Applets:Binomial and Poisson Distribution (Applet) - Revision history

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2026-03-16T13:48:57Z

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	The moments of the Poisson distribution can be derived directly from the corresponding equations of the Binomial distribution by taking the limits again:		The moments of the Poisson distribution can be derived directly from the corresponding equations of the Binomial distribution by taking the limits again:
	:$$m_1 =\lim_{\left.{I\hspace{0.05cm}\to\hspace{0.05cm}\infty, \hspace{0.2cm} {p\hspace{0.05cm}\to\hspace{0.05cm} 0}}\right.} \hspace{0.2cm} I \cdot p= \lambda,\hspace{0.8cm}		:$$m_1 =\lim_{\left.{I\hspace{0.05cm}\to\hspace{0.05cm}\infty, \hspace{0.2cm} {p\hspace{0.05cm}\to\hspace{0.05cm} 0}}\right.} \hspace{0.2cm} I \cdot p= \lambda,\hspace{0.8cm}\sigma =\lim_{\left.{I\hspace{0.05cm}\to\hspace{0.05cm}\infty, \hspace{0.2cm} {p\hspace{0.05cm}\to\hspace{0.05cm} 0}}\right.} \hspace{0.2cm} \sqrt{I \cdot p \cdot (1-p)} = \sqrt {\lambda}.$$
	\sigma =\lim_{\left.{I\hspace{0.05cm}\to\hspace{0.05cm}\infty, \hspace{0.2cm} {p\hspace{0.05cm}\to\hspace{0.05cm} 0}}\right.} \hspace{0.2cm} \sqrt{I \cdot p \cdot (1-p)} = \sqrt {\lambda}.$$

	We can see that for the Poisson distribution $\sigma^2 = m_1 = \lambda$ always holds. In contrast, the moments of the Binomial distribution always fulfill $\sigma^2 < m_1$.		We can see that for the Poisson distribution $\sigma^2 = m_1 = \lambda$ always holds. In contrast, the moments of the Binomial distribution always fulfill $\sigma^2 < m_1$.
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	==About the Authors==		==About the Authors==
	This interactive calculation was designed and realized at the  [http://www.lnt.ei.tum.de/startseite Lehrstuhl für Nachrichtentechnik]  of the  [https://www.tum.de/ Technische Universität München].		This interactive calculation was designed and realized at the  [http://www.lnt.ei.tum.de/startseite Lehrstuhl für Nachrichtentechnik]  of the  [https://www.tum.de/ Technische Universität München].
	*The original version was created in 2003 by  [[Biographies_and_Bibliographies/~~An_LNTwww_beteiligte_Studierende~~#Ji_Li_.28Bachelorarbeit_EI_2003.2C_Diplomarbeit_EI_2005.29\|Ji Li]] as part of her Diploma thesis using "FlashMX–Actionscript"  (Supervisor:  [[Biographies_and_Bibliographies/~~An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten~~#Prof._Dr.-Ing._habil.~~_G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29~~\|Günter Söder]]).		*The original version was created in 2003 by  [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#Ji_Li_.28Bachelorarbeit_EI_2003.2C_Diplomarbeit_EI_2005.29\|Ji Li]] as part of her Diploma thesis using "FlashMX–Actionscript"  (Supervisor:  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._Günter_Söder_(at_LNT_from_1974-2024)\|Günter Söder]]).
	*In 2018 this Applet was redesigned and updated to "HTML5" by  [[Biographies_and_Bibliographies/~~An_LNTwww_beteiligte_Studierende~~#Jimmy_He_.28Bachelorarbeit_2018.29\|Jimmy He]]  as part of his Bachelor's thesis (Supervisor:  [[Biographies_and_Bibliographies/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29\|Tasnád Kernetzky]]) .		*In 2018 this Applet was redesigned and updated to "HTML5" by  [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#Jimmy_He_.28Bachelorarbeit_2018.29\|Jimmy He]]  as part of his Bachelor's thesis (Supervisor:  [[Biographies_and_Bibliographies/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29\|Tasnád Kernetzky]]) .

	==Once again: Open Applet in new Tab==		==Once again: Open Applet in new Tab==

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	{{LntAppletLinkEn\|binomPoissonDistributions_en}}         [https://lntwww.lnt.ei.tum.de/Applets:Binomial-_und_Poissonverteilung_(Applet) '''English Applet with German WIKI description''']		{{LntAppletLinkEn\|binomPoissonDistributions_en}}         [https://lntwww.lnt.ei.tum.de/Applets:Binomial-_und_Poissonverteilung_(Applet) '''English Applet with German WIKI description''']
			[[de:Applets:Binomial_and_Poisson_Distribution_(Applet)]]

Maintenance script: Update domain links: en.lntwww.de → en.lntwww.lnt.ei.tum.de, www.lntwww.de → lntwww.lnt.ei.tum.de

2026-03-16T11:09:51Z

Update domain links: en.lntwww.de → en.lntwww.lnt.ei.tum.de, www.lntwww.de → lntwww.lnt.ei.tum.de

← Older revision		Revision as of 13:09, 16 March 2026
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	{{LntAppletLinkEn\|binomPoissonDistributions_en}}         [https://~~www~~.~~lntwww~~.de/Applets:Binomial-_und_Poissonverteilung_(Applet) '''English Applet with German WIKI description''']		{{LntAppletLinkEn\|binomPoissonDistributions_en}}         [https://lntwww.lnt.ei.tum.de/Applets:Binomial-_und_Poissonverteilung_(Applet) '''English Applet with German WIKI description''']


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	==Once again: Open Applet in new Tab==		==Once again: Open Applet in new Tab==

	{{LntAppletLinkEn\|binomPoissonDistributions_en}}         [https://~~www~~.~~lntwww~~.de/Applets:Binomial-_und_Poissonverteilung_(Applet) '''English Applet with German WIKI description''']		{{LntAppletLinkEn\|binomPoissonDistributions_en}}         [https://lntwww.lnt.ei.tum.de/Applets:Binomial-_und_Poissonverteilung_(Applet) '''English Applet with German WIKI description''']

Noah at 20:40, 26 March 2023

2023-03-26T20:40:48Z

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	This applet allows the calculation and graphical display of		This applet allows the calculation and graphical display of
	*the probabilities ${\rm Pr}(z=\mu)$ of a discrete random variable $z \in \{\mu \} = \{0, 1, 2, 3, \text{...} \}$, that determine its ''Probability Density Function'' (PDF) – here representation with Dirac functions ${\rm \delta}( z-\mu)$:		*the probabilities ${\rm Pr}(z=\mu)$ of a discrete random variable $z \in \{\mu \} = \{0, 1, 2, 3, \text{...} \}$, that determine its ''Probability Density Function'' (PDF) – here representation with Dirac delta functions ${\rm \delta}( z-\mu)$:
	:$$f_{z}(z)=\sum_{\mu=1}^{M}{\rm Pr}(z=\mu)\cdot {\rm \delta}( z-\mu),$$		:$$f_{z}(z)=\sum_{\mu=1}^{M}{\rm Pr}(z=\mu)\cdot {\rm \delta}( z-\mu),$$
	*the probabilities ${\rm Pr}(z \le \mu)$ of the ''Cumulative Distribution Function'' (CDF):		*the probabilities ${\rm Pr}(z \le \mu)$ of the ''Cumulative Distribution Function'' (CDF):
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	the new random variable $f$ ("~~Error~~ per block") calculates to:		the new random variable $f$ ("error per block") calculates to:
	:$$f=\sum_{i=1}^{I}e_i.$$		:$$f=\sum_{i=1}^{I}e_i.$$

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	We can see that for the Poisson distribution $\sigma^2 = m_1 = \lambda$ always holds. In contrast, the moments of the Binomial distribution always fulfill $\sigma^2 < m_1$.		We can see that for the Poisson distribution $\sigma^2 = m_1 = \lambda$ always holds. In contrast, the moments of the Binomial distribution always fulfill $\sigma^2 < m_1$.

	[[File: P_ID616__Sto_T_2_4_S2neu.png \|frame\| Moments of Poisson Distribution ~~\| rechts~~]]		[[File: P_ID616__Sto_T_2_4_S2neu.png \|frame\| Moments of Poisson Distribution]]
	{{GraueBox\|TEXT=		{{GraueBox\|TEXT=
	$\text{Example 2:}$		$\text{Example 2:}$

Hwang at 12:19, 18 January 2023

2023-01-18T12:19:04Z

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	We can derive the formulas for		We can derive the formulas for
	*the linear average:   $m_1 = I\cdot p,$		*the linear average:   $m_1 = I\cdot p,$
	*the ~~quadratic average~~:   $m_2 = (I^2-I)\cdot p^2+I\cdot p,$		*the second moment:   $m_2 = (I^2-I)\cdot p^2+I\cdot p,$
	*the variance and standard deviation:   $\sigma^2 = {m_2 - m_1^2} = {I \cdot p\cdot (1-p)} \hspace{0.3cm}\Rightarrow \hspace{0.3cm}		*the variance and standard deviation:   $\sigma^2 = {m_2 - m_1^2} = {I \cdot p\cdot (1-p)} \hspace{0.3cm}\Rightarrow \hspace{0.3cm}
	\sigma = \sqrt{I \cdot p\cdot (1-p)}.$		\sigma = \sqrt{I \cdot p\cdot (1-p)}.$

Javier: Text replacement - "”" to """

2021-05-28T14:48:47Z

Text replacement - "”" to """

← Older revision		Revision as of 16:48, 28 May 2021
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	the new random variable $f$ ("Error per block~~”~~) calculates to:		the new random variable $f$ ("Error per block") calculates to:
	:$$f=\sum_{i=1}^{I}e_i.$$		:$$f=\sum_{i=1}^{I}e_i.$$

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	'''(H)'''     Variation possibilities for the graphic display		'''(H)'''     Variation possibilities for the graphic display

	$\hspace{1.5cm}$"$+$~~”~~ (Zoom in),		$\hspace{1.5cm}$"$+$" (Zoom in),

	$\hspace{1.5cm}$ "$-$~~”~~ (Zoom out)		$\hspace{1.5cm}$ "$-$" (Zoom out)

	$\hspace{1.5cm}$ "$\rm o$~~”~~ (Reset)		$\hspace{1.5cm}$ "$\rm o$" (Reset)

	$\hspace{1.5cm}$ "$\leftarrow$~~”~~ (Move left), etc.		$\hspace{1.5cm}$ "$\leftarrow$" (Move left), etc.

	'''( I )'''     Output of ${\rm Pr} (z = \mu)$ and ${\rm Pr} (z \le \mu)$		'''( I )'''     Output of ${\rm Pr} (z = \mu)$ and ${\rm Pr} (z \le \mu)$
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	==About the Authors==		==About the Authors==
	This interactive calculation was designed and realized at the  [http://www.lnt.ei.tum.de/startseite Lehrstuhl für Nachrichtentechnik]  of the  [https://www.tum.de/ Technische Universität München].		This interactive calculation was designed and realized at the  [http://www.lnt.ei.tum.de/startseite Lehrstuhl für Nachrichtentechnik]  of the  [https://www.tum.de/ Technische Universität München].
	*The original version was created in 2003 by  [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Studierende#Ji_Li_.28Bachelorarbeit_EI_2003.2C_Diplomarbeit_EI_2005.29\|Ji Li]] as part of her Diploma thesis using "FlashMX–Actionscript~~”~~  (Supervisor:  [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29\|Günter Söder]]).		*The original version was created in 2003 by  [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Studierende#Ji_Li_.28Bachelorarbeit_EI_2003.2C_Diplomarbeit_EI_2005.29\|Ji Li]] as part of her Diploma thesis using "FlashMX–Actionscript"  (Supervisor:  [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29\|Günter Söder]]).
	*In 2018 this Applet was redesigned and updated to "HTML5" by  [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Studierende#Jimmy_He_.28Bachelorarbeit_2018.29\|Jimmy He]]  as part of his Bachelor's thesis (Supervisor:  [[Biographies_and_Bibliographies/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29\|Tasnád Kernetzky]]) .		*In 2018 this Applet was redesigned and updated to "HTML5" by  [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Studierende#Jimmy_He_.28Bachelorarbeit_2018.29\|Jimmy He]]  as part of his Bachelor's thesis (Supervisor:  [[Biographies_and_Bibliographies/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29\|Tasnád Kernetzky]]) .

Javier: Text replacement - "„" to """

2021-05-28T14:46:30Z

Text replacement - "„" to """

← Older revision		Revision as of 16:46, 28 May 2021
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	the new random variable $f$ (~~&bdquo;~~Error per block”) calculates to:		the new random variable $f$ ("Error per block”) calculates to:
	:$$f=\sum_{i=1}^{I}e_i.$$		:$$f=\sum_{i=1}^{I}e_i.$$

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	'''(H)'''     Variation possibilities for the graphic display		'''(H)'''     Variation possibilities for the graphic display

	$\hspace{1.5cm}$~~&bdquo;~~$+$” (Zoom in),		$\hspace{1.5cm}$"$+$” (Zoom in),

	$\hspace{1.5cm}$ ~~&bdquo;~~$-$” (Zoom out)		$\hspace{1.5cm}$ "$-$” (Zoom out)

	$\hspace{1.5cm}$ ~~&bdquo;~~$\rm o$” (Reset)		$\hspace{1.5cm}$ "$\rm o$” (Reset)

	$\hspace{1.5cm}$ ~~&bdquo;~~$\leftarrow$” (Move left), etc.		$\hspace{1.5cm}$ "$\leftarrow$” (Move left), etc.

	'''( I )'''     Output of ${\rm Pr} (z = \mu)$ and ${\rm Pr} (z \le \mu)$		'''( I )'''     Output of ${\rm Pr} (z = \mu)$ and ${\rm Pr} (z \le \mu)$
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	==About the Authors==		==About the Authors==
	This interactive calculation was designed and realized at the  [http://www.lnt.ei.tum.de/startseite Lehrstuhl für Nachrichtentechnik]  of the  [https://www.tum.de/ Technische Universität München].		This interactive calculation was designed and realized at the  [http://www.lnt.ei.tum.de/startseite Lehrstuhl für Nachrichtentechnik]  of the  [https://www.tum.de/ Technische Universität München].
	*The original version was created in 2003 by  [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Studierende#Ji_Li_.28Bachelorarbeit_EI_2003.2C_Diplomarbeit_EI_2005.29\|Ji Li]] as part of her Diploma thesis using ~~&bdquo;~~FlashMX–Actionscript”  (Supervisor:  [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29\|Günter Söder]]).		*The original version was created in 2003 by  [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Studierende#Ji_Li_.28Bachelorarbeit_EI_2003.2C_Diplomarbeit_EI_2005.29\|Ji Li]] as part of her Diploma thesis using "FlashMX–Actionscript”  (Supervisor:  [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29\|Günter Söder]]).
	*In 2018 this Applet was redesigned and updated to "HTML5" by  [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Studierende#Jimmy_He_.28Bachelorarbeit_2018.29\|Jimmy He]]  as part of his Bachelor's thesis (Supervisor:  [[Biographies_and_Bibliographies/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29\|Tasnád Kernetzky]]) .		*In 2018 this Applet was redesigned and updated to "HTML5" by  [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Studierende#Jimmy_He_.28Bachelorarbeit_2018.29\|Jimmy He]]  as part of his Bachelor's thesis (Supervisor:  [[Biographies_and_Bibliographies/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29\|Tasnád Kernetzky]]) .

Guenter at 10:01, 27 January 2021

2021-01-27T10:01:29Z

← Older revision		Revision as of 12:01, 27 January 2021
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	{{LntAppletLinkEn\|binomPoissonDistributions_en}}         [https://www.lntwww.de/Applets:Binomial-_und_Poissonverteilung_(Applet) '''German ~~Version~~''']		{{LntAppletLinkEn\|binomPoissonDistributions_en}}         [https://www.lntwww.de/Applets:Binomial-_und_Poissonverteilung_(Applet) '''English Applet with German WIKI description''']


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	==Once again: Open Applet in new Tab==		==Once again: Open Applet in new Tab==

	{{LntAppletLinkEn\|binomPoissonDistributions_en}}         [https://www.lntwww.de/Applets:Binomial-_und_Poissonverteilung_(Applet) '''German ~~Version~~''']		{{LntAppletLinkEn\|binomPoissonDistributions_en}}         [https://www.lntwww.de/Applets:Binomial-_und_Poissonverteilung_(Applet) '''English Applet with German WIKI description''']

Guenter at 10:56, 19 January 2021

2021-01-19T10:56:34Z

← Older revision		Revision as of 12:56, 19 January 2021
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	This section deals with the similarities and differences between Binomial and Poisson distributions.		This section deals with the similarities and differences between Binomial and Poisson distributions.

	[[File: ~~P_ID60__Sto_T_2_4_S3_neu~~.png \|frame\| Binomial vs. Poisson distribution]]		[[File: EN_Sto_T_2_4_S3.png \|frame\| Binomial vs. Poisson distribution]]

	The '''Binomial distribution''' is used to describe stochastic events, that have a fixed period $T$. For example the period of an ISDN (''Integrated Services Digital Network'') network with $64 \ \rm kbit/s$ is $T \approx 15.6 \ \rm \mu s$.		The '''Binomial distribution''' is used to describe stochastic events, that have a fixed period $T$. For example the period of an ISDN (''Integrated Services Digital Network'') network with $64 \ \rm kbit/s$ is $T \approx 15.6 \ \rm \mu s$.
	* Binary events such as the error-free $(e_i = 0)$/ faulty $(e_i = 1)$ transmission of individual symbols only occur in this time frame.		* Binary events such as the error-free $(e_i = 0)$/ faulty $(e_i = 1)$ transmission of individual symbols only occur in this time frame.
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	==Exercises==		==Exercises==

	In these exercises, the term '''Blue''' refers to distribution function 1 (marked blue in the applet) and the term '''Red''' refers to distribution function 2 (marked red in applet).		* First select the number  $(1,\text{...}, 7)$  of the exercise.  The number  $0$  corresponds to a "Reset":  Same setting as at program start.
			*A task description is displayed.  The parameter values are adjusted.  Solution after pressing "Show solution". <br>
			*In these exercises, the term  '''Blue'''  refers to distribution function 1 (marked blue in the applet) and the term  '''Red'''  refers to distribution function 2 (marked red in applet).

Guenter at 10:43, 19 January 2021

2021-01-19T10:43:31Z

← Older revision		Revision as of 12:43, 19 January 2021
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	{{LntAppletLinkEn\|binomPoissonDistributions_en}}		{{LntAppletLinkEn\|binomPoissonDistributions_en}}         [https://www.lntwww.de/Applets:Binomial-_und_Poissonverteilung_(Applet) '''German Version''']


	==Applet Description==		==Applet Description==
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	'''(H)'''     Variation possibilities for the graphic display		'''(H)'''     Variation possibilities for the graphic display

	$\hspace{1.5cm}$&bdquo;$+$” (Zoom in),		$\hspace{1.5cm}$&bdquo;$+$” (Zoom in),

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	==About the Authors==		==About the Authors==
	This interactive calculation was designed and realized at the [http://www.lnt.ei.tum.de/startseite Lehrstuhl für Nachrichtentechnik] of the [https://www.tum.de/ Technische Universität München].		This interactive calculation was designed and realized at the  [http://www.lnt.ei.tum.de/startseite Lehrstuhl für Nachrichtentechnik]  of the  [https://www.tum.de/ Technische Universität München].
	*The original version was created in 2003 by[[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Studierende#Ji_Li_.28Bachelorarbeit_EI_2003.2C_Diplomarbeit_EI_2005.29\|Ji Li]] as part of her Diploma thesis using &bdquo;FlashMX–Actionscript” (Supervisor: [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29\|Günter Söder]]).		*The original version was created in 2003 by  [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Studierende#Ji_Li_.28Bachelorarbeit_EI_2003.2C_Diplomarbeit_EI_2005.29\|Ji Li]] as part of her Diploma thesis using &bdquo;FlashMX–Actionscript”  (Supervisor:  [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29\|Günter Söder]]).
	*In 2018 this Applet was redesigned and updated to "HTML5" by [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Studierende#Jimmy_He_.28Bachelorarbeit_2018.29\|Jimmy He]] as part of his Bachelor's thesis (Supervisor: [[Biographies_and_Bibliographies/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29\|Tasnád Kernetzky]]) .		*In 2018 this Applet was redesigned and updated to "HTML5" by  [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Studierende#Jimmy_He_.28Bachelorarbeit_2018.29\|Jimmy He]]  as part of his Bachelor's thesis (Supervisor:  [[Biographies_and_Bibliographies/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29\|Tasnád Kernetzky]]) .

	==Once again: Open Applet in new Tab==		==Once again: Open Applet in new Tab==

	{{LntAppletLinkEn\|binomPoissonDistributions_en}}		{{LntAppletLinkEn\|binomPoissonDistributions_en}}         [https://www.lntwww.de/Applets:Binomial-_und_Poissonverteilung_(Applet) '''German Version''']