Theory of Stochastic Signals/Poisson Distribution - Revision history

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2026-03-16T12:28:51Z

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

Guenter at 17:58, 7 February 2024

2024-02-07T17:58:00Z

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Guenter at 17:04, 7 February 2024

2024-02-07T17:04:28Z

← Older revision		Revision as of 19:04, 7 February 2024
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	<br>		<br>
	{{BlaueBox\|TEXT=		{{BlaueBox\|TEXT=
	$\text{Definition:}$  The  »'''Poisson distribution'''«  is a limiting case of the  [[Theory_of_Stochastic_Signals/Binomial_Distribution#General_description_of_the_binomial_distribution\|~~$\text{~~binomial distribution}$]],  where		$\text{Definition:}$  The  »'''Poisson distribution'''«  is a limiting case of the  [[Theory_of_Stochastic_Signals/Binomial_Distribution#General_description_of_the_binomial_distribution\|»binomial distribution»]],  where
	*on the one hand,  the limit transitions  $I → ∞$  and  $p → 0$  are assumed,		*on the one hand,  the limit transitions  $I → ∞$  and  $p → 0$  are assumed,
	*additionally,  it is assumed that the product~~ ~~ $I · p = λ$~~  has a finite value.~~
			*additionally,  it is assumed that the product has following value:
			:$$I · p =\it λ.$$

			The parameter  $\it \lambda $  gives the average number of  »ones«  in a fixed time unit and is called the  »'''rate'''«. }}

	~~The parameter  $\it \lambda $  gives the average number of  "ones"  in a fixed unit of time and is called the  »'''rate'''«. }}~~

			$\text{Further,  it should be noted:}$
			*In contrast to the binomial distribution  $(0 ≤ μ ≤ I)$  here the random quantity can take on arbitrarily large  $($integer,  non-negative$)$  values.

	~~Further,  it should be noted:~~		*This means that the set of possible values is uncountable here.
	*In contrast to the binomial distribution  $(0 ≤ μ ≤ I)$  here the random quantity can take on arbitrarily large  (integer,  non-negative)  values.
	*This means that the set of possible values ~~here~~ is uncountable.		*But since no intermediate values can occur,  this is also called a  »discrete distribution«.
	*But since no intermediate values can occur,  this is also called a  "discrete distribution".


	{{BlueBox\|TEXT=		{{BlueBox\|TEXT=
	$\text{Calculation rule:}$		$\text{Calculation rule:}$  Considering above limit transitions for the  [[Theory_of_Stochastic_Signals/Binomial_Distribution#Probabilities_of_the_binomial_distribution\|»probabilities of the binomial distribution«]],  it follows for the  »'''Poisson distribution probabilities'''«:

	*Considering above limit transitions for the  [[Theory_of_Stochastic_Signals/Binomial_Distribution#Probabilities_of_the_binomial_distribution\|~~$\text{~~probabilities of the binomial distribution}$]],  it follows for the  »'''~~probabilities of~~ Poisson distribution'''«:
	:$$p_\mu = {\rm Pr} ( z=\mu ) = \lim_{I\to\infty} \cdot \frac{I !}{\mu ! \cdot (I-\mu )!} \cdot (\frac{\lambda}{I} )^\mu \cdot ( 1-\frac{\lambda}{I})^{I-\mu}.$$		:$$p_\mu = {\rm Pr} ( z=\mu ) = \lim_{I\to\infty} \cdot \frac{I !}{\mu ! \cdot (I-\mu )!} \cdot (\frac{\lambda}{I} )^\mu \cdot ( 1-\frac{\lambda}{I})^{I-\mu}.$$
	*From this,  after some algebraic transformations~~,  we obtain~~:		From this,  after some algebraic transformations:
	:$$p_\mu = \frac{ \lambda^\mu}{\mu!}\cdot {\rm e}^{-\lambda}.$$}}		:$$p_\mu = \frac{ \lambda^\mu}{\mu!}\cdot {\rm e}^{-\lambda}.$$}}


	[[File: EN_Sto_T_2_4_S1_neu.png \|frame\| Probabilities of the Poisson distribution<br>compared to the binomial probabilities\| right]]		{{GraueBox\|TEXT=
	~~{{GraueBox\|TEXT=~~		[[File: EN_Sto_T_2_4_S1_neu.png \|frame\| Probabilities of the Poisson distribution<br>compared to the binomial probabilities\| right]]
	$\text{Example 1:}$  ~~The~~ probabilities		$\text{Example 1:}$  In the graph on the right can be seen the probabilities of
	*~~of the~~ binomial distribution with  $I =6$,  $p = 0.4$,  and		*binomial distribution with  $I =6$,  $p = 0.4$,  $($blue arrows and labels$)$
	*~~of the~~ Poisson distribution with  $λ = 2.4$
			*Poisson distribution with  $λ = 2.4$   $($red arrows and labels$)$.


	~~can be seen in the graph on the right. ~~ You can recognize:		You can recognize:
	*Both distributions have the same mean  $m_1 = 2.4$.		#Both distributions have the same mean  $m_1 = 2.4$.
	*In the ~~Poisson~~ distribution  ~~(red arrows and labels)~~  the~~  "~~outer values"  are more probable than in the binomial distribution.		#In the binomial distribution,  all probabilities  ${\rm Pr}(z > 6) \equiv 0$.
	*In addition,  random variables  $z > 6$  are also possible with the Poisson distribution,  but their probabilities are also rather small at the chosen rate. }}		#In the Poisson distribution the outer values  are more probable than with the binomial distribution.
			#Random variables  $z > 6$  are also possible with the Poisson distribution,  but their probabilities are also rather small at the chosen rate. }}

Guenter at 16:29, 19 April 2023

2023-04-19T16:29:11Z

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	The parameter  $λ$  gives the average number of  "ones"  in a fixed unit of time and is called the  »'''rate'''«. }}		The parameter  $\it \lambda $  gives the average number of  "ones"  in a fixed unit of time and is called the  »'''rate'''«. }}

Guenter at 14:30, 13 December 2022

2022-12-13T14:30:03Z

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	With the interactive HTML 5/JavaScript applet  [[Applets:Binomial_and_Poisson_Distribution_(Applet)\|"Binomial and Poisson Distribution"]]		⇒   With the interactive HTML 5/JavaScript applet  [[Applets:Binomial_and_Poisson_Distribution_(Applet)\|"Binomial and Poisson Distribution"]]
	*you can determine the probabilities and means (moments) of the Poisson distribution for any  $λ$-values		*you can determine the probabilities and means (moments) of the Poisson distribution for any  $λ$-values
	*and visualize the similarities and differences compared to the binomial distribution.		*and visualize the similarities and differences compared to the binomial distribution.

Hwang at 17:32, 12 December 2022

2022-12-12T17:32:46Z

← Older revision		Revision as of 19:32, 12 December 2022
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	==Applications of the Poisson distribution==		==Applications of the Poisson distribution==
	<br>		<br>
	The Poisson distribution is the result of a so-called  [https://en.wikipedia.org/wiki/Poisson_point_process $\text{Poisson process}$].  Such a process is often used as a model for sequences of events that may occur at random times.  Examples of such events include.		The Poisson distribution is the result of a so-called  [https://en.wikipedia.org/wiki/Poisson_point_process $\text{Poisson process}$].  Such a process is often used as a model for sequences of events that may occur at random times.  Examples of such events include
	*the failure of equipment - an important task in reliability theory,		*the failure of equipment - an important task in reliability theory,
	*the shot noise in optical transmission,  and		*the shot noise in optical transmission,  and

Hwang at 17:25, 12 December 2022

2022-12-12T17:25:52Z

← Older revision		Revision as of 19:25, 12 December 2022
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	$\text{Calculation rule:}$		$\text{Calculation rule:}$

	*Considering above limit transitions for the  [[Theory_of_Stochastic_Signals/Binomial_Distribution#Probabilities_of_the_binomial_distribution\|$\text{probabilities of the binomial distribution}$]],  it follows for the  »'''~~Probabilities~~ of Poisson ~~Distribution~~'''«:		*Considering above limit transitions for the  [[Theory_of_Stochastic_Signals/Binomial_Distribution#Probabilities_of_the_binomial_distribution\|$\text{probabilities of the binomial distribution}$]],  it follows for the  »'''probabilities of Poisson distribution'''«:
	:$$p_\mu = {\rm Pr} ( z=\mu ) = \lim_{I\to\infty} \cdot \frac{I !}{\mu ! \cdot (I-\mu )!} \cdot (\frac{\lambda}{I} )^\mu \cdot ( 1-\frac{\lambda}{I})^{I-\mu}.$$		:$$p_\mu = {\rm Pr} ( z=\mu ) = \lim_{I\to\infty} \cdot \frac{I !}{\mu ! \cdot (I-\mu )!} \cdot (\frac{\lambda}{I} )^\mu \cdot ( 1-\frac{\lambda}{I})^{I-\mu}.$$
	*From this,  after some algebraic transformations,  we obtain:		*From this,  after some algebraic transformations,  we obtain:

Hwang at 17:05, 12 December 2022

2022-12-12T17:05:32Z

Guenter at 14:06, 16 February 2022

2022-02-16T14:06:21Z

← Older revision		Revision as of 16:06, 16 February 2022
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	*Both distributions have exactly the same mean  $m_1 = 2.4$.		*Both distributions have exactly the same mean  $m_1 = 2.4$.
	*For the Poisson distribution (marked red in the figure),  the standard deviation~~&nbsp (standard deviation)~~  $σ ≈ 1.55$.		*For the Poisson distribution (marked red in the figure),  the standard deviation  $σ ≈ 1.55$.
	*In contrast,  for the (blue) binomial distribution,  the standard deviation is only  $σ = 1.2$.}}		*In contrast,  for the (blue) binomial distribution,  the standard deviation is only  $σ = 1.2$.}}

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

2022-02-15T15:45:53Z

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

← Older revision		Revision as of 17:45, 15 February 2022
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	$\text{Calculation rule:}$		$\text{Calculation rule:}$

	*The  '''mean'''  and the  '''~~rms value~~'''  of the Poisson distribution are obtained directly from the  [[Theory_of_Stochastic_Signals/Binomial_Distribution#Moments_of_the_binomial_distribution\|corresponding equations of the binomial distribution]]  by twofold limiting:		*The  '''mean'''  and the  '''standard deviation'''  of the Poisson distribution are obtained directly from the  [[Theory_of_Stochastic_Signals/Binomial_Distribution#Moments_of_the_binomial_distribution\|corresponding equations of the binomial distribution]]  by twofold limiting:
	:$$m_1 =\lim_{\left.{I\hspace{0.05cm}\to\hspace{0.05cm}\infty \atop {p\hspace{0.05cm}\to\hspace{0.05cm} 0} }\right.} I \cdot p= \lambda,$$		:$$m_1 =\lim_{\left.{I\hspace{0.05cm}\to\hspace{0.05cm}\infty \atop {p\hspace{0.05cm}\to\hspace{0.05cm} 0} }\right.} I \cdot p= \lambda,$$
	:$$\sigma =\lim_{\left.{I\hspace{0.05cm}\to\hspace{0.05cm}\infty \atop {p\hspace{0.05cm}\to\hspace{0.05cm} 0} }\right.} \sqrt{I \cdot p \cdot (1-p)} = \sqrt {\lambda}.$$		:$$\sigma =\lim_{\left.{I\hspace{0.05cm}\to\hspace{0.05cm}\infty \atop {p\hspace{0.05cm}\to\hspace{0.05cm} 0} }\right.} \sqrt{I \cdot p \cdot (1-p)} = \sqrt {\lambda}.$$
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	*Both distributions have exactly the same mean  $m_1 = 2.4$.		*Both distributions have exactly the same mean  $m_1 = 2.4$.
	*For the Poisson distribution (marked red in the figure),  the ~~rms value~~&nbsp (standard deviation)  $σ ≈ 1.55$.		*For the Poisson distribution (marked red in the figure),  the standard deviation&nbsp (standard deviation)  $σ ≈ 1.55$.
	*In contrast,  for the (blue) binomial distribution,  the ~~rms value~~ is only  $σ = 1.2$.}}		*In contrast,  for the (blue) binomial distribution,  the standard deviation is only  $σ = 1.2$.}}

← Older revision		Revision as of 19:05, 12 December 2022
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	<br>		<br>
	{{BlaueBox\|TEXT=		{{BlaueBox\|TEXT=
	$\text{Definition:}$  The  '''Poisson distribution'''  is a limiting case of the  [[Theory_of_Stochastic_Signals/Binomial_Distribution#General_description_of_the_binomial_distribution\|binomial distribution]],  where		$\text{Definition:}$  The  »'''Poisson distribution'''«  is a limiting case of the  [[Theory_of_Stochastic_Signals/Binomial_Distribution#General_description_of_the_binomial_distribution\|$\text{binomial distribution}$]],  where
	*on the one hand,  the limit transitions  $I → ∞$  and  $p → 0$  are assumed,		*on the one hand,  the limit transitions  $I → ∞$  and  $p → 0$  are assumed,
	*additionally,  it is assumed that the product  $I · p = λ$  has a finite value.		*additionally,  it is assumed that the product  $I · p = λ$  has a finite value.


	The parameter  $λ$  gives the average number of  "ones"  in a fixed unit of time and is called the  '''rate'''. }}		The parameter  $λ$  gives the average number of  "ones"  in a fixed unit of time and is called the  »'''rate'''«. }}


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	$\text{Calculation rule:}$		$\text{Calculation rule:}$

	*Considering above limit transitions for the  [[Theory_of_Stochastic_Signals/Binomial_Distribution#Probabilities_of_the_binomial_distribution\|probabilities of the binomial distribution]],  it follows for the  '''Probabilities of Poisson Distribution''':		*Considering above limit transitions for the  [[Theory_of_Stochastic_Signals/Binomial_Distribution#Probabilities_of_the_binomial_distribution\|$\text{probabilities of the binomial distribution}$]],  it follows for the  »'''Probabilities of Poisson Distribution'''«:
	:$$p_\mu = {\rm Pr} ( z=\mu ) = \lim_{I\to\infty} \cdot \frac{I !}{\mu ! \cdot (I-\mu )!} \cdot (\frac{\lambda}{I} )^\mu \cdot ( 1-\frac{\lambda}{I})^{I-\mu}.$$		:$$p_\mu = {\rm Pr} ( z=\mu ) = \lim_{I\to\infty} \cdot \frac{I !}{\mu ! \cdot (I-\mu )!} \cdot (\frac{\lambda}{I} )^\mu \cdot ( 1-\frac{\lambda}{I})^{I-\mu}.$$
	*From this,  after some algebraic transformations,  we obtain:		*From this,  after some algebraic transformations,  we obtain:
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	$\text{Calculation rule:}$		$\text{Calculation rule:}$

	*The  '''mean'''  and the  '''standard deviation'''  of the Poisson distribution are obtained directly from the  [[Theory_of_Stochastic_Signals/Binomial_Distribution#Moments_of_the_binomial_distribution\|corresponding equations of the binomial distribution]]  by twofold limiting:		*The  »'''mean'''«  and the  »'''standard deviation'''«  of the Poisson distribution are obtained directly from the  [[Theory_of_Stochastic_Signals/Binomial_Distribution#Moments_of_the_binomial_distribution\|$\text{corresponding equations of the binomial distribution}$]]  by twofold limiting:
	:$$m_1 =\lim_{\left.{I\hspace{0.05cm}\to\hspace{0.05cm}\infty \atop {p\hspace{0.05cm}\to\hspace{0.05cm} 0} }\right.} I \cdot p= \lambda,$$		:$$m_1 =\lim_{\left.{I\hspace{0.05cm}\to\hspace{0.05cm}\infty \atop {p\hspace{0.05cm}\to\hspace{0.05cm} 0} }\right.} I \cdot p= \lambda,$$
	:$$\sigma =\lim_{\left.{I\hspace{0.05cm}\to\hspace{0.05cm}\infty \atop {p\hspace{0.05cm}\to\hspace{0.05cm} 0} }\right.} \sqrt{I \cdot p \cdot (1-p)} = \sqrt {\lambda}.$$		:$$\sigma =\lim_{\left.{I\hspace{0.05cm}\to\hspace{0.05cm}\infty \atop {p\hspace{0.05cm}\to\hspace{0.05cm} 0} }\right.} \sqrt{I \cdot p \cdot (1-p)} = \sqrt {\lambda}.$$
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	[[File: EN_Sto_T_2_4_S3.png \|right\|frame\| Scheme for binomial distribution  (red)  and Poisson distribution  (blue)]]		[[File: EN_Sto_T_2_4_S3.png \|right\|frame\| Scheme for binomial distribution  (red)  and Poisson distribution  (blue)]]

	The  '''binomial distribution'''  is suitable for the description of such stochastic events,  which are characterized by a given clock  $T$.   For example,  for  [[Examples_of_Communication_Systems/General_Description_of_ISDN\|ISDN]]  ("Integrated Services Digital Network")  with  $64 \ \rm kbit/s$   ⇒   the clock time  $T \approx 15.6 \ \rm µ s$.		The  »'''binomial distribution'''«  is suitable for the description of such stochastic events,  which are characterized by a given clock  $T$.   For example,  for  [[Examples_of_Communication_Systems/General_Description_of_ISDN\|$\text{ISDN}$]]  ("Integrated Services Digital Network")  with  $64 \ \rm kbit/s$   ⇒   the clock time  $T \approx 15.6 \ \rm µ s$.
	*'''Binary events only occur in this time grid'''.  Such events are,  for example,  error-free  $(e_i = 0)$  or errored  $(e_i = 1)$  transmission of individual symbols.		*'''Binary events only occur in this time grid'''.  Such events are,  for example,  error-free  $(e_i = 0)$  or errored  $(e_i = 1)$  transmission of individual symbols.
	*The binomial distribution now allows statistical statements about the number of transmission errors to be expected in a longer time interval  $T_{\rm I} = I ⋅ T$  according to the upper diagram of the graph  (time marked in blue).		*The binomial distribution now allows statistical statements about the number of transmission errors to be expected in a longer time interval  $T_{\rm I} = I ⋅ T$  according to the upper diagram of the graph  (time marked in blue).


	Also the  '''Poisson distribution'''  makes statements about the number of occurring binary events in a finite time interval:		Also the  »'''Poisson distribution'''«  makes statements about the number of occurring binary events in a finite time interval:
	*If one assumes the same observation period  $T_{\rm I}$  and increases the number  $I$  of subintervals more and more,  then the clock time  $T$,  at which a new binary event  ("0"  or  "1")  can occur,  becomes smaller and smaller.  In the limiting case  $T \to 0$.		*If one assumes the same observation period  $T_{\rm I}$  and increases the number  $I$  of subintervals more and more,  then the clock time  $T$,  at which a new binary event  ("0"  or  "1")  can occur,  becomes smaller and smaller.  In the limiting case  $T \to 0$.
	*This means:  In the Poisson distribution,  '''the binary events are possible'''  not only at discrete time points given by a time grid,  but  '''at any time'''.  The time diagram below illustrates this fact.		*This means:  In the Poisson distribution,  '''the binary events are possible'''  not only at discrete time points given by a time grid,  but  '''at any time'''.  The time diagram below illustrates this fact.
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	==Applications of the Poisson distribution==		==Applications of the Poisson distribution==
	<br>		<br>
	The Poisson distribution is the result of a so-called  [https://en.wikipedia.org/wiki/Poisson_point_process Poisson process].  Such a process is often used as a model for sequences of events that may occur at random times.  Examples of such events include.		The Poisson distribution is the result of a so-called  [https://en.wikipedia.org/wiki/Poisson_point_process $\text{Poisson process}$].  Such a process is often used as a model for sequences of events that may occur at random times.  Examples of such events include.
	*the failure of equipment - an important task in reliability theory,		*the failure of equipment - an important task in reliability theory,
	*the shot noise in optical transmission,  and		*the shot noise in optical transmission,  and
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	From this,  further characteristics can be derived:		From this,  further characteristics can be derived:
	*The distance  $τ$  between two connection requests satisfies the  [[Theory_of_Stochastic_Signals/Exponentially_Distributed_Random_Variables#One-sided_exponential_distribution\|exponential distribution]].		*The distance  $τ$  between two connection requests satisfies the  [[Theory_of_Stochastic_Signals/Exponentially_Distributed_Random_Variables#One-sided_exponential_distribution\|$\text{exponential distribution}$]].
	*The mean time interval between two connection requests is  ${\rm E}[\hspace{0.05cm}τ\hspace{0.05cm}] = 1/λ ≈ 0.667 \ \rm s$.}}		*The mean time interval between two connection requests is  ${\rm E}[\hspace{0.05cm}τ\hspace{0.05cm}] = 1/λ ≈ 0.667 \ \rm s$.}}