Theory of Stochastic Signals/Exponentially Distributed Random Variables - Revision history

Hwang at 20:22, 20 December 2022

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Hwang at 20:19, 20 December 2022

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Hwang at 20:12, 20 December 2022

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Hwang at 20:10, 20 December 2022

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Hwang at 20:03, 20 December 2022

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Hwang at 20:02, 20 December 2022

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Hwang at 20:01, 20 December 2022

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Hwang at 23:17, 12 November 2022

2022-11-12T23:17:02Z

Noah at 16:09, 15 February 2022

2022-02-15T16:09:39Z

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

2022-02-15T15:46:03Z

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

@@ Line 131: / Line 131: @@
 * The&nbsp; $k$&ndash;th order  moments&nbsp; $m_k$&nbsp; of the Laplace distribution agree with those of the exponential distribution for even&nbsp; $k$.
 * For odd&nbsp; $k$,&nbsp;  the&nbsp; (symmetric)&nbsp; Laplace distribution always yields&nbsp; $m_k= 0$.
-*For generation the Laplace distribution,&nbsp; one uses a between&nbsp; $±1$&nbsp; uniformly distributed random variable&nbsp; $v$&nbsp; $($where&nbsp; $v = 0$&nbsp; must be excluded$)$&nbsp; and the transformation characteristic curve
+*For generation of the Laplace distribution,&nbsp; one uses a between&nbsp; $±1$&nbsp; uniformly distributed random variable&nbsp; $v$&nbsp; $($where&nbsp; $v = 0$&nbsp; must be excluded$)$&nbsp; and the transformation characteristic curve
 :$$x=\frac{{\rm sign}(v)}{\lambda}\cdot \rm ln(\it v \rm ).$$

@@ Line 119: / Line 119: @@
 ::$$x = g_1(u)= \frac{1}{\lambda} \cdot {\rm ln} \left(\frac{1}{1 - u} \right) .$$
-*In a computer implementation,&nbsp; however,&nbsp; ensure that the critical value&nbsp; $1$&nbsp; is excluded for the uniformly distributed input variable&nbsp; $u$&nbsp; &nbsp;
+*In a computer implementation,&nbsp; however,&nbsp; ensure that the critical value&nbsp; $1$&nbsp; is excluded for the uniformly distributed input variable&nbsp; $u$.&nbsp;
 *This,&nbsp; however,&nbsp; has (almost) no effect on the final result. }}

@@ Line 46: / Line 46: @@
 *However,&nbsp; the above equation is only valid under the condition that the derivative&nbsp; $g\hspace{0.03cm}'(u) \ne 0$.
 *For a characteristic with horizontal sections&nbsp; $(g\hspace{0.05cm}'(u) = 0)$:&nbsp; Additional Dirac delta functions appear in the PDF if the input variable has components in these ranges.
-*The weights of these Dirac functions are equal to the probabilities that the input variable lies in these ranges.
+*The weights of these Dirac delta functions are equal to the probabilities that the input variable lies in these ranges.

@@ Line 38: / Line 38: @@
 {{BlaueBox|TEXT=
-$\text{Procedure:}$&nbsp; If a continuous valued random variable&nbsp; $u$&nbsp; possesses the PDF&nbsp; $f_{u}(u)$,&nbsp; then the probability density function of the random variable transformed at the nonlinear characteristic&nbsp; $x = g(u)$&nbsp; holds:
+$\text{Procedure:}$&nbsp; If a continuous-valued random variable&nbsp; $u$&nbsp; possesses the PDF&nbsp; $f_{u}(u)$,&nbsp; then the probability density function of the random variable transformed at the nonlinear characteristic&nbsp; $x = g(u)$&nbsp; holds:
 :$$f_{x}(x)=\frac{f_u(u)}{\mid g\hspace{0.05cm}'(u)\mid}\Bigg \vert_{\hspace{0.1cm} u=h(x)}.$$

@@ Line 16: / Line 16: @@
 The left sketch shows the&nbsp; "probability density function"&nbsp; $\rm (PDF)$&nbsp; of such an exponentially distributed random variable&nbsp; $x$.&nbsp; To be emphasized:
-#The larger the distribution parameter&nbsp; $λ$&nbsp; is,&nbsp; the steeper the decay occurs.
+#The larger the distribution parameter&nbsp; $λ$,&nbsp; &nbsp; the steeper the decay.
 #By definition&nbsp; $f_{x}(0) = λ/2$,&nbsp; i.e. the mean of left-hand limit&nbsp; $(0)$&nbsp; and right-hand limit&nbsp; $(\lambda)$.

@@ Line 9: / Line 9: @@
 {{BlaueBox|TEXT=
 $\text{Definition:}$&nbsp;
-A continuous random variable&nbsp; $x$&nbsp; is called&nbsp; (one-sided)&nbsp; '''exponentially distributed'''&nbsp; if it can take only non&ndash;negative values and the probability density function has the following shape  for&nbsp; $x>0$:
+A continuous random variable&nbsp; $x$&nbsp; is called&nbsp; (one-sided)&nbsp; &raquo;'''exponentially distributed'''&laquo;&nbsp; if it can take only non&ndash;negative values and the probability density function has the following shape  for&nbsp; $x>0$:
 :$$f_x(x)=\it \lambda\cdot\rm e^{\it -\lambda \hspace{0.05cm}\cdot \hspace{0.03cm} x}.$$}}
@@ Line 31: / Line 31: @@
 $\text{Example 1:}$&nbsp; The exponential distribution has great importance for reliability studies,&nbsp; and the term&nbsp; "lifetime distribution"&nbsp; is also commonly used in this context.
 #In these applications,&nbsp; the random variable is often the time&nbsp; $t$&nbsp; that elapses before a component fails.
-#Furthermore,&nbsp; it should be noted that the exponential distribution is closely related to the&nbsp; [[Theory_of_Stochastic_Signals/Poisson_Distribution|Poisson distribution]]. }}
+#Furthermore,&nbsp; it should be noted that the exponential distribution is closely related to the&nbsp; [[Theory_of_Stochastic_Signals/Poisson_Distribution|$\text{Poisson distribution}$]]. }}
 ==Transformation of random variables==
 <br>
-To generate such an exponentially distributed random variable on a digital computer,&nbsp; you can use e.g. a&nbsp; '''nonlinear transformation'''.&nbsp; The underlying principle is first stated here in general terms.
+To generate such an exponentially distributed random variable on a digital computer,&nbsp; you can use e.g. a&nbsp; &raquo;'''nonlinear transformation'''&laquo;.&nbsp; The underlying principle is first stated here in general terms.
 {{BlaueBox|TEXT=
@@ Line 75: / Line 75: @@
 :$$x=g_1(u) =\frac{1}{\lambda}\cdot \rm ln \ (\frac{1}{1-\it u}).$$
-*It can be shown that by this characteristic&nbsp; $x=g_1(u)$&nbsp; a one-sided exponentially distributed random variable&nbsp; $x$&nbsp; with the following PDF arises&nbsp; <br>(derivation see [[Theory_of_Stochastic_Signals/Exponentially_Distributed_Random_Variables#Derivation_of_the_corresponding_transformation_characteristic|next section]]):
+*It can be shown that by this characteristic&nbsp; $x=g_1(u)$&nbsp; a one-sided exponentially distributed random variable&nbsp; $x$&nbsp; with the following PDF arises&nbsp; <br>(derivation see [[Theory_of_Stochastic_Signals/Exponentially_Distributed_Random_Variables#Derivation_of_the_corresponding_transformation_characteristic|"next section"]]):
 :$$f_{x}(x)=\lambda\cdot\rm e^{\it -\lambda \hspace{0.05cm}\cdot \hspace{0.03cm} x}\hspace{0.2cm}{\rm for}\hspace{0.2cm} {\it x}>0.$$
 *Note:
@@ Line 125: / Line 125: @@
 ==Two-sided exponential distribution - Laplace distribution==
 <br>
-Closely related to the exponential distribution is the&nbsp; [https://en.wikipedia.org/wiki/Laplace_distribution Laplace distribution]&nbsp; with the probability density function
+Closely related to the exponential distribution is the&nbsp; [https://en.wikipedia.org/wiki/Laplace_distribution $\text{Laplace distribution}$]&nbsp; with the probability density function
 :$$f_{x}(x)=\frac{\lambda}{2}\cdot\rm e^{\it -\lambda \hspace{0.05cm} \cdot \hspace{0.05cm} | x|}.$$
@@ Line 135: / Line 135: @@
 Further notes:
-*From the&nbsp; [[Aufgaben:Exercise_3.8:_Amplification_and_Limitation| Exercise 3.8]]&nbsp; one can see further properties of the Laplace distribution.
+*From the&nbsp; [[Aufgaben:Exercise_3.8:_Amplification_and_Limitation|"Exercise 3.8"]]&nbsp; one can see further properties of the Laplace distribution.
 *With the HTML 5/JavaScript applet&nbsp; [[Applets:PDF,_CDF_and_Moments_of_Special_Distributions|"PDF, CDF and Moments of Special Distributions"]]&nbsp; you can display the characteristics of the exponential and the Laplace distribution.
 *In the&nbsp; (German language)&nbsp; learning video&nbsp; [[Wahrscheinlichkeit_und_WDF_(Lernvideo)|"Wahrscheinlichkeit und WDF"]] &nbsp; $\Rightarrow$ &nbsp; "Probability and PDF",&nbsp; it is shown which meaning the Laplace distribution has for the description of speech and music signals.

@@ Line 75: / Line 75: @@
 :$$x=g_1(u) =\frac{1}{\lambda}\cdot \rm ln \ (\frac{1}{1-\it u}).$$
-*It can be shown that by this characteristic&nbsp; $x=g_1(u)$&nbsp; a one-sided exponentially distributed random variable&nbsp; $x$&nbsp; with the following PDF arises&nbsp; <br>(derivation see [[Theory_of_Stochastic_Signals/Exponentially_Distributed_Random_Variables#Derivation_of_the_corresponding_transformation_characteristic|next page]]):
+*It can be shown that by this characteristic&nbsp; $x=g_1(u)$&nbsp; a one-sided exponentially distributed random variable&nbsp; $x$&nbsp; with the following PDF arises&nbsp; <br>(derivation see [[Theory_of_Stochastic_Signals/Exponentially_Distributed_Random_Variables#Derivation_of_the_corresponding_transformation_characteristic|next section]]):
 :$$f_{x}(x)=\lambda\cdot\rm e^{\it -\lambda \hspace{0.05cm}\cdot \hspace{0.03cm} x}\hspace{0.2cm}{\rm for}\hspace{0.2cm} {\it x}>0.$$
 *Note:
@@ Line 96: / Line 96: @@
 {{BlaueBox|TEXT=
 $\text{Task:}$&nbsp;
-#&nbsp; Now the transformation characteristic&nbsp; $x = g_1(u)= g(u)$&nbsp; already used on the last page is derived.
+#&nbsp; Now the transformation characteristic&nbsp; $x = g_1(u)= g(u)$&nbsp; already used in the last section is derived.
 #&nbsp; This forms from the uniformly distributed random variable&nbsp; $u$&nbsp; with PDF&nbsp; $f_{u}(u)$&nbsp; a one-sided exponentially distributed random variable&nbsp; $x$&nbsp; with PDF&nbsp; $f_{x}(x)$:

@@ Line 24: / Line 24: @@
 *The&nbsp; "moments"&nbsp; of the one-sided exponential distribution are generally equal to &nbsp;
 :$$m_k = k!/λ^k.$$
-*From this and from Steiner's theorem,nbsp; we get for the&nbsp; "mean"nbsp; and thenbsp; "standard deviation"&nbsp; (or&nbsp; "standard deviation"):
+*From this and from Steiner's theorem, we get for the "mean" and the "standard deviation":
 :$$m_1={1}/{\lambda},$$
 :$$\sigma=\sqrt{m_2-m_1^2}=\sqrt{\frac{2}{\lambda^2}-\frac{1}{\lambda^2}}={1}/{\lambda}.$$