Aufgaben:Exercise 3.11: Chebyshev's Inequality - Revision history

Guenter at 15:39, 13 March 2023

2023-03-13T15:39:05Z

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

2022-02-17T11:34:31Z

Text replacement - " rms " to " standard deviation "

Bene: Text replacement - "root mean square" to "standard deviation"

2022-02-17T11:01:02Z

Text replacement - "root mean square" to "standard deviation"

Guenter at 16:01, 8 February 2022

2022-02-08T16:01:41Z

Guenter at 11:59, 3 February 2022

2022-02-03T11:59:59Z

Guenter at 11:41, 3 February 2022

2022-02-03T11:41:37Z

Guenter at 11:31, 3 February 2022

2022-02-03T11:31:07Z

Noah at 21:41, 12 January 2022

2022-01-12T21:41:20Z

Guenter at 10:32, 20 December 2021

2021-12-20T10:32:35Z

Guenter: Guenter moved page Aufgaben:Aufgabe 3.11: Tschebyscheffsche Ungleichung to Aufgaben:Exercise 3.11: Chebyshev's Inequality

2021-12-20T10:30:04Z

Guenter moved page Aufgabe 3.11: Tschebyscheffsche Ungleichung to Exercise 3.11: Chebyshev's Inequality

@@ Line 3: / Line 3: @@
 }}
-[[File:EN_Sto_A_3_11_neu.png|right|frame|Exemplary Chebyshev's bound '''Korrektur''' Chebyshev]]
+[[File:EN_Sto_A_3_11_neu.png|right|frame|Exemplary Chebyshev bound]]
 [[File:P_ID921__Sto_A_3_11_b.png|frame|Values of the&nbsp; "complementary Gaussian error function"]]

← Older revision		Revision as of 11:34, 17 February 2022
Line 20:		Line 20:


−	From this plot it can be seen that the  "Chebyshev's Inequality"  is only a very rough bound.  <br>It should be used only if really only the mean and the ~~rms~~ are known from the random size.	+	From this plot it can be seen that the  "Chebyshev's Inequality"  is only a very rough bound.  <br>It should be used only if really only the mean and the standard deviation are known from the random size.

@@ Line 8: / Line 8: @@
 If nothing else is known about a random variable&nbsp; $x$&nbsp; than only
 *the mean value&nbsp; $m_x$,&nbsp; and
-*the root mean square&nbsp; $\sigma_x$,
+*the standard deviation&nbsp; $\sigma_x$,

@@ Line 3: / Line 3: @@
 }}
-[[File:EN_Sto_A_3_11.png|right|frame|Exemplary Chebyshev's bound '''Korrektur''' Chebyshev]]
+[[File:EN_Sto_A_3_11_neu.png|right|frame|Exemplary Chebyshev's bound '''Korrektur''' Chebyshev]]
 [[File:P_ID921__Sto_A_3_11_b.png|frame|Values of the&nbsp; "complementary Gaussian error function"]]

@@ Line 3: / Line 3: @@
 }}
-[[File:EN_Sto_A_3_11.png|right|frame|Exemplary Chebyshev's bound]]
+[[File:EN_Sto_A_3_11.png|right|frame|Exemplary Chebyshev's bound '''Korrektur''' Chebyshev]]
 [[File:P_ID921__Sto_A_3_11_b.png|frame|Values of the&nbsp; "complementary Gaussian error function"]]
@@ Line 64: / Line 64: @@
 *For&nbsp; $\varepsilon < \sigma_x$ &nbsp; &rArr; &nbsp; Chebyshev yields a probability greater&nbsp; $1$.&nbsp; This information is useless.
 *The last statement is true.&nbsp; For example,&nbsp; with the uniform distribution:
-:$${\rm Pr}(| x- m_x | \ge \varepsilon)=\left\{ \begin{array}{*{4}{c}} 1-{\varepsilon}/{\varepsilon_{\rm 0}} & \rm f\ddot{u}r\hspace{0.1cm}{\it \varepsilon<\varepsilon_{\rm 0}=\sqrt{\rm 3}\cdot\sigma_x},\rm 0 & \rm else. \end{array} \right. $$
+:$${\rm Pr}(| x- m_x | \ge \varepsilon)=\left\{ \begin{array}{*{4}{c}} 1-{\varepsilon}/{\varepsilon_{\rm 0}} & \rm for\hspace{0.1cm}{\it \varepsilon<\varepsilon_{\rm 0}=\sqrt{\rm 3}\cdot\sigma_x},\\\rm 0 & \rm else. \end{array} \right. $$
@@ Line 88: / Line 88: @@
 :$$k= 1\text{:}\hspace{0.5cm} {\rm Pr}(|x-m_x| \ge \sigma_{x})= \rm e^{-2}= \rm 13.53\%,$$
 :$$k= 2\text{:}\hspace{0.5cm} {\rm Pr}(|x-m_x| \ge 2 \cdot \sigma_{x})= \rm \rm e^{-3}=\rm 4.97\% ,$$
-:$$k= 3\text\text{:}\hspace{0.5cm} {\rm Pr}(|x-m_x| \ge 3 \cdot\sigma_{x})= \rm \rm e^{-4}\hspace{0.15cm}\underline{ =\rm 1.83\% },$$
+:$$k= 3\text{:}\hspace{0.5cm} {\rm Pr}(|x-m_x| \ge 3 \cdot\sigma_{x})= \rm \rm e^{-4}\hspace{0.15cm}\underline{ =\rm 1.83\% },$$
 :$$k= 4\text{:}\hspace{0.5cm} {\rm Pr}(|x-m_x| \ge 4 \cdot \sigma_{x}) = \rm e^{-5}= \rm 0.67\%.$$
 {{ML-Fuß}}

@@ Line 7: / Line 7: @@
 [[File:P_ID921__Sto_A_3_11_b.png|frame|Values of the&nbsp; "complementary Gaussian error function"]]
 If nothing else is known about a random variable&nbsp; $x$&nbsp; than only
-*the mean value&nbsp; $m_x$&nbsp; and
+*the mean value&nbsp; $m_x$,&nbsp; and
 *the root mean square&nbsp; $\sigma_x$,
@@ Line 59: / Line 59: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; Correct are <u>the proposed solutions 2 and 3</u>:
+'''(1)'''&nbsp; Correct are&nbsp; <u>the proposed solutions 2 and 3</u>:
-*The first statement is false. Chebyshev's inequality provides the bound here&nbsp; $1/9$.
+*The first statement is false.&nbsp; Here,&nbsp; Chebyshev's inequality provides the bound&nbsp; $1/9$.
-*For no distribution can the probability considered here be equal&nbsp; $1/4$&nbsp;.
+*For no distribution the probability considered here can be equal&nbsp; $1/4$.
-*For&nbsp; $\varepsilon < \sigma_x$&nbsp; Chebyshev yields a probability greater&nbsp; $1$.&nbsp; This information is useless.
+*For&nbsp; $\varepsilon < \sigma_x$ &nbsp; &rArr; &nbsp; Chebyshev yields a probability greater&nbsp; $1$.&nbsp; This information is useless.
-*The last statement is true.&nbsp; For example, in the uniform distribution:
+*The last statement is true.&nbsp; For example,&nbsp; with the uniform distribution:
 :$${\rm Pr}(| x- m_x | \ge \varepsilon)=\left\{ \begin{array}{*{4}{c}} 1-{\varepsilon}/{\varepsilon_{\rm 0}} & \rm f\ddot{u}r\hspace{0.1cm}{\it \varepsilon<\varepsilon_{\rm 0}=\sqrt{\rm 3}\cdot\sigma_x},\rm 0 & \rm else. \end{array} \right. $$
 '''(2)'''&nbsp; For the Gaussian distribution holds:
 :$$p_k={\rm Pr}(| x-m_x| \ge k\cdot\sigma_{x})=\rm 2\cdot \rm Q(\it k).$$
-*This results in the following numerical values&nbsp; (in brackets: &nbsp; bound according to Chebyshev):
+*This results in the following numerical values&nbsp; $($in brackets: &nbsp; bound according to Chebyshev$)$:
 :$$k= 1\text{:}\hspace{0.5cm} {\rm Pr}(|x-m_x| \ge \sigma_{x}) = 31.7 \% \hspace{0.3cm}(100 \%),$$
 :$$k= 2\text{:}\hspace{0.5cm} {\rm Pr}(|x-m_x| \ge 2 \cdot \sigma_{x}) = 4.54 \% \hspace{0.3cm}(25 \%),$$
@@ Line 75: / Line 76: @@
 :$$k= 4\text{:}\hspace{0.5cm} {\rm Pr}(|x-m_x| \ge 4 \cdot \sigma_{x}) = 0.0064 \% \hspace{0.3cm}(6.25 \%).$$
 '''(3)'''&nbsp; Without restricting generality, we set&nbsp; $\lambda = 1$
 &nbsp; &#8658; &nbsp; $m_x = \sigma_x = 1$.&nbsp; Then holds:
 :$${\rm Pr}(|x - m_x| \ge k\cdot\sigma_{x}) = {\rm Pr}(| x-1| \ge k).$$
-*Since in this special case the random variable is always&nbsp; $x >0$&nbsp;, it further holds:
+*Since in this special case the random variable is always&nbsp; $x >0$,&nbsp; it further holds:
 :$$p_k= {\rm Pr}( x \ge k+1)=\int_{k+\rm 1}^{\infty}\hspace{-0.15cm}
 {\rm e}^{-x}\, {\rm d} x={\rm e}^{-( k + 1)}.$$
 *This yields the following numerical values for the exponential distribution:
-:$$k= 1\text{:}\hspace{0.5cm} {\rm Pr}(|x-m_x| \ge \sigma_{x}) \rm e^{-2}= \rm 13.53\%,$$
+:$$k= 1\text{:}\hspace{0.5cm} {\rm Pr}(|x-m_x| \ge \sigma_{x})= \rm e^{-2}= \rm 13.53\%,$$
 :$$k= 2\text{:}\hspace{0.5cm} {\rm Pr}(|x-m_x| \ge 2 \cdot \sigma_{x})= \rm \rm e^{-3}=\rm 4.97\% ,$$
 :$$k= 3\text\text{:}\hspace{0.5cm} {\rm Pr}(|x-m_x| \ge 3 \cdot\sigma_{x})= \rm \rm e^{-4}\hspace{0.15cm}\underline{ =\rm 1.83\% },$$

@@ Line 3: / Line 3: @@
 }}
-[[File:EN_Sto_A_3_11.png|right|frame|Exemplary Chebyshevsch barrier]]
+[[File:EN_Sto_A_3_11.png|right|frame|Exemplary Chebyshev's bound]]
-If nothing else is known about a random size&nbsp; $x$&nbsp; than only
 *the mean value&nbsp; $m_x$&nbsp; and
-*the rms&nbsp; $\sigma_x$,
+*the root mean square&nbsp; $\sigma_x$,
-so the&nbsp;  <i>Chebyshev's Inequality</i>&nbsp; an upper bound on the probability that $x$&nbsp; deviates by more than one value from its mean value $\varepsilon$&nbsp; is given.
+so the&nbsp;  "Chebyshev's Inequality"&nbsp; gives an upper bound on the probability that $x$&nbsp; deviates by more than a value&nbsp; $\varepsilon$&nbsp; from its mean.&nbsp; This bound is:
 :$${\rm Pr}(|x-m_x|\ge \varepsilon) \le {\sigma_x^{\rm 2}}/{\varepsilon^{\rm 2}}.$$
 To explain:
-*In the graph, this upper bound is drawn in red.
+*In the graph,&nbsp; this upper bound is drawn in red.
-*The larger curve shows the actual probability in the uniform distribution.
+*The green curve shows the actual probability for the uniform distribution.
 *The blue points are for the exponential distribution.
-From this plot it can be seen that the&nbsp; <i>Chebyshev's Inequality</i>&nbsp; is only a very rough bound.&nbsp; <br>It should be used only if really only the mean and the rms are known from the random size.
+From this plot it can be seen that the&nbsp; "Chebyshev's Inequality"&nbsp; is only a very rough bound.&nbsp; <br>It should be used only if really only the mean and the rms are known from the random size.
 <br>
 Hints:
-*The exercise belongs to the chapter&nbsp; [[Theory_of_Stochastic_Signals/Further_Distributions|Further Distributions]].
+*The exercise belongs to the chapter&nbsp; [[Theory_of_Stochastic_Signals/Further_Distributions|"Further Distributions"]].
-*In particular, reference is made to the page&nbsp;  [[Theory_of_Stochastic_Signals/Further_Distributions#Chebyshev.27s_inequality|Chebyshev's inequality]]&nbsp;.
+*In particular,&nbsp; reference is made to the section&nbsp;  [[Theory_of_Stochastic_Signals/Further_Distributions#Chebyshev.27s_inequality|"Chebyshev's inequality"]]&nbsp;.
-*On the right, values of the complementary Gaussian error function&nbsp; ${\rm Q}(x)$&nbsp; are given.
+*On the right,&nbsp; values of the complementary Gaussian error function&nbsp; ${\rm Q}(x)$&nbsp; are given.
@@ Line 39: / Line 39: @@
 {Which of the following statements are true?
 |type="[]"}
-- Conceivably, a random variable with&nbsp; ${\rm Pr}(|x -m_x> | \ge 3\sigma_x) = 1/4$.
+- Conceivably,&nbsp; a random variable with&nbsp; ${\rm Pr}(|x -m_x | \ge 3\sigma_x) = 1/4$.
-+ "Chebyshev" yields f&uuml;r&nbsp; $\varepsilon < \sigma_x$&nbsp; no information.
++ "Chebyshev"&nbsp; yields for&nbsp; $\varepsilon < \sigma_x$&nbsp; no information.
-+ ${\rm Pr}(|x -m_x> | \ge \sigma_x)$&nbsp; is identically zero for large&nbsp; $\varepsilon$&nbsp; if&nbsp; $x$&nbsp; is bounded.
++ ${\rm Pr}(|x -m_x | \ge \sigma_x)$&nbsp; is identically zero for large&nbsp; $\varepsilon$&nbsp; if&nbsp; $x$&nbsp; is bounded.

@@ Line 1: / Line 1: @@
-{{quiz-Header|Buchseite=Stochastische Signaltheorie/Weitere Verteilungen
+{{quiz-Header|Buchseite=Theory_of_Stochastic_Signals/Further_Distributions
 }}
-[[File:EN_Sto_A_3_11.png|right|frame|Beispielhafte Tschebyscheffsch&ndash;Schranke]]
+[[File:EN_Sto_A_3_11.png|right|frame|Exemplary Chebyshevsch barrier]]
-Ist &uuml;ber eine Zufallsgr&ouml;&szlig;e&nbsp; $x$&nbsp; nichts weiter bekannt als nur
+If nothing else is known about a random size&nbsp; $x$&nbsp; than only
-*der Mittelwert&nbsp; $m_x$&nbsp; und
+*the mean value&nbsp; $m_x$&nbsp; and
-*die Streuung&nbsp; $\sigma_x$,
+*the rms&nbsp; $\sigma_x$,
-so gibt die&nbsp;  <i>Tschebyscheffsche Ungleichung</i>&nbsp; eine obere Schranke f&uuml;r die Wahrscheinlichkeit an, dass&nbsp; $x$&nbsp; betragsm&auml;&szlig;ig mehr als einen Wert&nbsp; $\varepsilon$&nbsp; von seinem Mittelwert&nbsp; $m_x$&nbsp; abweicht.
+so the&nbsp;  <i>Chebyshev's Inequality</i>&nbsp; an upper bound on the probability that $x$&nbsp; deviates by more than one value from its mean value $\varepsilon$&nbsp; is given.
-Diese Schranke lautet:
+This bound is:
-:$${\rm Pr}(|x-m_x|\ge \varepsilon) \le  {\sigma_x^{\rm 2}}/{\varepsilon^{\rm 2}}.$$
+:$${\rm Pr}(|x-m_x|\ge \varepsilon) \le {\sigma_x^{\rm 2}}/{\varepsilon^{\rm 2}}.$$
-Zur Erläuterung:
+To explain:
-*In der Grafik ist diese obere Schranke rot eingezeichnet.
+*In the graph, this upper bound is drawn in red.
-*Der gr&uuml;ne Kurvenverlauf zeigt die tatsächliche Wahrscheinlichkeit bei der Gleichverteilung.
+*The larger curve shows the actual probability in the uniform distribution.
-*Die blauen Punkte gelten f&uuml;r die Exponentialverteilung.
+*The blue points are for the exponential distribution.
-Aus dieser Darstellung ist zu erkennen, dass die&nbsp; <i>Tschebyscheffsche Ungleichung</i>&nbsp; nur eine sehr grobe Schranke darstellt.&nbsp; <br>Sie sollte nur dann verwendet werden, wenn von der Zufallsgr&ouml;&szlig;e wirklich nur der Mittelwert und die Streuung bekannt sind.
+From this plot it can be seen that the&nbsp; <i>Chebyshev's Inequality</i>&nbsp; is only a very rough bound.&nbsp; <br>It should be used only if really only the mean and the rms are known from the random size.
-[[File:P_ID921__Sto_A_3_11_b.png|frame|Werte der komplementären Gaußschen Fehlerfunktion]]
+[[File:P_ID921__Sto_A_3_11_b.png|frame|Values of the complementary Gaussian error function]]
 <br>
-''Hinweise:''
+Hints:
-*Die Aufgabe gehört zum  Kapitel&nbsp; [[Theory_of_Stochastic_Signals/Weitere_Verteilungen|Weitere Verteilungen]].
+*The exercise belongs to the chapter&nbsp; [[Theory_of_Stochastic_Signals/Further_Distributions|Further Distributions]].
-*Insbesondere wird auf die Seite&nbsp;  [[Theory_of_Stochastic_Signals/Weitere_Verteilungen#Tschebyscheffsche_Ungleichung|Tschebyscheffsche Ungleichung]]&nbsp; Bezug genommen .
+*In particular, reference is made to the page&nbsp;  [[Theory_of_Stochastic_Signals/Further_Distributions#Chebyshev.27s_inequality|Chebyshev's inequality]]&nbsp;.
-*Rechts sind Werte der komplementären Gaußschen Fehlerfunktion&nbsp; ${\rm Q}(x)$&nbsp; angegeben.
+*On the right, values of the complementary Gaussian error function&nbsp; ${\rm Q}(x)$&nbsp; are given.
@@ Line 34: / Line 34: @@
-===Fragebogen===
+===Questions===
 <quiz display=simple>
-{Welche der folgenden Aussagen sind zutreffend?
+{Which of the following statements are true?
 |type="[]"}
-- Vorstellbar ist eine Zufallsgröße mit&nbsp; ${\rm Pr}(|x -m_x> | \ge 3\sigma_x) = 1/4$.
+- Conceivably, a random variable with&nbsp; ${\rm Pr}(|x -m_x> | \ge 3\sigma_x) = 1/4$.
-+ "Tschebyscheff" liefert f&uuml;r&nbsp; $\varepsilon < \sigma_x$&nbsp; keine Information.
++ "Chebyshev" yields f&uuml;r&nbsp; $\varepsilon < \sigma_x$&nbsp; no information.
-+ ${\rm Pr}(|x -m_x> | \ge \sigma_x)$&nbsp; ist für große&nbsp; $\varepsilon$&nbsp; identisch Null, wenn&nbsp; $x$&nbsp; begrenzt ist.
++ ${\rm Pr}(|x -m_x> | \ge \sigma_x)$&nbsp; is identically zero for large&nbsp; $\varepsilon$&nbsp; if&nbsp; $x$&nbsp; is bounded.
-{Es gelte&nbsp; $k = 1, \ 2, \ 3, \ 4$.&nbsp; Geben Sie die Überschreitungswahrscheinlichkeit&nbsp; $p_k = {\rm Pr}(|x -m_x | \ge k \cdot \sigma_x)$&nbsp; für die <u>Gau&szlig;verteilung</u> an.&nbsp; Wie gro&szlig; ist&nbsp; $p_3$?
+{It holds&nbsp; $k = 1, \ 2, \ 3, \ 4$.&nbsp; Give the excess probability&nbsp; $p_k = {\rm Pr}(|x -m_x | \ge k \cdot \sigma_x)$&nbsp; for the <u>Gaussian distribution</u>.&nbsp; How large is&nbsp; $p_3$?
 |type="{}"}
 ${\rm Pr}(|x -m_x | \ge 3 \sigma_x) \ = \ $ { 0.26 3% } $\ \%$
-{Welche Überschreitungswahrscheinlichkeiten&nbsp;  $p_k$&nbsp; ergeben sich bei der&nbsp; <u>Exponentialverteilung</u>.&nbsp; Hier gilt &nbsp; $m_x = \sigma_x = 1/\lambda$.&nbsp; Wie gro&szlig; ist&nbsp; $p_3$?
+{What are the excess probabilities&nbsp; $p_k$&nbsp; for the&nbsp; <u>exponential distribution</u>.&nbsp; Here &nbsp; $m_x = \sigma_x = 1/\lambda$.&nbsp; What is&nbsp; $p_3$?
 |type="{}"}
 ${\rm Pr}(|x -m_x | \ge 3 \sigma_x) \ = \ $ { 1.83 3% } $\ \%$
@@ Line 57: / Line 57: @@
 </quiz>
-===Musterlösung===
+===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; Richtig sind <u>die Lösungsvorschläge 2 und 3</u>:
+'''(1)'''&nbsp; Correct are <u>the proposed solutions 2 and 3</u>:
-*Die erste Aussage ist falsch.&nbsp; Die Tschebyscheffsche Ungleichung liefert hier die Schranke&nbsp; $1/9$.
+*The first statement is false. Chebyshev's inequality provides the bound here&nbsp; $1/9$.
-*Bei keiner Verteilung kann die hier betrachtete Wahrscheinlichkeit gleich&nbsp; $1/4$&nbsp; sein.
+*For no distribution can the probability considered here be equal&nbsp; $1/4$&nbsp;.
-*Für&nbsp; $\varepsilon < \sigma_x$&nbsp; liefert Tschebyscheff eine Wahrscheinlichkeit gr&ouml;&szlig;er als&nbsp; $1$.&nbsp; Diese Information ist  nutzlos.
+*For&nbsp; $\varepsilon < \sigma_x$&nbsp; Chebyshev yields a probability greater&nbsp; $1$.&nbsp; This information is useless.
-*Die letzte Aussage ist zutreffend.&nbsp; Beispielsweise gilt bei der Gleichverteilung:
+*The last statement is true.&nbsp; For example, in the uniform distribution:
-:$${\rm Pr}(| x- m_x | \ge \varepsilon)=\left\{ \begin{array}{*{4}{c}} 1-{\varepsilon}/{\varepsilon_{\rm 0}} & \rm f\ddot{u}r\hspace{0.1cm}{\it \varepsilon<\varepsilon_{\rm 0}=\sqrt{\rm 3}\cdot\sigma_x},\\\rm 0 & \rm sonst. \end{array} \right. $$
+:$${\rm Pr}(| x- m_x | \ge \varepsilon)=\left\{ \begin{array}{*{4}{c}} 1-{\varepsilon}/{\varepsilon_{\rm 0}} & \rm f\ddot{u}r\hspace{0.1cm}{\it \varepsilon<\varepsilon_{\rm 0}=\sqrt{\rm 3}\cdot\sigma_x},\rm 0 & \rm else. \end{array} \right. $$
-'''(2)'''&nbsp; Bei der Gau&szlig;verteilung gilt:
+'''(2)'''&nbsp; For the Gaussian distribution holds:
 :$$p_k={\rm Pr}(| x-m_x| \ge k\cdot\sigma_{x})=\rm 2\cdot \rm Q(\it k).$$
-*Daraus ergeben sich folgende Zahlenwerte&nbsp; (in Klammern: &nbsp; Schranke nach Tschebyscheff):
+*This results in the following numerical values&nbsp; (in brackets: &nbsp; bound according to Chebyshev):
 :$$k= 1\text{:}\hspace{0.5cm} {\rm Pr}(|x-m_x| \ge \sigma_{x}) = 31.7 \% \hspace{0.3cm}(100 \%),$$
 :$$k= 2\text{:}\hspace{0.5cm} {\rm Pr}(|x-m_x| \ge 2 \cdot \sigma_{x}) = 4.54 \% \hspace{0.3cm}(25 \%),$$
@@ Line 75: / Line 75: @@
 :$$k= 4\text{:}\hspace{0.5cm} {\rm Pr}(|x-m_x| \ge 4 \cdot \sigma_{x}) = 0.0064 \% \hspace{0.3cm}(6.25 \%).$$
-'''(3)'''&nbsp; Ohne Einschr&auml;nkung der Allgemeing&uuml;ltigkeit setzen wir&nbsp; $\lambda = 1$
+'''(3)'''&nbsp; Without restricting generality, we set&nbsp; $\lambda = 1$
-&nbsp; &#8658; &nbsp; $m_x = \sigma_x = 1$.&nbsp; Dann gilt:
+&nbsp; &#8658; &nbsp; $m_x = \sigma_x = 1$.&nbsp; Then holds:
-:$${\rm Pr}(|x - m_x| \ge  k\cdot\sigma_{x}) = {\rm Pr}(| x-1| \ge  k).$$
+:$${\rm Pr}(|x - m_x| \ge k\cdot\sigma_{x}) = {\rm Pr}(| x-1| \ge k).$$
-*Da in diesem Sonderfall die Zufallsgröße stets&nbsp; $x >0$&nbsp; ist, gilt weiter:
+*Since in this special case the random variable is always&nbsp; $x >0$&nbsp;, it further holds:
 :$$p_k= {\rm Pr}( x \ge k+1)=\int_{k+\rm 1}^{\infty}\hspace{-0.15cm}
 {\rm e}^{-x}\, {\rm d} x={\rm e}^{-( k + 1)}.$$
-*Daraus ergeben sich für die Exponentialverteilung folgende Zahlenwerte:
+*This yields the following numerical values for the exponential distribution:
-:$$k= 1\text{:}\hspace{0.5cm} {\rm Pr}(|x-m_x| \ge \sigma_{x})  \rm e^{-2}= \rm 13.53\%,$$
+:$$k= 1\text{:}\hspace{0.5cm} {\rm Pr}(|x-m_x| \ge \sigma_{x}) \rm e^{-2}= \rm 13.53\%,$$
 :$$k= 2\text{:}\hspace{0.5cm} {\rm Pr}(|x-m_x| \ge 2 \cdot \sigma_{x})= \rm \rm e^{-3}=\rm 4.97\% ,$$
 :$$k= 3\text\text{:}\hspace{0.5cm} {\rm Pr}(|x-m_x| \ge 3 \cdot\sigma_{x})= \rm \rm e^{-4}\hspace{0.15cm}\underline{ =\rm 1.83\% },$$

← Older revision		Revision as of 10:32, 20 December 2021
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−	[[Category:Theory of Stochastic Signals: Exercises\|^3.7 ~~Weitere Verteilungen~~^]]	+	[[Category:Theory of Stochastic Signals: Exercises\|^3.7 Further Distributions^]]

← Older revision	Revision as of 10:30, 20 December 2021
(No difference)