular channel (non-frequency selective fading, attenuation, diffraction, and refraction effects only, no line-of-sight).		*Application:     Modeling of the cellular channel (non-frequency selective fading, attenuation, diffraction, and refraction effects only, no line-of-sight).

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	*Probability density function  $(\rm I_0$  denotes the modified zero-order Bessel function$)$:		*Probability density function  $(\rm I_0$  denotes the modified zero-order Bessel function$)$:
	:$$f_X(x) = \frac{x}{\lambda_X^2} \cdot {\rm exp} \big [ -\frac{x^2 + C_X^2}{2\cdot \lambda_X^2}\big ] \cdot {\rm I}_0 \left [ \frac{x \cdot C_X}{\lambda_X^2} \right ]\hspace{0.5cm}\text{with}\hspace{0.5cm}{\rm I }_0 (u) = {\rm J }_0 ({\rm j} \cdot u) =		:$$f_X(x) = \frac{x}{\lambda_X^2} \cdot {\rm exp} \big [ -\frac{x^2 + C_X^2}{2\cdot \lambda_X^2}\big ] \cdot {\rm I}_0 \left [ \frac{x \cdot C_X}{\lambda_X^2} \right ]\hspace{0.5cm}\text{with}\hspace{0.5cm}{\rm I }_0 (u) = {\rm J }_0 ({\rm j} \cdot u) =\sum_{k = 0}^{\infty} \frac{ (u/2)^{2k}}{k! \cdot \Gamma (k+1)}\hspace{0.05cm}.$$
	\sum_{k = 0}^{\infty} \frac{ (u/2)^{2k}}{k! \cdot \Gamma (k+1)}
	\hspace{0.05cm}.$$
	*Application:     Cellular channel modeling (non-frequency selective fading, attenuation, diffraction, and refraction effects only, with line-of-sight).		*Application:     Cellular channel modeling (non-frequency selective fading, attenuation, diffraction, and refraction effects only, with line-of-sight).

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	*Probability density function:		*Probability density function:
	:$$f_X(x) = \lambda_X \cdot k_X \cdot (\lambda_X \cdot x)^{k_X-1} \cdot {\rm e}^{(\lambda_X \cdot x)^{k_X}}		:$$f_X(x) = \lambda_X \cdot k_X \cdot (\lambda_X \cdot x)^{k_X-1} \cdot {\rm e}^{(\lambda_X \cdot x)^{k_X}}\hspace{0.05cm}.$$
	\hspace{0.05cm}.$$

	*Application:     PDF with adjustable skewness $S_X$; exponential distribution  $(k_X = 1)$  and Rayleigh distribution  $(k_X = 2)$  included as special cases.		*Application:     PDF with adjustable skewness $S_X$; exponential distribution  $(k_X = 1)$  and Rayleigh distribution  $(k_X = 2)$  included as special cases.
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	*Probability density function:		*Probability density function:
	:$$f_X(x) =		:$$f_X(x) =\left\{ \begin{array}{c} 2/(\pi \cdot {R_X}^2) \cdot \sqrt{{R_X}^2 - (x- m_X)^2} \\0 \end{array} \right.\hspace{0.15cm}\begin{array}{*{1}c} {\rm for}\hspace{0.1cm} \|x- m_X\|\hspace{-0.05cm} \le \hspace{-0.05cm}R_X,\\ {\rm for}\hspace{0.1cm} \|x- m_X\| \hspace{-0.05cm} > \hspace{-0.05cm} R_X \\ \end{array}.$$
	\left\{ \begin{array}{c} 2/(\pi \cdot {R_X}^2) \cdot \sqrt{{R_X}^2 - (x- m_X)^2} \\
	0 \end{array} \right.\hspace{0.15cm}
	\begin{array}{*{1}c} {\rm for}\hspace{0.1cm} \|x- m_X\|\hspace{-0.05cm} \le \hspace{-0.05cm}R_X,
	\\ {\rm for}\hspace{0.1cm} \|x- m_X\| \hspace{-0.05cm} > \hspace{-0.05cm} R_X \\ \end{array}.$$
	*Application:     PDF of Chebyshev nodes   ⇒   zeros of Chebyshev polynomials from numerics.		*Application:     PDF of Chebyshev nodes   ⇒   zeros of Chebyshev polynomials from numerics.

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	*Probability density function:		*Probability density function:
	:$$f_X(x) =		:$$f_X(x) =\left\{ \begin{array}{c} 3/(4 \cdot {R_X}^3) \cdot \big ({R_X}^2 - (x- m_X)^2\big ) \\0 \end{array} \right.\hspace{0.15cm}\begin{array}{*{1}c} {\rm for}\hspace{0.1cm} \|x\|\hspace{-0.05cm} \le \hspace{-0.05cm}R_X,\\ {\rm for}\hspace{0.1cm} \|x\| \hspace{-0.05cm} > \hspace{-0.05cm} R_X \\ \end{array}.$$
	\left\{ \begin{array}{c} 3/(4 \cdot {R_X}^3) \cdot \big ({R_X}^2 - (x- m_X)^2\big ) \\
	0 \end{array} \right.\hspace{0.15cm}
	\begin{array}{*{1}c} {\rm for}\hspace{0.1cm} \|x\|\hspace{-0.05cm} \le \hspace{-0.05cm}R_X,
	\\ {\rm for}\hspace{0.1cm} \|x\| \hspace{-0.05cm} > \hspace{-0.05cm} R_X \\ \end{array}.$$
	*Application:     PDF of eigenvalues of symmetric random matrices whose dimension approaches infinity.		*Application:     PDF of eigenvalues of symmetric random matrices whose dimension approaches infinity.

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	<br>		<br>
	This interactive calculation tool was designed and implemented at the  [https://www.ei.tum.de/en/lnt/home/ $\text{Institute for Communications Engineering}$]  at the  [https://www.tum.de/en $\text{Technical University of Munich}$].		This interactive calculation tool was designed and implemented at the  [https://www.ei.tum.de/en/lnt/home/ $\text{Institute for Communications Engineering}$]  at the  [https://www.tum.de/en $\text{Technical University of Munich}$].
	*The first version was created in 2005 by  [[Biographies_and_Bibliographies/~~An_LNTwww_beteiligte_Studierende~~#Bettina_Hirner_.28Diplomarbeit_LB_2005.29\|»Bettina Hirner«]]  as part of her diploma thesis with “FlashMX – Actionscript”  (Supervisor:  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_from_1974-2024.29\| »Günter Söder« ]]  and  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Klaus_Eichin_.28at_LNT_from_1972-2011.29\| »Klaus Eichin« ]]).		*The first version was created in 2005 by  [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#Bettina_Hirner_.28Diplomarbeit_LB_2005.29\|»Bettina Hirner«]]  as part of her diploma thesis with “FlashMX – Actionscript”  (Supervisor:  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_from_1974-2024.29\| »Günter Söder« ]]  and  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Klaus_Eichin_.28at_LNT_from_1972-2011.29\| »Klaus Eichin« ]]).

	*In 2019 the program was redesigned via HTML5/JavaScript by  [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#Matthias_Niller_.28Ingenieurspraxis_Math_2019.29\|»Matthias Niller«]]  (Ingenieurspraxis Mathematik, Supervisor:  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Benedikt_Leible_.28at_L.C3.9CT_since_2017.29\| »Benedikt Leible« ]]  and  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Tasn.C3.A1d_Kernetzky_.28at_L.C3.9CT_from_2014-2022.29\| »Tasnád Kernetzky« ]] ).		*In 2019 the program was redesigned via HTML5/JavaScript by  [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#Matthias_Niller_.28Ingenieurspraxis_Math_2019.29\|»Matthias Niller«]]  (Ingenieurspraxis Mathematik, Supervisor:  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Benedikt_Leible_.28at_L.C3.9CT_since_2017.29\| »Benedikt Leible« ]]  and  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Tasn.C3.A1d_Kernetzky_.28at_L.C3.9CT_from_2014-2022.29\| »Tasnád Kernetzky« ]] ).

Maintenance script: Add German interlanguage link

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	{{LntAppletLinkEnDe\|wdf-vtf_en\|wdf-vtf}}		{{LntAppletLinkEnDe\|wdf-vtf_en\|wdf-vtf}}
			[[de:Applets:WDF,_VTF_und_Momente_spezieller_Verteilungen_(Applet)]]

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← Older revision		Revision as of 15:39, 24 February 2025
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	<br>		<br>
	This interactive calculation tool was designed and implemented at the  [https://www.ei.tum.de/en/lnt/home/ $\text{Institute for Communications Engineering}$]  at the  [https://www.tum.de/en $\text{Technical University of Munich}$].		This interactive calculation tool was designed and implemented at the  [https://www.ei.tum.de/en/lnt/home/ $\text{Institute for Communications Engineering}$]  at the  [https://www.tum.de/en $\text{Technical University of Munich}$].
	*The first version was created in 2005 by  [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Studierende#Bettina_Hirner_.28Diplomarbeit_LB_2005.29\|»Bettina Hirner«]]  as part of her diploma thesis with “FlashMX – Actionscript”  (Supervisor:  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.~~28at_LNT_since_1974~~.29\| »Günter Söder« ]]  and  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Klaus_Eichin_.28at_LNT_from_1972-2011.29\| »Klaus Eichin« ]]).		*The first version was created in 2005 by  [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Studierende#Bettina_Hirner_.28Diplomarbeit_LB_2005.29\|»Bettina Hirner«]]  as part of her diploma thesis with “FlashMX – Actionscript”  (Supervisor:  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_from_1974-2024.29\| »Günter Söder« ]]  and  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Klaus_Eichin_.28at_LNT_from_1972-2011.29\| »Klaus Eichin« ]]).

	*In 2019 the program was redesigned via HTML5/JavaScript by  [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#Matthias_Niller_.28Ingenieurspraxis_Math_2019.29\|»Matthias Niller«]]  (Ingenieurspraxis Mathematik, Supervisor:  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Benedikt_Leible_.28at_L.C3.9CT_since_2017.29\| »Benedikt Leible« ]]  and  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Tasn.C3.A1d_Kernetzky_.28at_L.C3.9CT_from_2014-2022.29\| »Tasnád Kernetzky« ]] ).		*In 2019 the program was redesigned via HTML5/JavaScript by  [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#Matthias_Niller_.28Ingenieurspraxis_Math_2019.29\|»Matthias Niller«]]  (Ingenieurspraxis Mathematik, Supervisor:  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Benedikt_Leible_.28at_L.C3.9CT_since_2017.29\| »Benedikt Leible« ]]  and  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Tasn.C3.A1d_Kernetzky_.28at_L.C3.9CT_from_2014-2022.29\| »Tasnád Kernetzky« ]] ).

Höfler: Text replacement - "[[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Benedikt_Leible.2C_M.Sc._.28at_L.C3.9CT_since_2017.29|" to "[[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Benedikt_Leible_.28at_L.C3.9CT_since_2017.29|"

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← Older revision		Revision as of 15:48, 3 February 2025
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	*The first version was created in 2005 by  [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Studierende#Bettina_Hirner_.28Diplomarbeit_LB_2005.29\|»Bettina Hirner«]]  as part of her diploma thesis with “FlashMX – Actionscript”  (Supervisor:  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29\| »Günter Söder« ]]  and  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Klaus_Eichin_.28at_LNT_from_1972-2011.29\| »Klaus Eichin« ]]).		*The first version was created in 2005 by  [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Studierende#Bettina_Hirner_.28Diplomarbeit_LB_2005.29\|»Bettina Hirner«]]  as part of her diploma thesis with “FlashMX – Actionscript”  (Supervisor:  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29\| »Günter Söder« ]]  and  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Klaus_Eichin_.28at_LNT_from_1972-2011.29\| »Klaus Eichin« ]]).

	*In 2019 the program was redesigned via HTML5/JavaScript by  [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#Matthias_Niller_.28Ingenieurspraxis_Math_2019.29\|»Matthias Niller«]]  (Ingenieurspraxis Mathematik, Supervisor:  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#~~Benedikt_Leible~~.~~2C_M~~.~~Sc._~~.28at_L.C3.9CT_since_2017.29\| »Benedikt Leible« ]]  and  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Tasn.C3.A1d_Kernetzky_.28at_L.C3.9CT_from_2014-2022.29\| »Tasnád Kernetzky« ]] ).		*In 2019 the program was redesigned via HTML5/JavaScript by  [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#Matthias_Niller_.28Ingenieurspraxis_Math_2019.29\|»Matthias Niller«]]  (Ingenieurspraxis Mathematik, Supervisor:  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Benedikt_Leible_.28at_L.C3.9CT_since_2017.29\| »Benedikt Leible« ]]  and  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Tasn.C3.A1d_Kernetzky_.28at_L.C3.9CT_from_2014-2022.29\| »Tasnád Kernetzky« ]] ).

	*Last revision and English version 2021 by  [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#Carolin_Mirschina_.28Ingenieurspraxis_Math_2019.2C_danach_Werkstudentin.29\|»Carolin Mirschina«]]  in the context of a working student activity.		*Last revision and English version 2021 by  [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#Carolin_Mirschina_.28Ingenieurspraxis_Math_2019.2C_danach_Werkstudentin.29\|»Carolin Mirschina«]]  in the context of a working student activity.

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← Older revision		Revision as of 12:17, 26 October 2023
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	*The first version was created in 2005 by  [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Studierende#Bettina_Hirner_.28Diplomarbeit_LB_2005.29\|»Bettina Hirner«]]  as part of her diploma thesis with “FlashMX – Actionscript”  (Supervisor:  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29\| »Günter Söder« ]]  and  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Klaus_Eichin_.28at_LNT_from_1972-2011.29\| »Klaus Eichin« ]]).		*The first version was created in 2005 by  [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Studierende#Bettina_Hirner_.28Diplomarbeit_LB_2005.29\|»Bettina Hirner«]]  as part of her diploma thesis with “FlashMX – Actionscript”  (Supervisor:  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29\| »Günter Söder« ]]  and  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Klaus_Eichin_.28at_LNT_from_1972-2011.29\| »Klaus Eichin« ]]).

	*In 2019 the program was redesigned via HTML5/JavaScript by  [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#Matthias_Niller_.28Ingenieurspraxis_Math_2019.29\|»Matthias Niller«]]  (Ingenieurspraxis Mathematik, Supervisor:  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Benedikt_Leible.2C_M.Sc._.28at_L.C3.9CT_since_2017.29\| »Benedikt Leible« ]]  and  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#~~Tasn~~.C3.~~A1d_Kernetzky~~.~~2C_M.Sc~~._.28at_L.C3.~~9CT_since_2014~~.29\| »Tasnád Kernetzky« ]] ).		*In 2019 the program was redesigned via HTML5/JavaScript by  [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#Matthias_Niller_.28Ingenieurspraxis_Math_2019.29\|»Matthias Niller«]]  (Ingenieurspraxis Mathematik, Supervisor:  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Benedikt_Leible.2C_M.Sc._.28at_L.C3.9CT_since_2017.29\| »Benedikt Leible« ]]  and  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Tasn.C3.A1d_Kernetzky_.28at_L.C3.9CT_from_2014-2022.29\| »Tasnád Kernetzky« ]] ).

	*Last revision and English version 2021 by  [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#Carolin_Mirschina_.28Ingenieurspraxis_Math_2019.2C_danach_Werkstudentin.29\|»Carolin Mirschina«]]  in the context of a working student activity.		*Last revision and English version 2021 by  [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#Carolin_Mirschina_.28Ingenieurspraxis_Math_2019.2C_danach_Werkstudentin.29\|»Carolin Mirschina«]]  in the context of a working student activity.

Guenter at 10:46, 5 April 2023

2023-04-05T10:46:19Z

← Older revision		Revision as of 12:46, 5 April 2023
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	===Gaussian distributed random variables===		===Gaussian distributed random variables===

	[[File:EN_Sto_T_3_5_S2_v2.png \|right\|frame\|Gaussian random variable:  PDF and CDF ~~'''KORREKTUR'''~~]]		[[File:EN_Sto_T_3_5_S2_v2.png \|right\|frame\|Gaussian random variable:  PDF and CDF]]
	'''(1)'''    »'''Probability density function'''«   $($axisymmetric around  $m_X)$		'''(1)'''    »'''Probability density function'''«   $($axisymmetric around  $m_X)$
	:$$f_X(x) = \frac{1}{\sqrt{2\pi}\cdot\sigma_X}\cdot {\rm e}^{-(X-m_X)^2 /(2\sigma_X^2) }.$$		:$$f_X(x) = \frac{1}{\sqrt{2\pi}\cdot\sigma_X}\cdot {\rm e}^{-(X-m_X)^2 /(2\sigma_X^2) }.$$
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	<br>		<br>
	This interactive calculation tool was designed and implemented at the  [https://www.ei.tum.de/en/lnt/home/ $\text{Institute for Communications Engineering}$]  at the  [https://www.tum.de/en $\text{Technical University of Munich}$].		This interactive calculation tool was designed and implemented at the  [https://www.ei.tum.de/en/lnt/home/ $\text{Institute for Communications Engineering}$]  at the  [https://www.tum.de/en $\text{Technical University of Munich}$].
	*The first version was created in 2005 by  [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Studierende#Bettina_Hirner_.28Diplomarbeit_LB_2005.29\|»Bettina Hirner«]]  as part of her diploma thesis with “FlashMX – Actionscript”  (Supervisor:  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29\| Günter Söder ]]  and  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Klaus_Eichin_.28at_LNT_from_1972-2011.29\| Klaus Eichin ]]).		*The first version was created in 2005 by  [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Studierende#Bettina_Hirner_.28Diplomarbeit_LB_2005.29\|»Bettina Hirner«]]  as part of her diploma thesis with “FlashMX – Actionscript”  (Supervisor:  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29\| »Günter Söder« ]]  and  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Klaus_Eichin_.28at_LNT_from_1972-2011.29\| »Klaus Eichin« ]]).

	*In 2019 the program was redesigned via HTML5/JavaScript by  [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#Matthias_Niller_.28Ingenieurspraxis_Math_2019.29\|»Matthias Niller«]]  (Ingenieurspraxis Mathematik, Supervisor:  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Benedikt_Leible.2C_M.Sc._.28at_L.C3.9CT_since_2017.29\| Benedikt Leible ]]  and  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28at_L.C3.9CT_since_2014.29\| Tasnád Kernetzky ]] ).		*In 2019 the program was redesigned via HTML5/JavaScript by  [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#Matthias_Niller_.28Ingenieurspraxis_Math_2019.29\|»Matthias Niller«]]  (Ingenieurspraxis Mathematik, Supervisor:  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Benedikt_Leible.2C_M.Sc._.28at_L.C3.9CT_since_2017.29\| »Benedikt Leible« ]]  and  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28at_L.C3.9CT_since_2014.29\| »Tasnád Kernetzky« ]] ).

	*Last revision and English version 2021 by  [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#Carolin_Mirschina_.28Ingenieurspraxis_Math_2019.2C_danach_Werkstudentin.29\|»Carolin Mirschina«]]  in the context of a working student activity.		*Last revision and English version 2021 by  [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#Carolin_Mirschina_.28Ingenieurspraxis_Math_2019.2C_danach_Werkstudentin.29\|»Carolin Mirschina«]]  in the context of a working student activity.

Guenter at 10:43, 5 April 2023

2023-04-05T10:43:48Z

← Older revision		Revision as of 12:43, 5 April 2023
Line 146:		Line 146:
	===Gaussian distributed random variables===		===Gaussian distributed random variables===

	[[File:~~Gauss_WDF_VTF~~.png \|right\|frame\|Gaussian random variable:  PDF and CDF '''KORREKTUR''']]		[[File:EN_Sto_T_3_5_S2_v2.png \|right\|frame\|Gaussian random variable:  PDF and CDF '''KORREKTUR''']]
	'''(1)'''    »'''Probability density function'''«   $($axisymmetric around  $m_X)$		'''(1)'''    »'''Probability density function'''«   $($axisymmetric around  $m_X)$
	:$$f_X(x) = \frac{1}{\sqrt{2\pi}\cdot\sigma_X}\cdot {\rm e}^{-(X-m_X)^2 /(2\sigma_X^2) }.$$		:$$f_X(x) = \frac{1}{\sqrt{2\pi}\cdot\sigma_X}\cdot {\rm e}^{-(X-m_X)^2 /(2\sigma_X^2) }.$$

Guenter at 15:23, 13 March 2023

2023-03-13T15:23:17Z

← Older revision		Revision as of 17:23, 13 March 2023
Line 325:		Line 325:


	$\text{(D) Wigner semicircle distribution}$     [https://en.wikipedia.org/wiki/Wigner_semicircle_distribution $\text{More detailed description}$] ~~'''KORREKTUR: link'''~~		$\text{(D) Wigner semicircle distribution}$     [https://en.wikipedia.org/wiki/Wigner_semicircle_distribution $\text{More detailed description}$]

	*Probability density function:		*Probability density function:
Line 515:		Line 515:
	<br>		<br>
	This interactive calculation tool was designed and implemented at the  [https://www.ei.tum.de/en/lnt/home/ $\text{Institute for Communications Engineering}$]  at the  [https://www.tum.de/en $\text{Technical University of Munich}$].		This interactive calculation tool was designed and implemented at the  [https://www.ei.tum.de/en/lnt/home/ $\text{Institute for Communications Engineering}$]  at the  [https://www.tum.de/en $\text{Technical University of Munich}$].
	*The first version was created in 2005 by  [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Studierende#Bettina_Hirner_.28Diplomarbeit_LB_2005.29\|~~$\text{~~Bettina Hirner}$]]  as part of her diploma thesis with “FlashMX – Actionscript”  (Supervisor:  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29\|~~$\text{~~Günter Söder}$]]  and  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Klaus_Eichin_.28at_LNT_from_1972-2011.29\|~~$\text{~~Klaus Eichin}$]]).		*The first version was created in 2005 by  [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Studierende#Bettina_Hirner_.28Diplomarbeit_LB_2005.29\|»Bettina Hirner«]]  as part of her diploma thesis with “FlashMX – Actionscript”  (Supervisor:  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29\| Günter Söder ]]  and  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Klaus_Eichin_.28at_LNT_from_1972-2011.29\| Klaus Eichin ]]).

	*In 2019 the program was redesigned via HTML5/JavaScript by  [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#Matthias_Niller_.28Ingenieurspraxis_Math_2019.29\|~~$\text{~~Matthias Niller}$]]  (Ingenieurspraxis Mathematik, Supervisor:  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Benedikt_Leible.2C_M.Sc._.28at_L.C3.9CT_since_2017.29\|~~$\text{~~Benedikt Leible}$]]  and  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28at_L.C3.9CT_since_2014.29\|~~$\text{~~Tasnád Kernetzky}$]] ).		*In 2019 the program was redesigned via HTML5/JavaScript by  [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#Matthias_Niller_.28Ingenieurspraxis_Math_2019.29\|»Matthias Niller«]]  (Ingenieurspraxis Mathematik, Supervisor:  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Benedikt_Leible.2C_M.Sc._.28at_L.C3.9CT_since_2017.29\| Benedikt Leible ]]  and  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28at_L.C3.9CT_since_2014.29\| Tasnád Kernetzky ]] ).

	*Last revision and English version 2021 by  [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#Carolin_Mirschina_.28Ingenieurspraxis_Math_2019.2C_danach_Werkstudentin.29\|~~$\text{~~Carolin Mirschina}$]]  in the context of a working student activity.  ~~Translation using DEEPL.com (free version).~~		*Last revision and English version 2021 by  [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#Carolin_Mirschina_.28Ingenieurspraxis_Math_2019.2C_danach_Werkstudentin.29\|»Carolin Mirschina«]]  in the context of a working student activity.

	*The conversion of this applet was financially supported by  [https://www.ei.tum.de/studium/studienzuschuesse/ $\text{Studienzuschüsse}$]  (TUM Department of Electrical and Computer Engineering).  We thank.		*The conversion of this applet was financially supported by  [https://www.ei.tum.de/studium/studienzuschuesse/ $\text{Studienzuschüsse}$]  (TUM Department of Electrical and Computer Engineering).  We thank.

Guenter at 15:17, 13 March 2023

2023-03-13T15:17:41Z

← Older revision		Revision as of 17:17, 13 March 2023
Line 325:		Line 325:


	$\text{(D) Wigner semicircle distribution}$     [https://de.~~qwertyu~~.~~wiki~~/wiki/Wigner_semicircle_distribution $\text{More detailed description}$] '''KORREKTUR: link'''		$\text{(D) Wigner semicircle distribution}$     [https://en.wikipedia.org/wiki/Wigner_semicircle_distribution $\text{More detailed description}$] '''KORREKTUR: link'''

	*Probability density function:		*Probability density function:

Hwang at 13:15, 23 February 2023

2023-02-23T13:15:32Z

← Older revision		Revision as of 15:15, 23 February 2023
Line 146:		Line 146:
	===Gaussian distributed random variables===		===Gaussian distributed random variables===

	[[File:Gauss_WDF_VTF.png \|right\|frame\|Gaussian random variable:  PDF and CDF]]		[[File:Gauss_WDF_VTF.png \|right\|frame\|Gaussian random variable:  PDF and CDF '''KORREKTUR''']]
	'''(1)'''    »'''Probability density function'''«   $($axisymmetric around  $m_X)$		'''(1)'''    »'''Probability density function'''«   $($axisymmetric around  $m_X)$
	:$$f_X(x) = \frac{1}{\sqrt{2\pi}\cdot\sigma_X}\cdot {\rm e}^{-(X-m_X)^2 /(2\sigma_X^2) }.$$		:$$f_X(x) = \frac{1}{\sqrt{2\pi}\cdot\sigma_X}\cdot {\rm e}^{-(X-m_X)^2 /(2\sigma_X^2) }.$$

← Older revision		Revision as of 15:48, 3 February 2025
Line 517:		Line 517:
	*The first version was created in 2005 by  [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Studierende#Bettina_Hirner_.28Diplomarbeit_LB_2005.29\|»Bettina Hirner«]]  as part of her diploma thesis with “FlashMX – Actionscript”  (Supervisor:  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29\| »Günter Söder« ]]  and  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Klaus_Eichin_.28at_LNT_from_1972-2011.29\| »Klaus Eichin« ]]).		*The first version was created in 2005 by  [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Studierende#Bettina_Hirner_.28Diplomarbeit_LB_2005.29\|»Bettina Hirner«]]  as part of her diploma thesis with “FlashMX – Actionscript”  (Supervisor:  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29\| »Günter Söder« ]]  and  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Klaus_Eichin_.28at_LNT_from_1972-2011.29\| »Klaus Eichin« ]]).

	*In 2019 the program was redesigned via HTML5/JavaScript by  [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#Matthias_Niller_.28Ingenieurspraxis_Math_2019.29\|»Matthias Niller«]]  (Ingenieurspraxis Mathematik, Supervisor:  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#~~Benedikt_Leible~~.~~2C_M~~.~~Sc._~~.28at_L.C3.9CT_since_2017.29\| »Benedikt Leible« ]]  and  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Tasn.C3.A1d_Kernetzky_.28at_L.C3.9CT_from_2014-2022.29\| »Tasnád Kernetzky« ]] ).		*In 2019 the program was redesigned via HTML5/JavaScript by  [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#Matthias_Niller_.28Ingenieurspraxis_Math_2019.29\|»Matthias Niller«]]  (Ingenieurspraxis Mathematik, Supervisor:  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Benedikt_Leible_.28at_L.C3.9CT_since_2017.29\| »Benedikt Leible« ]]  and  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Tasn.C3.A1d_Kernetzky_.28at_L.C3.9CT_from_2014-2022.29\| »Tasnád Kernetzky« ]] ).

	*Last revision and English version 2021 by  [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#Carolin_Mirschina_.28Ingenieurspraxis_Math_2019.2C_danach_Werkstudentin.29\|»Carolin Mirschina«]]  in the context of a working student activity.		*Last revision and English version 2021 by  [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#Carolin_Mirschina_.28Ingenieurspraxis_Math_2019.2C_danach_Werkstudentin.29\|»Carolin Mirschina«]]  in the context of a working student activity.

← Older revision		Revision as of 12:17, 26 October 2023
Line 517:		Line 517:
	*The first version was created in 2005 by  [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Studierende#Bettina_Hirner_.28Diplomarbeit_LB_2005.29\|»Bettina Hirner«]]  as part of her diploma thesis with “FlashMX – Actionscript”  (Supervisor:  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29\| »Günter Söder« ]]  and  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Klaus_Eichin_.28at_LNT_from_1972-2011.29\| »Klaus Eichin« ]]).		*The first version was created in 2005 by  [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Studierende#Bettina_Hirner_.28Diplomarbeit_LB_2005.29\|»Bettina Hirner«]]  as part of her diploma thesis with “FlashMX – Actionscript”  (Supervisor:  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29\| »Günter Söder« ]]  and  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Klaus_Eichin_.28at_LNT_from_1972-2011.29\| »Klaus Eichin« ]]).

	*In 2019 the program was redesigned via HTML5/JavaScript by  [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#Matthias_Niller_.28Ingenieurspraxis_Math_2019.29\|»Matthias Niller«]]  (Ingenieurspraxis Mathematik, Supervisor:  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Benedikt_Leible.2C_M.Sc._.28at_L.C3.9CT_since_2017.29\| »Benedikt Leible« ]]  and  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#~~Tasn~~.C3.~~A1d_Kernetzky~~.~~2C_M.Sc~~._.28at_L.C3.~~9CT_since_2014~~.29\| »Tasnád Kernetzky« ]] ).		*In 2019 the program was redesigned via HTML5/JavaScript by  [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#Matthias_Niller_.28Ingenieurspraxis_Math_2019.29\|»Matthias Niller«]]  (Ingenieurspraxis Mathematik, Supervisor:  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Benedikt_Leible.2C_M.Sc._.28at_L.C3.9CT_since_2017.29\| »Benedikt Leible« ]]  and  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Tasn.C3.A1d_Kernetzky_.28at_L.C3.9CT_from_2014-2022.29\| »Tasnád Kernetzky« ]] ).

	*Last revision and English version 2021 by  [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#Carolin_Mirschina_.28Ingenieurspraxis_Math_2019.2C_danach_Werkstudentin.29\|»Carolin Mirschina«]]  in the context of a working student activity.		*Last revision and English version 2021 by  [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#Carolin_Mirschina_.28Ingenieurspraxis_Math_2019.2C_danach_Werkstudentin.29\|»Carolin Mirschina«]]  in the context of a working student activity.