Applets:Binomial and Poisson Distribution (Applet) - Versionsgeschichte

Höfler: Textersetzung - „Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29“ durch „Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_von_1974-2024.29“

2025-02-24T09:43:33Z

Textersetzung - „Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29“ durch „Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_von_1974-2024.29“

Höfler am 26. Oktober 2023 um 09:35 Uhr

2023-10-26T09:35:07Z

Tasnad am 9. Juni 2020 um 20:42 Uhr

2020-06-09T20:42:18Z

Jimmy: /* Applet Manual */

2018-03-12T04:27:14Z

Applet Manual

Jimmy: /* Zur Handhabung des Applets */

2018-03-12T04:17:29Z

Zur Handhabung des Applets

Jimmy am 12. März 2018 um 04:07 Uhr

2018-03-12T04:07:51Z

Guenter am 28. Februar 2018 um 15:07 Uhr

2018-02-28T15:07:42Z

Guenter am 28. Februar 2018 um 14:55 Uhr

2018-02-28T14:55:41Z

Guenter am 28. Februar 2018 um 14:20 Uhr

2018-02-28T14:20:02Z

Tasnad am 26. Februar 2018 um 21:31 Uhr

2018-02-26T21:31:45Z

@@ Zeile 263: / Zeile 263: @@
 ==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].
-*The original version was created in 2003 by[[Biografien_und_Bibliografien/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&ndash;Actionscript&rdquo;  (Supervisor: [[Biografien_und_Bibliografien/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[[Biografien_und_Bibliografien/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&ndash;Actionscript&rdquo;  (Supervisor: [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_von_1974-2024.29|Günter Söder]]).
 *In 2018 this Applet was redesigned and updated to &quot;HTML5&quot; by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Jimmy_He_.28Bachelorarbeit_2018.29|Jimmy He]] as part of his Bachelor's thesis (Supervisor: [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_LÜT-Angehörige#Dr.-Ing._Tasn.C3.A1d_Kernetzky_.28bei_L.C3.9CT_von_2014-2022.29|Tasnád Kernetzky]]) .

@@ Zeile 264: / Zeile 264: @@
 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[[Biografien_und_Bibliografien/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&ndash;Actionscript&rdquo;  (Supervisor: [[Biografien_und_Bibliografien/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 &quot;HTML5&quot; by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Jimmy_He_.28Bachelorarbeit_2018.29|Jimmy He]] as part of his Bachelor's thesis (Supervisor: [[Biografien_und_Bibliografien/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 &quot;HTML5&quot; by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Jimmy_He_.28Bachelorarbeit_2018.29|Jimmy He]] as part of his Bachelor's thesis (Supervisor: [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_LÜT-Angehörige#Dr.-Ing._Tasn.C3.A1d_Kernetzky_.28bei_L.C3.9CT_von_2014-2022.29|Tasnád Kernetzky]]) .
 ==Once again: Open Applet in new Tab==
 {{LntAppletLinkEn|binomPoissonDistributions_en}}

@@ Zeile 1: / Zeile 1: @@
-{{LntAppletLinkEn|verteilungen}}
+{{LntAppletLinkEn|binomPoissonDistributions_en}}
 ==Applet Description==
@@ Zeile 268: / Zeile 268: @@
 ==Once again: Open Applet in new Tab==
-{{LntAppletLinkEn|verteilungen}}
+{{LntAppletLinkEn|binomPoissonDistributions_en}}

@@ Zeile 230: / Zeile 230: @@
 ==Applet Manual==
 [[Datei:Handhabung_binomial.png|left|600px]]
-&nbsp; &nbsp; '''(A)''' &nbsp; &nbsp; Vorauswahl für blauen Parametersatz
+&nbsp; &nbsp; '''(A)''' &nbsp; &nbsp; Preselection for blue parameter set
 &nbsp; &nbsp; '''(B)''' &nbsp; &nbsp; Parameter input: Sliders $I$ and $p$

@@ Zeile 228: / Zeile 228: @@
-==Zur Handhabung des Applets==
+==Applet Manual==
 [[Datei:Handhabung_binomial.png|left|600px]]
 &nbsp; &nbsp; '''(A)''' &nbsp; &nbsp; Vorauswahl für blauen Parametersatz
-&nbsp; &nbsp; '''(B)''' &nbsp; &nbsp; Parametereingabe $I$ und $p$ per Slider
+&nbsp; &nbsp; '''(B)''' &nbsp; &nbsp; Parameter input: Sliders $I$ and $p$
-&nbsp; &nbsp; '''(C)''' &nbsp; &nbsp; Vorauswahl für roten Parametersatz
+&nbsp; &nbsp; '''(C)''' &nbsp; &nbsp; Preselection for Red parameter set
-&nbsp; &nbsp; '''(D)''' &nbsp; &nbsp; Parametereingabe $\lambda$ per Slider
+&nbsp; &nbsp; '''(D)''' &nbsp; &nbsp; Parameter input: Slider $\lambda$
-&nbsp; &nbsp; '''(E)''' &nbsp; &nbsp; Graphische Darstellung der Verteilungen
+&nbsp; &nbsp; '''(E)''' &nbsp; &nbsp; Graphic display of the Distribution
-&nbsp; &nbsp; '''(F)''' &nbsp; &nbsp; Momentenausgabe für blauen Parametersatz
+&nbsp; &nbsp; '''(F)''' &nbsp; &nbsp; Output of moments for blue parameter set
-&nbsp; &nbsp; '''(G)''' &nbsp; &nbsp; Momentenausgabe für roten Parametersatz
+&nbsp; &nbsp; '''(G)''' &nbsp; &nbsp; Output of moments for redparameter set
-&nbsp; &nbsp; '''(H)''' &nbsp; &nbsp; Variation der grafischen Darstellung
+&nbsp; &nbsp; '''(H)''' &nbsp; &nbsp; Variation possibilities for the graphic display
-$\hspace{1.5cm}$&bdquo;$+$&rdquo; (Vergrößern),
+$\hspace{1.5cm}$ &bdquo;$-$&rdquo; (Zoom out)
-$\hspace{1.5cm}$ &bdquo;$-$&rdquo; (Verkleinern)
+$\hspace{1.5cm}$ &bdquo;$\rm o$&rdquo; (Reset)
-$\hspace{1.5cm}$ &bdquo;$\rm o$&rdquo; (Zurücksetzen)
+$\hspace{1.5cm}$ &bdquo;$\leftarrow$&rdquo; (Move left),  etc.
-$\hspace{1.5cm}$ &bdquo;$\leftarrow$&rdquo; (Verschieben nach links),  usw.
+&nbsp; &nbsp; '''( I )''' &nbsp; &nbsp; Output of ${\rm Pr} (z = \mu)$ and ${\rm Pr} (z  \le \mu)$
-&nbsp; &nbsp; '''( I )''' &nbsp; &nbsp; Ausgabe von ${\rm Pr} (z = \mu)$ und ${\rm Pr} (z  \le \mu)$
+&nbsp; &nbsp; '''(J)''' &nbsp; &nbsp; Exercises: Exercise selection, description and solution
 <br clear=all>
-<br>'''Andere Möglichkeiten zur Variation der grafischen Darstellung''':
+<br>'''Other options for graphic display''':
-*Gedrückte Shifttaste und Scrollen:  Zoomen im Koordinatensystem,
+*Hold shift and scroll: Zoom in on/out of coordinate system,
-*Gedrückte Shifttaste und linke Maustaste: Verschieben des Koordinatensystems.
+*Hold shift and left click: Move the coordinate system.
 ==About the Authors==

← Nächstältere Version		Version vom 12. März 2018, 04:07 Uhr
Zeile 226:		Zeile 226:

	$\hspace{1.85cm}\text{In contrast, the Poisson distribution is never symmetric, since it extends to infinity!}$		$\hspace{1.85cm}\text{In contrast, the Poisson distribution is never symmetric, since it extends to infinity!}$
		+
		+
		+	==Zur Handhabung des Applets==
		+	[[Datei:Handhabung_binomial.png\|left\|600px]]
		+	'''(A)'''     Vorauswahl für blauen Parametersatz
		+
		+	'''(B)'''     Parametereingabe $I$ und $p$ per Slider
		+
		+	'''(C)'''     Vorauswahl für roten Parametersatz
		+
		+	'''(D)'''     Parametereingabe $\lambda$ per Slider
		+
		+	'''(E)'''     Graphische Darstellung der Verteilungen
		+
		+	'''(F)'''     Momentenausgabe für blauen Parametersatz
		+
		+	'''(G)'''     Momentenausgabe für roten Parametersatz
		+
		+	'''(H)'''     Variation der grafischen Darstellung
		+
		+	$\hspace{1.5cm}$&bdquo;$+$” (Vergrößern),
		+
		+	$\hspace{1.5cm}$ &bdquo;$-$” (Verkleinern)
		+
		+	$\hspace{1.5cm}$ &bdquo;$\rm o$” (Zurücksetzen)
		+
		+	$\hspace{1.5cm}$ &bdquo;$\leftarrow$” (Verschieben nach links), usw.
		+
		+	'''( I )'''     Ausgabe von ${\rm Pr} (z = \mu)$ und ${\rm Pr} (z \le \mu)$
		+
		+	'''(J)'''     Bereich für die Versuchsdurchführung
		+	<br clear=all>
		+	<br>'''Andere Möglichkeiten zur Variation der grafischen Darstellung''':
		+	*Gedrückte Shifttaste und Scrollen: Zoomen im Koordinatensystem,
		+	*Gedrückte Shifttaste und linke Maustaste: Verschieben des Koordinatensystems.
		+

	==About the Authors==		==About the Authors==

@@ Zeile 195: / Zeile 195: @@
 $\hspace{1.0cm}\Rightarrow\hspace{0.3cm}\text{Average:}\hspace{0.2cm}m_\text{1} = I \cdot p\hspace{0.3cm} \Rightarrow\hspace{0.3cm} m_\text{1, Blue}  = 5 \cdot 0.4\underline{ = 2 =}  \ m_\text{1, Red} = 10 \cdot 0.2; $
-$\hspace{1.85cm}\text{Standard deviation:}\hspace{0.4cm}\sigma = \sqrt{I \cdot p \cdot (1-p)} = \sqrt{m_1 \cdot (1-p)}\hspace{0.3cm}\Rightarrow\hspace{0.3cm} \sigma_{\rm Blue} = \sqrt{2 \cdot 0.6} =1.095 \le \sigma_{\rm Red} = \sqrt{2 \cdot 0.8} = 1.265.$
+$\hspace{1.85cm}\text{Standard deviation:}\hspace{0.4cm}\sigma = \sqrt{I \cdot p \cdot (1-p)} = \sqrt{m_1 \cdot (1-p)}\hspace{0.3cm}\Rightarrow\hspace{0.3cm} \sigma_{\rm Blue} = \sqrt{2 \cdot 0.6} =1.095 < \sigma_{\rm Red} = \sqrt{2 \cdot 0.8} = 1.265.$
 {{BlaueBox|TEXT=

@@ Zeile 147: / Zeile 147: @@
 [[Datei:  P_ID60__Sto_T_2_4_S3_neu.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.
-* With the Binomial distribution, it is possible to make statistical statements about the number of expected transmission erros in a period $T_{\rm I} = I · T$, as is shown in the time figure above (marked blue).
+* With the Binomial distribution, it is possible to make statistical statements about the number of expected erros in a period $T_{\rm I} = I · T$, as is shown in the time figure above (marked blue).
-* For very large values of $I$ and very small values of $p$, the Binomial distribution can be approximated by the ''Poisson distribution'' with rate $\lambda = I \cdot p$..
+* For very large values of $I$ and very small values of $p$, the Binomial distribution can be approximated by the ''Poisson distribution'' with rate $\lambda = I \cdot p$.
-* If at the same time $I · p \gg 1$, the Poisson distribution as well as the Binomial distribution turn into a discrete Gaussian distribution according to the ''de Moivre-Laplace theorem''.
+* If at the same time $I · p \gg 1$, the Poisson distribution as well as the Binomial distribution turn into a discrete Gaussian distribution according to the ''de Moivre-Laplace Theorem''.

@@ Zeile 6: / Zeile 6: @@
 *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) &ndash; here representation with Dirac functions ${\rm \delta}( z-\mu)$:
 :$$f_{z}(z)=\sum_{\mu=1}^{M}{\rm Pr}(z=\mu)\cdot {\rm \delta}( z-\mu),$$
-*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):
 :$$F_{z}(\mu)={\rm Pr}(z\le\mu).$$
@@ Zeile 49: / Zeile 49: @@
 We can derive the formulas for
-*the linear average:  $m_1 = I\cdot p,$
+*the linear average:  &nbsp; $m_1 = I\cdot p,$
-*the quadratic average: $m_2 = (I^2-I)\cdot p^2+I\cdot p,$
+*the quadratic average: &nbsp; $m_2 = (I^2-I)\cdot p^2+I\cdot p,$
-*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:  &nbsp; $\sigma^2 = {m_2 - m_1^2} = {I \cdot p\cdot (1-p)} \hspace{0.3cm}\Rightarrow \hspace{0.3cm}
 \sigma =  \sqrt{I \cdot p\cdot (1-p)}.$
@@ Zeile 57: / Zeile 57: @@
 '''Applications of the Binomial Distribution'''
-The Binomial distribution has a variety of uses in telecommunications as well as in other disciplines :
+The Binomial distribution has a variety of uses in telecommunications as well as in other disciplines:
 *It characterizes the distribution of rejected parts (Ausschussstücken) in statistical quality control.
 *The simulated bit error rate of a digital transmission system is technically a binomially distributed random variable.
@@ Zeile 66: / Zeile 66: @@
 $\text{Example 1:}$&nbsp;
 When transfering  blocks of $I =5$ binary symbols through a channel, that
-*distorts a symbol with probability $p = 0.1$ &nbsp; &rArr; &nbsp; Random variable $e_i = 1$, and
+*distorts a symbol with probability $p = 0.1$ &nbsp; &rArr; &nbsp; random variable $e_i = 1$, and
-*transfers the symbol undistorted with probability $1 - p = 0.9$  &nbsp; &rArr; &nbsp; Random variable $e_i = 0$,
+*transfers the symbol undistorted with probability $1 - p = 0.9$  &nbsp; &rArr; &nbsp; random variable $e_i = 0$,
@@ Zeile 74: / Zeile 74: @@
 $f$ can now take integer values between $\mu = 0$ (all symbols are correct) and $\mu = I = 5$ (all five symbols are erroneous). We describe the probability of  $\mu$ errors as $p_μ = {\rm Pr}(f = \mu)$.
-*The case that all five symbols are transmitted correctly occurs with the probability of $p_0 = 0.9^{5} ≈ 0.5905$. This can also be seen from the binomial formula for $μ = 0$ , considering the definition „10 over 0“ = 1.
+*The case that all five symbols are transmitted correctly occurs with the probability of $p_0 = 0.9^{5} ≈ 0.5905$. This can also be seen from the binomial formula for $μ = 0$ , considering the definition $5\text{ over } 0 = 1$.
 *A single error $(f = 1)$ occurs with the probability $p_1 = 5\cdot 0.1\cdot 0.9^4\approx 0.3281$. The first factor indicates, that there are $5\text{ over } 1 = 5$ possibe error positions. The other two factors take into account, that one symbol was erroneous and the other four are correct when $f =1$.
 *For $f =2$ there are $5\text{ over } 2 = (5 \cdot 4)/(1 \cdot 2) = 10$ combinations and you get a probability of $p_2 = 10\cdot 0.1^2\cdot 0.9^3\approx 0.0729$.
@@ Zeile 82: / Zeile 82: @@
 A second calculation option would be $p_{\rm R} =  p_{3}  + p_{4} + p_{5}$ with the approximation $p_{\rm R} \approx p_{3} = 0.81\%.$
-The average number of errors in a block is $m_f = 5 \cdot 0.1 = 0.5$ and the variance of the random variable $f$ is $\sigma_f^2 = 5 \cdot 0.1 \cdot 0.9= 0.45  \; \Rightarrow \sigma_f \approx 0.671.$}}
+The average number of errors in a block is $m_f = 5 \cdot 0.1 = 0.5$ and the variance of the random variable $f$ is $\sigma_f^2 = 5 \cdot 0.1 \cdot 0.9= 0.45$ &nbsp; &rArr; &nbsp;  standard deviation $\sigma_f \approx 0.671.$}}
 ===Properties of the Poisson Distribution===
-The ''Poisson Distribution'' is a special case of the Binomial Distribution, where
+The ''Poisson distribution'' is a special case of the Binomial distribution, where
 * $I → \infty$ and $p →0$.
 *Additionally, the parameter $λ = I · p$ must be finite.
 The parameter $λ$ indicates the average number of "ones" in a specified time unit and is called ''rate''.
-Unlike the Binomial distribution where $0 ≤ μ ≤ I$, here, the random variable can assume arbitrarily large non-negative integers, which means that the number of possible valuess is not countable. However, since no intermediate values can occur, the posson distribution is still a "discrete distribution".
+Unlike the Binomial distribution where $0 ≤ μ ≤ I$, here, the random variable can assume arbitrarily large non-negative integers, which means that the number of possible values is not countable. However, since no intermediate values can occur, the Poisson distribution is still a "discrete distribution".
 '''Probabilities of the Poisson Distribution'''
-With the limits $I → \infty$ and $p →0$, the likelihood of occurence of the Poisson distributed random variable $z$ can be derived from the probabilities of the binomial distribution:
+With the limits $I → \infty$ and $p →0$, the likelihood of occurence of the Poisson distributed random variable $z$ can be derived from the probabilities of the Binomial distribution:
 :$$p_\mu = {\rm Pr} ( z=\mu ) = \lim_{I\to\infty} \cdot \frac{I !}{\mu ! \cdot (I-\mu  )!} \cdot (\frac{\lambda}{I}  )^\mu \cdot  ( 1-\frac{\lambda}{I})^{I-\mu}.$$
 After some algebraic transformations we finally obtain
@@ Zeile 105: / Zeile 106: @@
 '''Moments of the Poisson Distribution'''
-The moment 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:
-:$$m_1 =\lim_{\left.{I\hspace{0.05cm}\to\hspace{0.05cm}\infty, \hspace{0.2cm}   {p\hspace{0.05cm}\to\hspace{0.05cm} 0}}\right.} \hspace{0.2cm} I \cdot p= \lambda,$$
+:$$m_1 =\lim_{\left.{I\hspace{0.05cm}\to\hspace{0.05cm}\infty, \hspace{0.2cm}   {p\hspace{0.05cm}\to\hspace{0.05cm} 0}}\right.} \hspace{0.2cm} I \cdot p= \lambda,\hspace{0.8cm}
-:$$\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 $\sigma^2 < m_1$.