Theory of Stochastic Signals/Expected Values and Moments - Revision history

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Hwang at 20:05, 13 December 2022

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@@ Line 126: / Line 126: @@
 <br>
 {{BlaueBox|TEXT=
-$\text{Definition:}$&nbsp; Another special case of an expected value is the&nbsp; &raquo;'''characteristic function'''&laquo;,&nbsp; where  for the valuation function is here to be set&nbsp; $g(x) = {\rm e}^{\hspace{0.03cm}{\rm j} \hspace{0.03cm}{\it Ω}\hspace{0.05cm}x}$:
+$\text{Definition:}$&nbsp; Another special case of an expected value is the&nbsp; &raquo;'''characteristic function'''&laquo;,&nbsp; where  for the valuation function is to be set&nbsp; $g(x) = {\rm e}^{\hspace{0.03cm}{\rm j} \hspace{0.03cm}{\it Ω}\hspace{0.05cm}x}$:
 :$$C_x({\it \Omega}) = {\rm E}\big[{\rm e}^{ {\rm j} \hspace{0.05cm} {\it \Omega} \hspace{0.05cm} x}\big] = \int_{-\infty}^{+\infty} {\rm e}^{ {\rm j} \hspace{0.05cm} {\it \Omega} \hspace{0.05cm} x}\cdot f_{\rm x}(x) \hspace{0.1cm}{\rm d}x.$$
-A comparison with the chapter&nbsp; [[Signal_Representation/Fourier_Transform_and_its_Inverse|"Fourier Transform and Inverse Fourier Transform"]]&nbsp; in the book&nbsp;  "Signal Representation"&nbsp; shows that the characteristic function can be interpreted as the inverse Fourier transform of the probability density function:
+* A comparison with the chapter&nbsp; [[Signal_Representation/Fourier_Transform_and_its_Inverse|&raquo;Fourier Transform and Inverse Fourier Transform&laquo;]]&nbsp; in the book&nbsp;  &raquo;Signal Representation&laquo;&nbsp; shows that the characteristic function can be interpreted as the&nbsp; &raquo;inverse Fourier transform of the probability density function&laquo;:
 :$$C_x ({\it \Omega}) \hspace{0.3cm}  \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet \hspace{0.3cm} f_{x}(x).$$}}
 If the random variable&nbsp; $x$&nbsp; is dimensionless,&nbsp; then the argument&nbsp; $\it Ω$&nbsp; of the characteristic function is also without unit.
-*The symbol&nbsp; $\it Ω$&nbsp; was chosen because the argument here has some relation to the angular frequency in the second Fourier integral&nbsp; <br>(compared to the representation in the&nbsp; $f$&ndash;domain,&nbsp; however,&nbsp; the factor&nbsp; $2\pi$&nbsp; is missing in the exponent).
+#The symbol&nbsp; $\it Ω$&nbsp; was chosen because the argument here has some relation to the angular frequency in the second Fourier integral&nbsp; <br>$($compared to the representation in the&nbsp; $f$&ndash;domain,&nbsp; however,&nbsp; the factor&nbsp; $2\pi$&nbsp; is missing in the exponent$)$.
-*But it is again insistently pointed out that&nbsp; &ndash; if one wants to establish a relation to systems theory &ndash;&nbsp; $C_x({\it Ω})$&nbsp; would correspond to the&nbsp; "time function"&nbsp; and&nbsp; $f_{x}(x)$&nbsp; to the&nbsp; "spectral function".
+#But it is again insistently pointed out that&nbsp;  $C_x({\it Ω})$&nbsp; would correspond to the&nbsp; &raquo;time function&laquo;&nbsp; and&nbsp; $f_{x}(x)$&nbsp; to the&nbsp; &raquo;spectral function&laquo;, if one wants to establish a relation to systems theory.
 {{BlaueBox|TEXT=
-$\text{Calculation possibility:}$&nbsp; Developing the complex function&nbsp; ${\rm e}^{\hspace{0.03cm}{\rm j} \hspace{0.03cm}{\it Ω}\hspace{0.05cm}x }$&nbsp; into a&nbsp; "power series"&nbsp; and interchanges expectation value formation and summation,&nbsp; the series representation of the characteristic function follows:
+$\text{Calculation possibility:}$&nbsp;
 :$$C_x ( {\it \Omega}) = 1 + \sum_{k=1}^{\infty}\hspace{0.2cm}\frac{m_k}{k!} \cdot ({\rm j} \hspace{0.01cm}{\it \Omega})^k .$$
-The&nbsp; [[Aufgaben:Exercise_3.4:_Characteristic_Function|"Exercise 3.4"]]&nbsp; shows more properties of the characteristic function. }}
+*Developing the complex function&nbsp; ${\rm e}^{\hspace{0.03cm}{\rm j} \hspace{0.03cm}{\it Ω}\hspace{0.05cm}x }$&nbsp; into a&nbsp; &raquo;power series&raquo;&nbsp; and interchanges expectation value formation and summation,&nbsp; the series representation of the characteristic function follows:
 {{GraueBox|TEXT=
-$\text{Example 3:}$&nbsp;
+$\text{Example 3:}$&nbsp; For a symmetric binary&nbsp; $($two&ndash;point distributed$)$&nbsp; random variable&nbsp; $x ∈ \{\pm1\}$&nbsp; with probabilities&nbsp;
-*For a symmetric binary&nbsp; (two&ndash;point distributed)&nbsp; random variable&nbsp; $x ∈ \{\pm1\}$&nbsp; with probabilities&nbsp; ${\rm Pr}(-1) = {\rm Pr}(+1) = 1/2$&nbsp; the characteristic function is cosine.
+:$${\rm Pr}(-1) = {\rm Pr}(+1) = 1/2$$
-*The analogue in systems theory is that the spectrum of a cosine signal with angular frequency&nbsp; ${\it Ω}_{\hspace{0.03cm}0}$&nbsp; consists of two Dirac delta functions at&nbsp; $±{\it Ω}_{\hspace{0.03cm}0}$. }}
+the characteristic function is cosine.&nbsp; The analogue in systems theory is that the spectrum of a cosine signal with angular frequency&nbsp; ${\it Ω}_{\hspace{0.03cm}0}$&nbsp; consists of two Dirac delta functions at&nbsp; $±{\it Ω}_{\hspace{0.03cm}0}$. }}
 {{GraueBox|TEXT=
-$\text{Example 4:}$&nbsp; A uniform distribution between&nbsp; $±y_0$&nbsp; has the following characteristic function according to the&nbsp; [[Signal_Representation/Fourier_Transform_Theorems|$\text{laws of Fourier transform}$]]:
+$\text{Example 4:}$&nbsp; A uniform distribution between&nbsp; $±y_0$&nbsp; has the following characteristic function according to the&nbsp; [[Signal_Representation/Fourier_Transform_Theorems|&raquo;Fourier transform laws&laquo;]]:
 :$$C_y({\it \Omega}) = \frac{1}{2 y_0} \cdot \int_{-y_0}^{+y_0} {\rm e}^{ {\rm j} \hspace{0.05cm} {\it \Omega} \hspace{0.05cm} y} \,{\rm d}y = \frac{ {\rm e}^{ {\rm j} \hspace{0.05cm} y_0 \hspace{0.05cm}{\it \Omega} } - {\rm e}^{ - {\rm j} \hspace{0.05cm} y_0 \hspace{0.05cm} {\it \Omega} } }{2 {\rm j} \cdot y_0 \cdot {\it \Omega} } = \frac{ {\rm sin}(y_0 \cdot {\it \Omega})}{ y_0 \cdot {\it \Omega} } = {\rm si}(y_0 \cdot {\it \Omega}).
 $$
-We already know the function&nbsp; ${\rm si}(x) = \sin(x)/x = {\rm sinc}(x/\pi) $&nbsp; from the book&nbsp; [[Signal_Representation/Special_Cases_of_Pulses#Rectangular_pulse|"Signal Representation"]].}}
+We already know the function&nbsp; ${\rm si}(x) = \sin(x)/x = {\rm sinc}(x/\pi) $&nbsp; from the book&nbsp; [[Signal_Representation/Special_Cases_of_Pulses#Rectangular_pulse|&raquo;Signal Representation&laquo;]].}}
 ==Exercises for the chapter==

@@ Line 106: / Line 106: @@
 ==Moment calculation as time average==
 <br>
-The expected value calculation according to the previous equations of this section corresponds to a&nbsp; "ensemble averaging",&nbsp; that is,&nbsp; averaging over all possible values&nbsp; $x_\mu$.
+The expected value calculation according to the previous equations of this section corresponds to a&nbsp; &raquo;'''ensemble averaging'''&laquo;,&nbsp; that is,&nbsp; averaging over all possible values&nbsp; $x_\mu$.
 However,&nbsp; the moments&nbsp; $m_k$&nbsp; can also be determined as&nbsp; &raquo;'''time averages'''&laquo;&nbsp; if the stochastic process generating the random variable is stationary and ergodic:
-*The exact definition for such a stationary and ergodic random process can be found in&nbsp; [[Theory_of_Stochastic_Signals/Auto-Correlation_Function#Random_processes|"Chapter 4.4"]].
+#The exact definition for such a stationary and ergodic random process can be found in&nbsp; [[Theory_of_Stochastic_Signals/Auto-Correlation_Function#Random_processes|$\text{Chapter 4.4}$]].
-*Time-averaging is always denoted by a sweeping line in the following.
+#In the following&nbsp; &raquo;time-averaging&laquo;&nbsp; is always denoted by a sweeping line.
-*For discrete time,&nbsp; the random signal&nbsp; $x(t)$&nbsp; is replaced by the random sequence&nbsp; $〈x_ν〉$.
+#For discrete time,&nbsp; the random signal&nbsp; $x(t)$&nbsp; is replaced by the random sequence&nbsp; $〈x_ν〉$.
-*For finite sequence length,&nbsp; these time averages are with&nbsp; $ν = 1, 2,\hspace{0.05cm}\text{...}\hspace{0.05cm} , N$:
+#For finite sequence length,&nbsp; these time averages are with&nbsp; $ν = 1, 2,\hspace{0.05cm}\text{...}\hspace{0.05cm} , N$:
-:$$m_k=\overline{x_{\nu}^{k}}=\frac{1}{N} \cdot \sum\limits_{\nu=1}^{N}x_{\nu}^{k},$$
+::$$m_k=\overline{x_{\nu}^{k}}=\frac{1}{N} \cdot \sum\limits_{\nu=1}^{N}x_{\nu}^{k},$$
-:$$m_1=\overline{x_{\nu}}=\frac{1}{N} \cdot \sum\limits_{\nu=1}^{N}x_{\nu},$$
+::$$m_1=\overline{x_{\nu}}=\frac{1}{N} \cdot \sum\limits_{\nu=1}^{N}x_{\nu},$$
-:$$m_2=\overline{x_{\nu}^{2}}=\frac{1}{N} \cdot \sum\limits_{\nu=1}^{N}x_{\nu}^{2}.$$
+::$$m_2=\overline{x_{\nu}^{2}}=\frac{1}{N} \cdot \sum\limits_{\nu=1}^{N}x_{\nu}^{2}.$$
-If the moments&nbsp; (or expected values)&nbsp; are to be determined by simulation,&nbsp; in practice this is usually done by time averaging.&nbsp; The corresponding computational algorithm differs only mariginally for discrete valued and continuous valued random variables.
+If the moments&nbsp; $($or expected values$)$&nbsp; are to be determined by simulation,&nbsp; in practice this is usually done by time averaging.&nbsp; The corresponding computational algorithm differs only mariginally for value-discrete and value-continuous random variables.
-The topic of this chapter is illustrated with examples in the&nbsp; (German language)&nbsp; learning video<br> &nbsp; &nbsp; &nbsp;  [[Momentenberechnung_bei_diskreten_Zufallsgrößen_(Lernvideo)|"Momentenberechnung bei diskreten Zufallsgrößen"]] &nbsp; $\Rightarrow$ &nbsp; "Moment calculation for discrete random variables".
+&rArr; &nbsp; The topic of this chapter is illustrated with examples in the&nbsp; (German language)&nbsp; learning video<br> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;  [[Momentenberechnung_bei_diskreten_Zufallsgrößen_(Lernvideo)|&raquo;Momentenberechnung bei diskreten Zufallsgrößen&raquo;]] &nbsp; $\Rightarrow$ &nbsp; &raquo;Moment calculation for discrete random variables&laquo;.

@@ Line 71: / Line 71: @@
 ==Some common central moments==
 <br>
-From the&nbsp; [[Theory_of_Stochastic_Signals/Expected_Values_and_Moments#Central_moments|$\text{last definition}$]]&nbsp; the following additional characteristics can be derived:
+From the&nbsp; [[Theory_of_Stochastic_Signals/Expected_Values_and_Moments#Central_moments|&raquo;last definition&laquo;]]&nbsp; the following additional characteristics can be derived:
 {{BlaueBox|TEXT=
 $\text{Definition:}$&nbsp; The&nbsp; &raquo;'''variance'''&laquo;&nbsp; $σ^2$&nbsp; of the considered random variable is the second order central moment &nbsp; &rArr; &nbsp; $\mu_2.$
-*The variance&nbsp; $σ^2$&nbsp; corresponds physically to the&nbsp; "switching power"&nbsp; and&nbsp; $σ$&nbsp; gives the&nbsp; standard deviation.
+#The variance&nbsp; $σ^2$&nbsp; corresponds physically to the&nbsp; &raquo;alternating current power&laquo;&nbsp; and&nbsp; $σ$&nbsp; gives the&nbsp; standard deviation.
-*From the linear and the second moment,&nbsp; the variance can be calculated according to&nbsp; "Steiner's theorem"&nbsp; in the following way:
+#From the first and the second moment,&nbsp; the variance can be calculated according to&nbsp; &raquo;Steiner's theorem&laquo;&nbsp; in the following way:
-:$$\sigma^{2} = m_2 - m_1^{2}.$$}}
+::$$\sigma^{2} = m_2 - m_1^{2}.$$}}
 {{BlaueBox|TEXT=
 $\text{Definition:}$&nbsp; The&nbsp; &raquo;'''Charlier's skewness'''&laquo;&nbsp; $S$&nbsp; denotes the third central moment related to&nbsp; $σ^3$.
-*For symmetric probability density function,&nbsp; this parameter is always&nbsp; $S=0$.
+#For symmetrical probability density function,&nbsp; this parameter is always&nbsp; $S=0$.
-*The larger&nbsp; $S = \mu_3/σ^3$&nbsp; is,&nbsp; the more asymmetric is the PDF around the mean&nbsp; $m_1$.
+#The larger&nbsp; $S = \mu_3/σ^3$,&nbsp; the more asymmetrical is the PDF around the mean&nbsp; $m_1$.
-*For example,&nbsp; for the&nbsp; [[Theory_of_Stochastic_Signals/Exponentially_Distributed_Random_Variables#One-sided_exponential_distribution|$\text{exponential distribution}$]]&nbsp; the (positive) skewness&nbsp; $S =2$,&nbsp; and this is independent of the distribution parameter&nbsp; $λ$.
+#For example,&nbsp; for the&nbsp; [[Theory_of_Stochastic_Signals/Exponentially_Distributed_Random_Variables#One-sided_exponential_distribution|&raquo;exponential distribution&laquo;]]&nbsp; the&nbsp; $($positive$)$&nbsp; skewness&nbsp; $S =2$,&nbsp; and this is independent of the distribution parameter&nbsp; $λ$.
-*For positive skewness&nbsp; $(S > 0)$&nbsp; one speaks of a&nbsp; "right&ndash;skewed"&nbsp; or of a&nbsp; "left&ndash;sloping distribution";&nbsp; this slopes flatter on the right side than on the left.
+#For positive skewness&nbsp; $(S > 0)$&nbsp; one speaks of a&nbsp; &raquo;right&ndash;skewed&laquo;&nbsp; or of a&nbsp; &raquo;left&ndash;sloping distribution&laquo;;&nbsp; this slopes flatter on the right side than on the left.
-*When the skewness is negative&nbsp; $(S < 0)$&nbsp; there is a&nbsp; "left&ndash;skewed"&nbsp; or a&nbsp; "right&ndash;steep distribution";&nbsp; such a distribution falls flatter on the left side than on the right}}
+#When the skewness&nbsp; $S < 0$&nbsp; there is a&nbsp; &raquo;left&ndash;skewed&laquo;&nbsp; or a&nbsp; &raquo;right&ndash;steep distribution&laquo;;&nbsp; such a distribution falls flatter on the left side than on the right.}}
 {{BlueBox|TEXT=
 $\text{Definition:}$&nbsp; The fourth-order central moment is also used for statistical analysis;&nbsp; The quotient&nbsp; $K = \mu_4/σ^4$ is called&nbsp; &raquo;'''kurtosis'''&laquo;.
-*For a&nbsp;[[Theory_of_Stochastic_Signals/Gaussian_Distributed_Random_Variables#Probability_density_function_.E2.80.93_Cumulative_density_function|$\text{Gaussian distributed random variable}$]]&nbsp; this always yields the value&nbsp; $K = 3$.
+#For a&nbsp;[[Theory_of_Stochastic_Signals/Gaussian_Distributed_Random_Variables#Probability_density_function_.E2.80.93_Cumulative_density_function|&raquo;Gaussian distributed random variable&laquo;]]&nbsp; this always yields the value&nbsp; $K = 3$.
-*Using also the so-called&nbsp; &raquo;'''excess'''&laquo;&nbsp; $\gamma = K - 3$,&nbsp; also known under the term&nbsp; "overkurtosis".
+#Using also the so-called&nbsp; &raquo;'''excess'''&laquo;&nbsp; $\gamma = K - 3$,&nbsp; also known under the term&nbsp; &raquo;overkurtosis&laquo;.
-*This parameter can be used,&nbsp; for example,&nbsp; to check whether a random variable at hand is approximately Gaussian:&nbsp; $\gamma \approx 0$. }}
+#This parameter can be used,&nbsp; for example,&nbsp; to check whether a random variable at hand is approximately Gaussian &nbsp; &rArr; &nbsp; excess&nbsp; $\gamma \approx 0$. }}
 {{GraueBox|TEXT=
 $\text{Example 2:}$&nbsp;
-*If the PDF has fewer offshoots than the Gaussian distribution,&nbsp; the kurtosis&nbsp; $K < 3$.&nbsp; For example,&nbsp; for the [[Theory_of_Stochastic_Signals/Uniformly_Distributed_Random_Variables|$\text{uniform distribution}$]]:&nbsp; $K = 1.8$&nbsp; &rArr; &nbsp; $\gamma = - 1.2$.
+*If the PDF has fewer offshoots than the Gaussian distribution,&nbsp; the kurtosis&nbsp; $K < 3$.&nbsp; For example,&nbsp; for the [[Theory_of_Stochastic_Signals/Uniformly_Distributed_Random_Variables|&raquo;uniform distribution&laquo;]]:&nbsp; $K = 1.8$&nbsp; &rArr; &nbsp; $\gamma = - 1.2$.
-*In contrast,&nbsp; $K > 3$&nbsp; indicates that the spurs are more pronounced than for the Gaussian distribution.&nbsp;For example,&nbsp; for the&nbsp; [[Theory_of_Stochastic_Signals/Exponentially_Distributed_Random_Variables#One-sided_exponential_distribution|$\text{exponential distribution}$]]:&nbsp;&nbsp; $K = 9$.
-*The&nbsp; [[Theory_of_Stochastic_Signals/Exponentially_Distributed_Random_Variables#Two-sided_exponential_distribution_-_Laplace_distribution|$\text{Laplace distribution}$]]&nbsp; ⇒ &nbsp; "two-sided exponential distribution"&nbsp; results in a slightly smaller kurtosis&nbsp; $K = 6$&nbsp; and the excess $\gamma = 3$.}}
+*In contrast,&nbsp; $K > 3$&nbsp; indicates that the offshoots are more pronounced than for the Gaussian distribution.&nbsp;For example,&nbsp; for the&nbsp; [[Theory_of_Stochastic_Signals/Exponentially_Distributed_Random_Variables#One-sided_exponential_distribution|&raquo;exponential distribution&laquo;]]:&nbsp;&nbsp; $K = 9$.
 ==Moment calculation as time average==

@@ Line 9: / Line 9: @@
 The probability density function&nbsp; $\rm (PDF)$,&nbsp; like the cumulative distribution function&nbsp; $\rm (CDF)$,&nbsp; provides very extensive information about the random variable under consideration.&nbsp; Less,&nbsp; but more compact information is provided by the so-called&nbsp; &raquo;expected values&laquo;&nbsp; and&nbsp; &raquo;moments&laquo;.
-*Their calculation possibilities have already been given for discrete valued random variables in the chapter&nbsp; [[Theory_of_Stochastic_Signals/Moments_of_a_Discrete_Random_Variable|"Moments of a Discrete Random Variable"]]&nbsp;.
+*Their calculation possibilities have already been given for value-discrete random variables in the chapter&nbsp; [[Theory_of_Stochastic_Signals/Moments_of_a_Discrete_Random_Variable|&raquo;Moments of a Discrete Random Variable&laquo;]].
-*Now these integrative descriptive quantities&nbsp; "expected value"&nbsp; and&nbsp; "moment"&nbsp; are considered in the context of the probability density function (PDF) of continuous valued random variables and thus formulated more generally.
 {{BlaueBox|TEXT=
-$\text{Definition:}$&nbsp; The&nbsp; &raquo;'''expected value'''&laquo;&nbsp; with respect to any weighting function&nbsp; $g(x)$&nbsp; can be calculated with the PDF&nbsp; $f_{\rm x}(x)$&nbsp; in the following way:
+$\text{Definitions:}$&nbsp;
 :$${\rm E}\big[g (x ) \big] = \int_{-\infty}^{+\infty} g(x)\cdot f_{x}(x) \,{\rm d}x.$$
 *Substituting into this equation for&nbsp; $g(x) = x^k$&nbsp; we get the&nbsp; &raquo;'''moment of $k$-th order'''&laquo;:
@@ Line 21: / Line 23: @@
 From this equation follows.
-*with&nbsp; $k = 1$&nbsp; for the&nbsp; "first order moment"&nbsp; or the&nbsp; "(linear)&nbsp; mean":
+*with&nbsp; $k = 1$&nbsp; for the&nbsp; &raquo;first order moment&laquo; &nbsp; &rArr; &nbsp; &raquo;$($linear$)$&nbsp; mean&laquo;:
 :$$m_1 = {\rm E}\big[x \big] = \int_{-\infty}^{ \rm +\infty} x\cdot f_{x} (x ) \,{\rm d}x,$$
-*with&nbsp; $k = 2$&nbsp; for the&nbsp; "second order moment"&nbsp; or the&nbsp; "second moment":
+*with&nbsp; $k = 2$&nbsp; for the&nbsp; &raquo;second order moment&laquo;:
 :$$m_2 = {\rm E}\big[x^{\rm 2} \big] = \int_{-\infty}^{ \rm +\infty} x^{ 2}\cdot f_{ x} (x) \,{\rm d}x.$$
-For a discrete valued&nbsp;  $M$&ndash;level random variable,&nbsp; the formulas given here again yield the equations already given in the second chapter&nbsp; $($calculation as a ensemble average$)$:
+For a&nbsp;  $M$&ndash;level random variable,&nbsp; the formulas given here again yield the equations already given in the second chapter&nbsp; $($&raquo;Calculation as an ensemble average&laquo;$)$:
 :$$m_1 = \sum\limits_{\mu=1}^{ M}\hspace{0.15cm}p_\mu\cdot x_\mu,$$
 :$$m_2 = \sum\limits_{\mu= 1}^{ M}\hspace{0.15cm}p_\mu\cdot x_\mu^2.$$
-Here it is taken into account that the integral over the Dirac delta function&nbsp; $δ(x)$&nbsp; is equal to&nbsp; $1$&nbsp;.
+Here it is taken into account that the integral over the Dirac delta function&nbsp; $δ(x)$&nbsp; is equal to&nbsp; $1$.&nbsp; In relation to signals,&nbsp; the following terms are also common:
+* $m_1$&nbsp; indicates the&nbsp; &raquo;DC component&laquo;.
 In relation to signals,&nbsp; the following terms are also common:
-* $m_1$&nbsp; indicates the&nbsp; "DC component",
+* $m_2$&nbsp; corresponds to the&nbsp; &raquo;signal power&laquo;&nbsp;  $($referred to the unit resistance&nbsp; $1 \ Ω)$.
-If one wants to indicate the power in&nbsp;  "Watt"&nbsp; $\rm (W)$&nbsp;, then&nbsp; $m_2$&nbsp; must still be divided by the resistance value&nbsp; $R$.
+For example,&nbsp; if&nbsp; $x$&nbsp; denotes a voltage, then according to these equations&nbsp; $m_1$&nbsp; has the unit&nbsp; "${\rm V}$"&nbsp; and&nbsp; $m_2$&nbsp; the unit&nbsp; "${\rm V}^2$".&nbsp; If one wants to indicate the power in&nbsp;  &raquo;Watt&laquo;&nbsp; $\rm (W)$,&nbsp; then&nbsp; $m_2$&nbsp; must still be divided by the resistance value&nbsp; $R$.
@@ Line 48: / Line 44: @@
 <br>
 {{BlaueBox|TEXT=
-$\text{Definition:}$&nbsp; Especially important in statistics are the&nbsp; &raquo;'''central moments'''&laquo;,&nbsp;  which,&nbsp;  in contrast to the conventional moments,&nbsp;  are each related to the mean&nbsp; $m_1$:
+$\text{Definition:}$&nbsp; Especially important in statistics are the&nbsp; &raquo;'''central moments'''&laquo;,&nbsp;  which  in contrast to the conventional moments  are each related to the mean&nbsp; $m_1$:
 :$$\mu_k = {\rm E}\big[(x-m_{\rm 1})^k\big] = \int_{-\infty}^{+\infty} (x-m_{\rm 1})^k\cdot f_x(x) \,\rm d \it x.$$}}
@@ Line 54: / Line 50: @@
-The noncentered moments&nbsp; $m_k$&nbsp; can be directly converted to the centered moments&nbsp; $\mu_k$:
+The non&ndash;centered moments&nbsp; $m_k$&nbsp; can be directly converted to the centered moments&nbsp; $\mu_k$:
 :$$\mu_k = \sum\limits_{\kappa= 0}^{k} \left( \begin{array}{*{2}{c} } k \\ \kappa \\ \end{array} \right)\cdot m_\kappa \cdot (-m_1)^{k-\kappa}.$$
-*According to the general equations of the&nbsp; [[Theory_of_Stochastic_Signals/Expected_Values_and_Moments#Moment_calculation_as_ensemble_average|$\text{last section}$]]&nbsp; the formal quantities&nbsp; $m_0 = 1$&nbsp; and&nbsp; $\mu_0 = 1$&nbsp; result.
+*According to the general equations of the&nbsp; [[Theory_of_Stochastic_Signals/Expected_Values_and_Moments#Moment_calculation_as_ensemble_average|&raquo;last section&laquo;]]&nbsp; the formal quantities&nbsp; $m_0 = 1$&nbsp; and&nbsp; $\mu_0 = 1$&nbsp; result.
-*For the first order central moment, according to the above definition, always&nbsp; $\mu_1 = 0$ holds.

@@ Line 7: / Line 7: @@
 ==Moment calculation as ensemble average==
 <br>
-The probability density function&nbsp; $\rm (PDF)$,&nbsp; like the cumulative distribution function&nbsp; $\rm (CDF)$,&nbsp; provides very extensive information about the random variable under consideration.&nbsp; Less,&nbsp; but more compact information is provided by the so-called&nbsp; "expected values"&nbsp; and&nbsp; "moments".
+The probability density function&nbsp; $\rm (PDF)$,&nbsp; like the cumulative distribution function&nbsp; $\rm (CDF)$,&nbsp; provides very extensive information about the random variable under consideration.&nbsp; Less,&nbsp; but more compact information is provided by the so-called&nbsp; &raquo;expected values&laquo;&nbsp; and&nbsp; &raquo;moments&laquo;.
 *Their calculation possibilities have already been given for discrete valued random variables in the chapter&nbsp; [[Theory_of_Stochastic_Signals/Moments_of_a_Discrete_Random_Variable|"Moments of a Discrete Random Variable"]]&nbsp;.