Theory of Stochastic Signals/Poisson Distribution - Revision history

Guenter at 17:58, 7 February 2024

2024-02-07T17:58:00Z

Guenter at 17:04, 7 February 2024

2024-02-07T17:04:28Z

Guenter at 16:29, 19 April 2023

2023-04-19T16:29:11Z

Guenter at 14:30, 13 December 2022

2022-12-13T14:30:03Z

Hwang at 17:32, 12 December 2022

2022-12-12T17:32:46Z

Hwang at 17:25, 12 December 2022

2022-12-12T17:25:52Z

Hwang at 17:05, 12 December 2022

2022-12-12T17:05:32Z

Guenter at 14:06, 16 February 2022

2022-02-16T14:06:21Z

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

2022-02-15T15:45:53Z

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

Guenter at 14:18, 14 December 2021

2021-12-14T14:18:20Z

@@ Line 33: / Line 33: @@
 {{GraueBox|TEXT=
-[[File: EN_Sto_T_2_4_S1_neu.png |frame| Probabilities of the Poisson distribution<br>compared to the binomial probabilities| right]]
+[[File: EN_Sto_T_2_4_S1_neu.png |frame| Binomial and Poisson  probabilities| right]]
 $\text{Example 1:}$&nbsp; In the graph on the right can be seen the probabilities of
 *binomial distribution with&nbsp; $I =6$,&nbsp; $p = 0.4$,&nbsp; $($blue arrows and labels$)$&nbsp;
@@ Line 44: / Line 44: @@
 #In the binomial distribution,&nbsp; all probabilities&nbsp; ${\rm Pr}(z > 6) \equiv 0$.
 #In the Poisson distribution the outer values&nbsp; are more probable than with the binomial distribution.
-#Random variables&nbsp; $z > 6$&nbsp; are also possible with the Poisson distribution,&nbsp; but their probabilities are also rather small at the chosen rate. }}
+#Random variables&nbsp; $z > 6$&nbsp; are also possible with the Poisson distribution.&nbsp;
@@ Line 50: / Line 51: @@
 <br>
 {{BlueBox|TEXT=
-$\text{Calculation rule:}$&nbsp;
+$\text{Calculation rule:}$&nbsp; &raquo;'''Mean'''&laquo;&nbsp; and&nbsp; &raquo;'''standard deviation'''&laquo;&nbsp;  are obtained directly from the&nbsp; [[Theory_of_Stochastic_Signals/Binomial_Distribution#Moments_of_the_binomial_distribution|&raquo;corresponding equations of the binomial distribution&laquo;]]&nbsp; by twofold limiting:
 :$$m_1 =\lim_{\left.{I\hspace{0.05cm}\to\hspace{0.05cm}\infty \atop {p\hspace{0.05cm}\to\hspace{0.05cm} 0} }\right.} I \cdot p= \lambda,$$
 :$$\sigma =\lim_{\left.{I\hspace{0.05cm}\to\hspace{0.05cm}\infty \atop {p\hspace{0.05cm}\to\hspace{0.05cm} 0} }\right.} \sqrt{I \cdot p \cdot (1-p)} = \sqrt {\lambda}.$$
-*From this it can be seen that in the Poisson distribution the variance is always&nbsp; $σ^2 = m_1 = λ$. }}
+From this it can be seen that with the Poisson distribution the variance is always&nbsp;
-[[File: P_ID616__Sto_T_2_4_S2neu.png |frame| Moments of the Poisson distribution | right]]
+{{GraueBox|TEXT=
-{{GraueBox|TEXT=
+[[File: P_ID616__Sto_T_2_4_S2neu.png |frame| Moments of the Poisson distribution | right]]
-$\text{Example 2:}$&nbsp;
+$\text{Example 2:}$&nbsp; As in&nbsp; $\text{Example 1}$,&nbsp; here we compare
+*the binomial distribution with&nbsp; $I =6$,&nbsp; $p = 0.4$,&nbsp; $($blue arrows and labels$)$&nbsp; and
-As in&nbsp; $\text{Example 1}$,&nbsp; here we compare:
-*the binomial distribution with&nbsp; $I =6$,&nbsp; $p = 0.4$,&nbsp; and
+*the Poisson distribution with&nbsp; $λ = 2.4$ &nbsp; $($red arrows and labels$)$.&nbsp;
 One can see from the accompanying sketch:
-*Both distributions have exactly the same mean&nbsp; $m_1 = 2.4$.
+#Both distributions have exactly the same mean&nbsp; $m_1 = 2.4$.
-*For the Poisson distribution (marked red in the figure),&nbsp; the standard deviation&nbsp; $σ ≈ 1.55$.
+#For the red Poisson distribution,&nbsp; the standard deviation&nbsp; $σ ≈ 1.55$.
-*In contrast,&nbsp; for the (blue) binomial distribution,&nbsp; the standard deviation is only&nbsp; $σ = 1.2$.}}
+#In contrast,&nbsp; for the blue binomial distribution,&nbsp; the standard deviation is only&nbsp; $σ = 1.2$.
-&rArr; &nbsp; With the interactive HTML 5/JavaScript applet&nbsp; [[Applets:Binomial_and_Poisson_Distribution_(Applet)|"Binomial and Poisson Distribution"]]&nbsp;
+&rArr; &nbsp; With the interactive HTML 5/JavaScript applet&nbsp; [[Applets:Binomial_and_Poisson_Distribution_(Applet)|&raquo;Binomial and Poisson Distribution&raquo;]],&nbsp; you can
-*you can determine the probabilities and means (moments) of the Poisson distribution for any&nbsp; $λ$-values
+*determine the probabilities and moments of the Poisson distribution for any&nbsp; $λ$-values
 *and visualize the similarities and differences compared to the binomial distribution.
@@ Line 84: / Line 84: @@
 Now both the similarities and the differences between binomial and Poisson distributed random variables shall be worked out again.
-[[File:  EN_Sto_T_2_4_S3.png |right|frame| Scheme for binomial distribution&nbsp; (red)&nbsp; and Poisson distribution&nbsp; (blue)]]
+The&nbsp; &raquo;'''binomial distribution'''&laquo;&nbsp; is suitable for the description of such stochastic events,&nbsp; which are characterized by a given clock&nbsp; $T$. &nbsp; For example,&nbsp; for&nbsp; [[Examples_of_Communication_Systems/General_Description_of_ISDN|'''ISDN''']]&nbsp; $($&raquo;Integrated Services Digital Network&raquo;$)$&nbsp; with&nbsp; $64 \ \rm kbit/s$ &nbsp; &rArr; &nbsp; the clock time&nbsp; $T \approx 15.6 \ \rm &micro; s$.
-The&nbsp; &raquo;'''binomial distribution'''&laquo;&nbsp; is suitable for the description of such stochastic events,&nbsp; which are characterized by a given clock&nbsp; $T$. &nbsp; For example,&nbsp; for&nbsp; [[Examples_of_Communication_Systems/General_Description_of_ISDN|$\text{ISDN}$]]&nbsp; ("Integrated Services Digital Network")&nbsp; with&nbsp; $64 \ \rm kbit/s$ &nbsp; &rArr; &nbsp; the clock time&nbsp; $T \approx 15.6 \ \rm &micro; s$.
-*'''Binary events only occur in this time grid'''.&nbsp; Such events are,&nbsp; for example,&nbsp; error-free&nbsp; $(e_i = 0)$&nbsp; or errored&nbsp; $(e_i = 1)$&nbsp; transmission of individual symbols.
+#'''Binary events only occur in this time grid'''.&nbsp; Such events are,&nbsp; for example,&nbsp; error-free&nbsp; $(e_i = 0)$&nbsp; or errored&nbsp; $(e_i = 1)$&nbsp; 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&nbsp; $T_{\rm I} = I ⋅ T$&nbsp; according to the upper diagram of the graph&nbsp; (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&nbsp; $T_{\rm I} = I ⋅ T$&nbsp; according to the upper diagram of the graph&nbsp; $($time marked in blue$)$.
 Also the&nbsp; &raquo;'''Poisson distribution'''&laquo;&nbsp; makes statements about the number of occurring binary events in a finite time interval:
-*If one assumes the same observation period&nbsp; $T_{\rm I}$&nbsp; and increases the number&nbsp; $I$&nbsp; of subintervals more and more,&nbsp; then the clock time&nbsp; $T$,&nbsp; at which a new binary event&nbsp; ("0"&nbsp; or&nbsp; "1")&nbsp; can occur,&nbsp; becomes smaller and smaller.&nbsp; In the limiting case&nbsp; $T \to 0$.
+#If one assumes the same observation period&nbsp; $T_{\rm I}$&nbsp; and increases the number&nbsp; $I$&nbsp; of subintervals more and more,&nbsp; then the clock time&nbsp; $T$,&nbsp; at which a new binary event&nbsp; $(0$&nbsp; or&nbsp; $1)$&nbsp; can occur,&nbsp; becomes smaller and smaller.&nbsp; In the limiting case:&nbsp; $T \to 0$.
-*This means:&nbsp; In the Poisson distribution,&nbsp; '''the binary events are possible'''&nbsp; not only at discrete time  points given by a time grid,&nbsp; but&nbsp; '''at any time'''.&nbsp; The time diagram below illustrates this fact.
+#This means:&nbsp; In the Poisson distribution,&nbsp; '''the binary events are possible'''&nbsp; not only at discrete time  points given by a time grid,&nbsp; but&nbsp; '''at any time'''.&nbsp;
-*In order to obtain on average during time&nbsp; $T_{\rm I}$&nbsp; exactly as many&nbsp; "ones"&nbsp; as in the binomial distribution&nbsp; (in the example:&nbsp; six),&nbsp; however,&nbsp; the characteristic probability&nbsp; $p = {\rm Pr}( e_i = 1)$&nbsp; related to the infinitesimally small time interval&nbsp; $T$&nbsp; must tend to zero.
+#In order to obtain during time&nbsp; $T_{\rm I}$&nbsp; on average exactly as many&nbsp; on average &raquo;ones&laquo;&nbsp; as in the binomial distribution&nbsp; $($in the example:&nbsp; six$)$,&nbsp; the characteristic probability related to the infinitesimally small  interval&nbsp; $T$&nbsp; must tend to&nbsp; $p = {\rm Pr}( e_i = 1)=0$.
 ==Applications of the Poisson distribution==
 <br>
-The Poisson distribution is the result of a so-called&nbsp; [https://en.wikipedia.org/wiki/Poisson_point_process $\text{Poisson process}$].&nbsp; Such a process is often used as a model for sequences of events that may occur at random times.&nbsp; Examples of such events include
+The Poisson distribution is the result of a so-called&nbsp; [https://en.wikipedia.org/wiki/Poisson_point_process &raquo;Poisson process&laquo;].&nbsp; Such a process is often used as a model for sequences of events that may occur at random times.&nbsp; Examples of such events include
-*the failure of equipment - an important task in reliability theory,
+#the prediction of equipmente failure &ndash; an important task in reliability theory,
-*the shot noise in optical transmission,&nbsp;  and
+#the shot noise in optical transmission,&nbsp;  and
-*the start of telephone calls in a switching center&nbsp; ("teletraffic engineering").
+#the start of telephone calls in a switching center&nbsp; $($&raquo;teletraffic engineering&laquo;$)$.
 {{GraueBox|TEXT=
-$\text{Example 3:}$&nbsp; If ninety switching requests per minute&nbsp; $($&rArr; &nbsp; $λ = 1.5 \text{ per second})$&nbsp; are received by a switching center on a long&ndash;term average,&nbsp; the probabilities&nbsp; $p_\mu$&nbsp; that exactly&nbsp; $\mu$&nbsp; connections occur in any one-second period are:
+$\text{Example 3:}$&nbsp; If ninety switching requests per minute&nbsp; $(λ = 1.5 \text{ per second})$&nbsp; are received by a switching center on a long&ndash;term average,&nbsp; the probabilities&nbsp; $p_\mu$&nbsp; that exactly&nbsp; $\mu$&nbsp; connections occur in any one-second period:
 :$$p_\mu = \frac{1.5^\mu}{\mu!}\cdot {\rm e}^{-1.5}.$$
@@ Line 112: / Line 113: @@
 From this,&nbsp; further characteristics can be derived:
-*The distance&nbsp; $τ$&nbsp; between two connection requests satisfies the&nbsp; [[Theory_of_Stochastic_Signals/Exponentially_Distributed_Random_Variables#One-sided_exponential_distribution|$\text{exponential distribution}$]].
+*The distance&nbsp; $τ$&nbsp; between two connection requests satisfies the&nbsp; [[Theory_of_Stochastic_Signals/Exponentially_Distributed_Random_Variables#One-sided_exponential_distribution|&raquo;exponential distribution&laquo;]].
-*The mean time interval between two connection requests is&nbsp; ${\rm E}[\hspace{0.05cm}τ\hspace{0.05cm}] = 1/λ ≈ 0.667 \ \rm s$.}}

@@ Line 8: / Line 8: @@
 <br>
 {{BlaueBox|TEXT=
-$\text{Definition:}$&nbsp; The&nbsp; &raquo;'''Poisson distribution'''&laquo;&nbsp; is a limiting case of the&nbsp; [[Theory_of_Stochastic_Signals/Binomial_Distribution#General_description_of_the_binomial_distribution|$\text{binomial distribution}$]],&nbsp; where
+$\text{Definition:}$&nbsp; The&nbsp; &raquo;'''Poisson distribution'''&laquo;&nbsp; is a limiting case of the&nbsp; [[Theory_of_Stochastic_Signals/Binomial_Distribution#General_description_of_the_binomial_distribution|&raquo;binomial distribution&raquo;]],&nbsp; where
 *on the one hand,&nbsp; the limit transitions&nbsp; $I → ∞$&nbsp; and&nbsp; $p → 0$&nbsp; are assumed,
-*additionally,&nbsp; it is assumed that the product&nbsp; $I · p = λ$&nbsp; has a finite value.
-Further,&nbsp; it should be noted:
+*This means that the set of possible values is uncountable here.
-*In contrast to the binomial distribution&nbsp; $(0 ≤ μ ≤ I)$&nbsp; here the random quantity can take on arbitrarily large&nbsp; (integer,&nbsp; non-negative)&nbsp; values.
-*This means that the set of possible values here is uncountable.
+*But since no intermediate values can occur,&nbsp; this is also called a&nbsp; &raquo;discrete distribution&laquo;.
 {{BlueBox|TEXT=
-$\text{Calculation rule:}$&nbsp;
+$\text{Calculation rule:}$&nbsp; Considering above limit transitions for the&nbsp; [[Theory_of_Stochastic_Signals/Binomial_Distribution#Probabilities_of_the_binomial_distribution|&raquo;probabilities of the binomial distribution&laquo;]],&nbsp; it follows for the&nbsp; &raquo;'''Poisson distribution probabilities'''&laquo;:
 :$$p_\mu = {\rm Pr} ( z=\mu ) = \lim_{I\to\infty} \cdot \frac{I !}{\mu ! \cdot (I-\mu )!} \cdot (\frac{\lambda}{I} )^\mu \cdot ( 1-\frac{\lambda}{I})^{I-\mu}.$$
-*From this,&nbsp; after some algebraic transformations,&nbsp; we obtain:
+From this,&nbsp; after some algebraic transformations:
 :$$p_\mu = \frac{ \lambda^\mu}{\mu!}\cdot {\rm e}^{-\lambda}.$$}}
-[[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]]
-$\text{Example 1:}$&nbsp; The probabilities
+$\text{Example 1:}$&nbsp; In the graph on the right can be seen the probabilities of
-*of the binomial distribution with&nbsp; $I =6$,&nbsp; $p = 0.4$,&nbsp; and
+*binomial distribution with&nbsp; $I =6$,&nbsp; $p = 0.4$,&nbsp; $($blue arrows and labels$)$&nbsp;
-*of the Poisson distribution with&nbsp; $λ = 2.4$
-can be seen in the graph on the right.&nbsp; You can recognize:
+You can recognize:
-*Both distributions have the same mean&nbsp; $m_1 = 2.4$.
+#Both distributions have the same mean&nbsp; $m_1 = 2.4$.
-*In the Poisson distribution&nbsp; (red arrows and labels)&nbsp; the&nbsp; "outer values"&nbsp; are more probable than in the binomial distribution.
+#In the binomial distribution,&nbsp; all probabilities&nbsp; ${\rm Pr}(z > 6) \equiv 0$.
-*In addition,&nbsp; random variables&nbsp; $z > 6$&nbsp; are also possible with the Poisson distribution,&nbsp; but their probabilities are also rather small at the chosen rate. }}
+#In the Poisson distribution the outer values&nbsp; are more probable than with the binomial distribution.

← Older revision		Revision as of 16:29, 19 April 2023
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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'''«. }}

@@ Line 72: / Line 72: @@
-With the interactive HTML 5/JavaScript applet&nbsp; [[Applets:Binomial_and_Poisson_Distribution_(Applet)|"Binomial and Poisson Distribution"]]&nbsp;
+&rArr; &nbsp; With the interactive HTML 5/JavaScript applet&nbsp; [[Applets:Binomial_and_Poisson_Distribution_(Applet)|"Binomial and Poisson Distribution"]]&nbsp;
 *you can determine the probabilities and means (moments) of the Poisson distribution for any&nbsp; $λ$-values
 *and visualize the similarities and differences compared to the binomial distribution.

@@ Line 96: / Line 96: @@
 ==Applications of the Poisson distribution==
 <br>
-The Poisson distribution is the result of a so-called&nbsp; [https://en.wikipedia.org/wiki/Poisson_point_process $\text{Poisson process}$].&nbsp; Such a process is often used as a model for sequences of events that may occur at random times.&nbsp; Examples of such events include.
+The Poisson distribution is the result of a so-called&nbsp; [https://en.wikipedia.org/wiki/Poisson_point_process $\text{Poisson process}$].&nbsp; Such a process is often used as a model for sequences of events that may occur at random times.&nbsp; Examples of such events include
 *the failure of equipment - an important task in reliability theory,
 *the shot noise in optical transmission,&nbsp;  and

@@ Line 49: / Line 49: @@
 $\text{Calculation rule:}$&nbsp;
-*The&nbsp; '''mean'''&nbsp; and the&nbsp; '''rms value'''&nbsp; of the Poisson distribution are obtained directly from the&nbsp; [[Theory_of_Stochastic_Signals/Binomial_Distribution#Moments_of_the_binomial_distribution|corresponding equations of the binomial distribution]]&nbsp; by twofold limiting:
+*The&nbsp; '''mean'''&nbsp; and the&nbsp; '''standard deviation'''&nbsp; of the Poisson distribution are obtained directly from the&nbsp; [[Theory_of_Stochastic_Signals/Binomial_Distribution#Moments_of_the_binomial_distribution|corresponding equations of the binomial distribution]]&nbsp; by twofold limiting:
 :$$m_1 =\lim_{\left.{I\hspace{0.05cm}\to\hspace{0.05cm}\infty \atop {p\hspace{0.05cm}\to\hspace{0.05cm} 0} }\right.} I \cdot p= \lambda,$$
 :$$\sigma =\lim_{\left.{I\hspace{0.05cm}\to\hspace{0.05cm}\infty \atop {p\hspace{0.05cm}\to\hspace{0.05cm} 0} }\right.} \sqrt{I \cdot p \cdot (1-p)} = \sqrt {\lambda}.$$
@@ Line 68: / Line 68: @@
 *Both distributions have exactly the same mean&nbsp; $m_1 = 2.4$.
-*For the Poisson distribution (marked red in the figure),&nbsp; the rms value&nbsp (standard deviation)&nbsp; $σ ≈ 1.55$.
+*For the Poisson distribution (marked red in the figure),&nbsp; the standard deviation&nbsp (standard deviation)&nbsp; $σ ≈ 1.55$.
-*In contrast,&nbsp; for the (blue) binomial distribution,&nbsp; the rms value is only&nbsp; $σ = 1.2$.}}
+*In contrast,&nbsp; for the (blue) binomial distribution,&nbsp; the standard deviation is only&nbsp; $σ = 1.2$.}}

@@ Line 109: / Line 109: @@
 From this,&nbsp; further characteristics can be derived:
-*The distance&nbsp; $τ$&nbsp; between two placement requests satisfies the&nbsp; [[Theory_of_Stochastic_Signals/Exponentially_Distributed_Random_Variables#One-sided_exponential_distribution|exponential distribution]].
+*The distance&nbsp; $τ$&nbsp; between two connection requests satisfies the&nbsp; [[Theory_of_Stochastic_Signals/Exponentially_Distributed_Random_Variables#One-sided_exponential_distribution|exponential distribution]].
-*The mean time interval between two placement requests is&nbsp; ${\rm E}[\hspace{0.05cm}τ\hspace{0.05cm}] = 1/λ ≈ 0.667 \ \rm s$.}}
+*The mean time interval between two connection requests is&nbsp; ${\rm E}[\hspace{0.05cm}τ\hspace{0.05cm}] = 1/λ ≈ 0.667 \ \rm s$.}}