Theory of Stochastic Signals/Linear Combinations of Random Variables - Revision history

Hwang at 14:02, 21 December 2022

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Hwang at 13:58, 21 December 2022

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Guenter at 16:27, 31 March 2022

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Guenter at 12:51, 25 February 2022

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Noah at 22:15, 17 February 2022

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Noah at 22:24, 25 January 2022

2022-01-25T22:24:52Z

@@ Line 17: / Line 17: @@
 :$$m_x =A \cdot m_u + B \cdot m_v + C =C,$$
 :$$m_y =D \cdot m_u + E \cdot m_v + F =F.$$
-Thus,&nbsp; the coefficients&nbsp; $C$&nbsp; and&nbsp; $F$&nbsp; give only the mean values of&nbsp; $x$&nbsp; and&nbsp; $y$.&nbsp; Both are always set to zero on the following pages.
+Thus,&nbsp; the coefficients&nbsp; $C$&nbsp; and&nbsp; $F$&nbsp; give only the mean values of&nbsp; $x$&nbsp; and&nbsp; $y$.&nbsp; Both are always set to zero in the following sections.
 ==Resulting correlation coefficient==
 <br>

@@ Line 20: / Line 20: @@
 ==Resulting correlation coefficient==
 <br>
-Let us now consider the&nbsp; '''variances'''&nbsp; of the random variables according to the linear combinations.
+Let us now consider the&nbsp; &raquo;'''variances'''&laquo;&nbsp; of the random variables according to the linear combinations.
 *For the random variable&nbsp; $x$&nbsp; holds independently of the parameter&nbsp; $C$:
 :$$\sigma _x ^2 = {\rm E}\big[x ^{\rm 2}\big] = A^{\rm 2} \cdot {\rm E}\big[u^{\rm 2}\big] + B^{\rm 2} \cdot {\rm E}\big[v^{\rm 2}\big] + {\rm 2} \cdot A \cdot B \cdot {\rm E}\big[u \cdot v\big].$$
@@ Line 31: / Line 31: @@
 :$$\sigma _y ^2 =(D^2 + E^2) \cdot \sigma ^2.$$
-The&nbsp; '''covariance'''&nbsp; $μ_{xy}$&nbsp; is identical to the joint moment&nbsp; $m_{xy}$ for zero mean random variables&nbsp; $x$&nbsp; and&nbsp; $y$ &nbsp; ⇒ &nbsp; $C = F = 0$:
+The&nbsp; &raquo;'''covariance'''&laquo;&nbsp; $μ_{xy}$&nbsp; is identical to the joint moment&nbsp; $m_{xy}$ for zero mean random variables&nbsp; $x$&nbsp; and&nbsp; $y$ &nbsp; ⇒ &nbsp; $C = F = 0$:
 :$$\mu_{xy } = m_{xy } = {\rm E}\big[x \cdot y\big] = {\rm E}\big[(A \cdot u + B \cdot v)\cdot (D \cdot u + E \cdot v)\big].$$
 Note here that&nbsp; ${\rm E}\big[ \text{...} \big]$&nbsp; denotes an expected value,&nbsp; while&nbsp; $E$&nbsp; describes a coefficient.
@@ Line 66: / Line 66: @@
 ==Generation of correlated random variables==
 <br>
-The&nbsp; [[Theory_of_Stochastic_Signals/Linear_Combinations_of_Random_Variables#Resulting_correlation_coefficient|previously used equations]]&nbsp; can be used to generate a two-dimensional random variable&nbsp; $(x,\hspace{0.08cm} y)$&nbsp; with given characteristics&nbsp; $σ_x$,&nbsp; $σ_y$&nbsp; and&nbsp; $ρ_{xy}$.
+The&nbsp; [[Theory_of_Stochastic_Signals/Linear_Combinations_of_Random_Variables#Resulting_correlation_coefficient|$\text{previously used equations}$]]&nbsp; can be used to generate a two-dimensional random variable&nbsp; $(x,\hspace{0.08cm} y)$&nbsp; with given characteristics&nbsp; $σ_x$,&nbsp; $σ_y$&nbsp; and&nbsp; $ρ_{xy}$.
 *If no further preconditions are met other than these three nominal values, one of the four coefficients&nbsp; $A, \ B, \ D$&nbsp; and&nbsp; $E$&nbsp; is arbitrary.

@@ Line 80: / Line 80: @@
 $(1)$ &nbsp; To generate a two-dimensional random variable with desired characteristics &nbsp; $σ_x =1$,&nbsp; $σ_y = 1.55$&nbsp; and&nbsp; $ρ_{xy} = -0.8$ &nbsp; the parameter set is suitable, for example:
-[[File:P_ID419__Sto_T_4_3_S3_neu.png|right |frame| Two-dimensional random variables generated by linear combination]]
+[[File:EN_Sto_T_4_3_S3_v1.png|right |frame| Two-dimensional random variables generated by linear combination]]
 :$$A = -0.8, \; B = 0.6, \; D = 1.55, \; E = 0.$$
 *This parameter set underlies the left graph.

@@ Line 100: / Line 100: @@
 [[Aufgaben:Exercise_4.7Z:_Generation_of_a_joint_PDF|Exercise 4.7Z: Generation of a Joint PDF]]
-[[Aufgaben:Exercise_4.8:_Diamond_shaped_joint_PDF|Exercise 4.8: Diamond shaped Joint PDF]]
+[[Aufgaben:Exercise_4.8:_Diamond-shaped_Joint_PDF|Exercise 4.8: Diamond-shaped Joint PDF]]
 [[Aufgaben:Exercise_4.8Z:_AWGN_Channel|Exercise 4.8Z: AWGN Channel]]

@@ Line 98: / Line 98: @@
 [[Aufgaben:Exercise_4.7:_Weighted_Sum_and_Difference|Exercise 4.7: Weighted Sum and Difference]]
-[[Aufgaben:Exercise_4.7Z:_Generation_of_a_joint_PDF|Exercise 4.7Z: Generation of a joint PDF]]
+[[Aufgaben:Exercise_4.7Z:_Generation_of_a_joint_PDF|Exercise 4.7Z: Generation of a Joint PDF]]
-[[Aufgaben:Exercise_4.8:_Diamond_shaped_joint_PDF|Exercise 4.8: Diamond shaped joint PDF]]
+[[Aufgaben:Exercise_4.8:_Diamond_shaped_joint_PDF|Exercise 4.8: Diamond shaped Joint PDF]]
 [[Aufgaben:Exercise_4.8Z:_AWGN_Channel|Exercise 4.8Z: AWGN Channel]]

@@ Line 98: / Line 98: @@
 [[Aufgaben:Exercise_4.7:_Weighted_Sum_and_Difference|Exercise 4.7: Weighted Sum and Difference]]
-[[Aufgaben:Exercise_4.7Z:_Generation_of_a_"2D-PDF"|Exercise 4.7Z: Generation of a "2D-PDF"]]
+[[Aufgaben:Exercise_4.7Z:_Generation_of_a_"2D-PDF"|Exercise 4.7Z: Generation of a joint PDF]]
-[[Aufgaben:Exercise_4.8:_Diamond_shaped_"2D-PDF"|Exercise 4.8: Diamond shaped "2D-PDF"]]
+[[Aufgaben:Exercise_4.8:_Diamond_shaped_"2D-PDF"|Exercise 4.8: Diamond shaped joint PDF]]
 [[Aufgaben:Exercise_4.8Z:_AWGN_Channel|Exercise 4.8Z: AWGN Channel]]

@@ Line 7: / Line 7: @@
 ==Prerequisites and mean values==
 <br>
-Throughout the chapter "Linear Combinations of Random Variables" we make the following assumptions:
+Throughout the chapter&nbsp; "Linear Combinations of Random Variables"&nbsp; we make the following assumptions:
 *The random variables&nbsp; $u$&nbsp; and&nbsp; $v$&nbsp; are zero mean each &nbsp; ⇒ &nbsp; $m_u = m_v = 0$&nbsp; and also statistically independent of each other &nbsp; ⇒ &nbsp; $ρ_{uv} = 0$.
 *The two random variables&nbsp; $u$&nbsp; and&nbsp; $v$&nbsp; each have equal standard deviation&nbsp; $σ$.&nbsp; No statement is made about the nature of the distribution.
-*Let the two random variables&nbsp; $x$&nbsp; and&nbsp; $y$&nbsp; be linear combinations of&nbsp; $u$&nbsp; and&nbsp; $v$, where:
+*Let the two random variables&nbsp; $x$&nbsp; and&nbsp; $y$&nbsp; be linear combinations of&nbsp; $u$&nbsp; and&nbsp; $v$,&nbsp; where:
 :$$x=A \cdot u + B \cdot v + C,$$
 :$$y=D \cdot u + E \cdot v + F.$$
-Thus, for the (linear) mean values of the new random variables&nbsp; $x$&nbsp; and&nbsp; $y$&nbsp; we obtain according to the general rules of calculation for expected values:
+Thus,&nbsp; for the&nbsp; (linear)&nbsp; mean values of the new random variables&nbsp; $x$&nbsp; and&nbsp; $y$&nbsp; we obtain according to the general rules of calculation for expected values:
 :$$m_x =A \cdot m_u + B \cdot m_v + C =C,$$
 :$$m_y =D \cdot m_u + E \cdot m_v + F =F.$$
-Thus, the coefficients&nbsp; $C$&nbsp; and&nbsp; $F$&nbsp; give only the mean values of&nbsp; $x$&nbsp; and&nbsp; $y$&nbsp; Both are always set to zero in the following pages.
+Thus,&nbsp; the coefficients&nbsp; $C$&nbsp; and&nbsp; $F$&nbsp; give only the mean values of&nbsp; $x$&nbsp; and&nbsp; $y$.&nbsp; Both are always set to zero on the following pages.
 ==Resulting correlation coefficient==
 <br>
@@ Line 24: / Line 24: @@
 :$$\sigma _x ^2 = {\rm E}\big[x ^{\rm 2}\big] = A^{\rm 2} \cdot {\rm E}\big[u^{\rm 2}\big] + B^{\rm 2} \cdot {\rm E}\big[v^{\rm 2}\big] + {\rm 2} \cdot A \cdot B \cdot {\rm E}\big[u \cdot v\big].$$
-*The expected values of&nbsp; $u^2$&nbsp; and&nbsp; $v^2$&nbsp; are by definition equal to&nbsp; $σ^2$, respectively, because&nbsp; $u$&nbsp; and&nbsp; $v$&nbsp; are zero mean.
+*The expected values of&nbsp; $u^2$&nbsp; and&nbsp; $v^2$&nbsp; are by definition equal to&nbsp; $σ^2$,&nbsp;  because&nbsp; $u$&nbsp; and&nbsp; $v$&nbsp; are zero mean.
-*Since&nbsp; $u$&nbsp; and&nbsp; $v$&nbsp; are moreover assumed to be statistically independent, one can also write for the expected value of the product:
+*Since&nbsp; $u$&nbsp; and&nbsp; $v$&nbsp; are moreover assumed to be statistically independent,&nbsp; one can write for the expected value of the product:
 :$${\rm E}\big[u \cdot v\big] = {\rm E}\big[u\big] \cdot {\rm E}\big[v\big] = m_u \cdot m_v = \rm 0.$$
-*Thus, for the variances of the random variables formed by linear combinations, we obtain:
+*Thus,&nbsp; for the variances of the random variables formed by linear combinations,&nbsp; we obtain:
 :$$\sigma _x ^2 =(A^2 + B^2) \cdot \sigma ^2,$$
 :$$\sigma _y ^2 =(D^2 + E^2) \cdot \sigma ^2.$$
-The&nbsp; '''covariance'''&nbsp; $μ_{xy}$&nbsp; is identical to the joint moment&nbsp; $m_{xy}$ for zero mean random variables&nbsp; $x$&nbsp; and&nbsp; $y$ &nbsp; ⇒ &nbsp; $C = F = 0$&nbsp; :
+The&nbsp; '''covariance'''&nbsp; $μ_{xy}$&nbsp; is identical to the joint moment&nbsp; $m_{xy}$ for zero mean random variables&nbsp; $x$&nbsp; and&nbsp; $y$ &nbsp; ⇒ &nbsp; $C = F = 0$:
 :$$\mu_{xy } = m_{xy } = {\rm E}\big[x \cdot y\big] = {\rm E}\big[(A \cdot u + B \cdot v)\cdot (D \cdot u + E \cdot v)\big].$$
-Note here that&nbsp; ${\rm E}\big[ \text{...} \big]$&nbsp; denotes an expected value, while&nbsp; $E$&nbsp; describes a variable.
+Note here that&nbsp; ${\rm E}\big[ \text{...} \big]$&nbsp; denotes an expected value,&nbsp; while&nbsp; $E$&nbsp; describes a coefficient.
 {{BlaueBox|TEXT=
-$\text{Conclusion:}$&nbsp;After evaluating this equation in an analogous manner to above, it follows:
+$\text{Conclusion:}$&nbsp;After evaluating this equation in an analogous manner to above,&nbsp; it follows:
 :$$\mu_{xy } = (A \cdot D + B \cdot E) \cdot \sigma^{\rm 2 }
-\hspace{0.3cm}\rightarrow\hspace{0.3cm}
+\hspace{0.3cm}\Rightarrow\hspace{0.3cm}
 \rho_{xy } = \frac{\rho_{xy } }{\sigma_x \cdot \sigma_y} = \frac {A \cdot D + B \cdot E}{\sqrt{(A^{\rm 2}+B^{\rm 2})(D^{\rm 2}+E^{\rm 2} ) } }. $$}}
 We now exclude two special cases:
-*$A = B = 0$&nbsp; &rArr; &nbsp; $x ≡ 0$&nbsp; as well as.
+*$A = B = 0$&nbsp; &rArr; &nbsp; $x ≡ 0$,
 *$D = E = 0$&nbsp; &rArr; &nbsp; $y ≡ 0$.
@@ Line 52: / Line 52: @@
 $\text{Example 1:}$&nbsp;
 If we set&nbsp; $A = E = 0$,&nbsp; we get the correlation coefficient&nbsp; $ρ_{xy} = 0$.&nbsp; This result is insightful:
-*Now&nbsp; $x$&nbsp; depends only on&nbsp; $v$&nbsp; and&nbsp; $y$&nbsp; depends exclusively on&nbsp; $u$&nbsp; .
+*Now&nbsp; $x$&nbsp; depends only on&nbsp; $v$&nbsp; and&nbsp; $y$&nbsp; depends exclusively on&nbsp; $u$.
-*But since&nbsp; $u$&nbsp; and&nbsp; $v$&nbsp; were assumed to be statistically independent, there are also no relationships between&nbsp; $x$&nbsp; and&nbsp; $y$.
+*But since&nbsp; $u$&nbsp; and&nbsp; $v$&nbsp; were assumed to be statistically independent,&nbsp; there are also no relationships between&nbsp; $x$&nbsp; and&nbsp; $y$.
-*Similarly, &nbsp;$ρ_{xy} = 0$&nbsp; results for the combination&nbsp; $B = D = 0$.}}
+*Similarly,&nbsp; &nbsp;$ρ_{xy} = 0$&nbsp; results for the combination&nbsp; $B = D = 0$.}}
 {{GraueBox|TEXT=
-$\text{Example 2:}$&nbsp; The constellation &nbsp;$B = E = 0$&nbsp; leads to the fact that both&nbsp; $x$&nbsp; and&nbsp; $y$&nbsp; depend only on&nbsp; $u$&nbsp; Then the correlation coefficient is obtained:
+$\text{Example 2:}$&nbsp; The constellation &nbsp;$B = E = 0$&nbsp; leads to the fact that both&nbsp; $x$&nbsp; and&nbsp; $y$&nbsp; depend only on&nbsp; $u$.&nbsp; Then the following correlation coefficient is obtained:
 :$$\rho_{xy } = \frac {A \cdot D }{\sqrt{A^{\rm 2}\cdot D^{\rm 2} } } = \frac {A \cdot D }{\vert A\vert \cdot D\vert } =\pm 1. $$
-*If&nbsp; $A$&nbsp; and&nbsp; $D$&nbsp; have the same sign, then&nbsp; $ρ_{xy} = +1$.
+*If&nbsp; $A$&nbsp; and&nbsp; $D$&nbsp; have the same sign,&nbsp; then&nbsp; $ρ_{xy} = +1$.
-*For different signs, the correlation coefficient is&nbsp; $-1$.
+*For different signs,&nbsp; the correlation coefficient is&nbsp; $-1$.
-*Also, for &nbsp;$A = D = 0$&nbsp; the coefficient&nbsp; $ρ_{xy} = ±1$ if&nbsp; $B \ne 0$&nbsp; and&nbsp; $E \ne 0$&nbsp; holds. }}
+*Also for &nbsp;$A = D = 0:$&nbsp; The coefficient&nbsp; $ρ_{xy} = ±1$,&nbsp; if&nbsp; $B \ne 0$&nbsp; and&nbsp; $E \ne 0$&nbsp; holds. }}
 ==Generation of correlated random variables==
 <br>
-The&nbsp; [[Theory_of_Stochastic_Signals/Linear_Combinations_of_Random_Variables#Resulting_correlation_coefficient|previously used equations]]&nbsp; can be used to generate a two-dimensional random variable&nbsp; $(x, y)$&nbsp; with given characteristics&nbsp; $σ_x$,&nbsp; $σ_y$&nbsp; and&nbsp; $ρ_{xy}$&nbsp; .
+The&nbsp; [[Theory_of_Stochastic_Signals/Linear_Combinations_of_Random_Variables#Resulting_correlation_coefficient|previously used equations]]&nbsp; can be used to generate a two-dimensional random variable&nbsp; $(x,\hspace{0.08cm} y)$&nbsp; with given characteristics&nbsp; $σ_x$,&nbsp; $σ_y$&nbsp; and&nbsp; $ρ_{xy}$.
 *If no further preconditions are met other than these three nominal values, one of the four coefficients&nbsp; $A, \ B, \ D$&nbsp; and&nbsp; $E$&nbsp; is arbitrary.
@@ Line 77: / Line 77: @@
 {{GraueBox|TEXT=
 $\text{Example 3:}$&nbsp;
-We always assume zero mean Gaussian random variables&nbsp; $u$&nbsp; and&nbsp; $v$&nbsp; Both have variance&nbsp; $σ^2 = 1$.
+We always assume zero mean Gaussian random variables&nbsp; $u$&nbsp; and&nbsp; $v$.&nbsp; Both have variance&nbsp; $σ^2 = 1$.
-$(1)$ &nbsp; To generate a 2D random variable with the desired characteristics&nbsp; $σ_x =1$,&nbsp; $σ_y = 1.55$&nbsp; and&nbsp; $ρ_{xy} = -0.8$&nbsp; the parameter set is suitable, for example.
+$(1)$ &nbsp; To generate a two-dimensional random variable with desired characteristics &nbsp; $σ_x =1$,&nbsp; $σ_y = 1.55$&nbsp; and&nbsp; $ρ_{xy} = -0.8$ &nbsp; the parameter set is suitable, for example:
-[[File:P_ID419__Sto_T_4_3_S3_neu.png|right |frame| 2D random variables generated by linear combination]]
+[[File:P_ID419__Sto_T_4_3_S3_neu.png|right |frame| Two-dimensional random variables generated by linear combination]]
 :$$A = -0.8, \; B = 0.6, \; D = 1.55, \; E = 0.$$
-*This set of parameters underlies the left graph.
+*This parameter set underlies the left graph.
-*The regression line&nbsp; $K(x)$&nbsp; is shown in red.
+*The regression line&nbsp; $(\rm RL)$&nbsp; is shown in red.
 *It runs at an angle of about&nbsp; $-50^\circ$.
 *Drawn in purple is the ellipse major axis, which lies slightly above the regression line.
@@ Line 91: / Line 91: @@
 $(2)$ &nbsp; The parameter set for the right graph is:
 :$$A = -0.625,\; B = 0.781,\; D = 1.501,\; E = -0.390.$$
-*In a statistical sense, the same result is obtained even though the two point clouds differ in detail.
+*In a statistical sense,&nbsp; the same result is obtained even though the two point clouds differ in detail.
-*In particular, with respect to regression line and ellipse major axis, there is no difference with respect to the parameter set&nbsp; $(1)$.  }}
+*With respect to regression line&nbsp; $(\rm RL)$&nbsp; and ellipse major axis&nbsp; $(\rm EA)$,&nbsp; there is no difference with respect to the parameter set&nbsp; $(1)$.  }}
 ==Exercises for the chapter==

@@ Line 1: / Line 1: @@
 {{Header
-|Untermenü=Zufallsgrößen mit statistischen Bindungen
+|Untermenü=Random Variables with Statistical Dependence
-|Vorherige Seite=Zweidimensionale Gaußsche Zufallsgrößen
+|Vorherige Seite=Two-Dimensional Gaussian Random Variables
-|Nächste Seite=Autokorrelationsfunktion (AKF)
+|Nächste Seite=Auto-Correlation Function
 }}
-==Voraussetzungen und Mittelwerte==
+==Prerequisites and mean values==
 <br>
-Im gesamten Kapitel "Linearkombinationen von Zufallsgrößen"  gehen wir von folgenden Annahmen aus:
+Throughout the chapter "Linear Combinations of Random Variables" we make the following assumptions:
-*Die Zufallsgrößen&nbsp; $u$&nbsp; und&nbsp; $v$&nbsp; seien jeweils mittelwertfrei &nbsp;  ⇒  &nbsp; $m_u = m_v = 0$&nbsp; und zudem statistisch unabhängig voneinander  &nbsp; ⇒  &nbsp; $ρ_{uv} = 0$.
+*The random variables&nbsp; $u$&nbsp; and&nbsp; $v$&nbsp; are zero mean each &nbsp; ⇒ &nbsp; $m_u = m_v = 0$&nbsp; and also statistically independent of each other &nbsp; ⇒ &nbsp; $ρ_{uv} = 0$.
-*Die beiden Zufallsgrößen&nbsp; $u$&nbsp; und&nbsp; $v$&nbsp; besitzen jeweils gleiche Streuung&nbsp; $σ$.&nbsp; Über die Art der Verteilung wird keine Aussage getroffen.
+*The two random variables&nbsp; $u$&nbsp; and&nbsp; $v$&nbsp; each have equal standard deviation&nbsp; $σ$.&nbsp; No statement is made about the nature of the distribution.
-*Die beiden Zufallsgrößen&nbsp; $x$&nbsp; und&nbsp; $y$&nbsp; seien Linearkombinationen von&nbsp; $u$&nbsp; und&nbsp; $v$, wobei gilt:
+*Let the two random variables&nbsp; $x$&nbsp; and&nbsp; $y$&nbsp; be linear combinations of&nbsp; $u$&nbsp; and&nbsp; $v$, where:
 :$$x=A \cdot u + B \cdot v + C,$$
 :$$y=D \cdot u + E \cdot v + F.$$
-Für die (linearen) Mittelwerte der neuen Zufallsgrößen&nbsp; $x$&nbsp; und&nbsp; $y$&nbsp; erhält man somit nach den allgemeinen Rechenregeln für Erwartungswerte:
+Thus, for the (linear) mean values of the new random variables&nbsp; $x$&nbsp; and&nbsp; $y$&nbsp; we obtain according to the general rules of calculation for expected values:
 :$$m_x =A \cdot m_u + B \cdot m_v + C =C,$$
 :$$m_y =D \cdot m_u + E \cdot m_v + F =F.$$
-Die Koeffizienten&nbsp; $C$&nbsp; und&nbsp; $F$&nbsp; geben somit lediglich die Mittelwerte von&nbsp; $x$&nbsp; und&nbsp; $y$&nbsp; an.&nbsp; Beide werden auf den folgenden Seiten stets zu Null gesetzt.
+Thus, the coefficients&nbsp; $C$&nbsp; and&nbsp; $F$&nbsp; give only the mean values of&nbsp; $x$&nbsp; and&nbsp; $y$&nbsp; Both are always set to zero in the following pages.
+==Resulting correlation coefficient==
 <br>
-Betrachten wir nun die&nbsp; '''Varianzen'''&nbsp; der Zufallsgrößen nach den Linearkombinationen.
+ResuLet us now consider the&nbsp; '''variances'''&nbsp; of the random variables according to the linear combinations.
-*Für die Zufallsgröße&nbsp; $x$&nbsp; gilt unabhängig vom Parameter&nbsp; $C$:
+*For the random variable&nbsp; $x$&nbsp; holds independently of the parameter&nbsp; $C$:
-:$$\sigma _x ^2 = {\rm E}\big[x ^{\rm 2}\big] = A^{\rm 2} \cdot {\rm E}\big[u^{\rm 2}\big]  + B^{\rm 2} \cdot {\rm E}\big[v^{\rm 2}\big] + {\rm 2} \cdot A \cdot B \cdot {\rm E}\big[u \cdot v\big].$$
+:$$\sigma _x ^2 = {\rm E}\big[x ^{\rm 2}\big] = A^{\rm 2} \cdot {\rm E}\big[u^{\rm 2}\big] + B^{\rm 2} \cdot {\rm E}\big[v^{\rm 2}\big] + {\rm 2} \cdot A \cdot B \cdot {\rm E}\big[u \cdot v\big].$$
-*Die Erwartungswerte von&nbsp; $u^2$&nbsp; und&nbsp; $v^2$&nbsp; sind definitionsgemäß jeweils gleich&nbsp; $σ^2$, weil&nbsp; $u$&nbsp; und&nbsp; $v$&nbsp; mittelwertfrei sind.
+*The expected values of&nbsp; $u^2$&nbsp; and&nbsp; $v^2$&nbsp; are by definition equal to&nbsp; $σ^2$, respectively, because&nbsp; $u$&nbsp; and&nbsp; $v$&nbsp; are zero mean.
-*Da&nbsp; $u$&nbsp; und&nbsp; $v$&nbsp; zudem als statistisch unabhängig vorausgesetzt werden, kann man für den Erwartungswert des Produktes auch schreiben:
+*Since&nbsp; $u$&nbsp; and&nbsp; $v$&nbsp; are moreover assumed to be statistically independent, one can also write for the expected value of the product:
 :$${\rm E}\big[u \cdot v\big] = {\rm E}\big[u\big] \cdot {\rm E}\big[v\big] = m_u \cdot m_v = \rm 0.$$
-*Damit erhält man für die Varianzen der durch Linearkombinationen gebildeten Zufallsgrößen:
+*Thus, for the variances of the random variables formed by linear combinations, we obtain:
 :$$\sigma _x ^2 =(A^2 + B^2) \cdot \sigma ^2,$$
 :$$\sigma _y ^2 =(D^2 + E^2) \cdot \sigma ^2.$$
-Die&nbsp; '''Kovarianz'''&nbsp; $μ_{xy}$&nbsp; ist bei mittelwertfreien Zufallsgrößen&nbsp; $x$&nbsp; und&nbsp; $y$   &nbsp; ⇒  &nbsp;  $C = F = 0$&nbsp; identisch mit dem gemeinsamen Moment&nbsp; $m_{xy}$:
+The&nbsp; '''covariance'''&nbsp; $μ_{xy}$&nbsp; is identical to the joint moment&nbsp; $m_{xy}$ for zero mean random variables&nbsp; $x$&nbsp; and&nbsp; $y$ &nbsp; ⇒ &nbsp; $C = F = 0$&nbsp; :
 :$$\mu_{xy } = m_{xy } = {\rm E}\big[x \cdot y\big] = {\rm E}\big[(A \cdot u + B \cdot v)\cdot (D \cdot u + E \cdot v)\big].$$
-Beachten Sie hierbei, dass&nbsp; ${\rm E}\big[ \text{...} \big]$&nbsp; einen Erwartungswert bezeichnet, während&nbsp; $E$&nbsp; eine Variable beschreibt.
+Note here that&nbsp; ${\rm E}\big[ \text{...} \big]$&nbsp; denotes an expected value, while&nbsp; $E$&nbsp; describes a variable.
 {{BlaueBox|TEXT=
-$\text{Fazit:}$&nbsp;Nach Auswertung dieser Gleichung in analoger Weise zu oben folgt daraus:
+$\text{Conclusion:}$&nbsp;After evaluating this equation in an analogous manner to above, it follows:
-:$$\mu_{xy } =  (A \cdot D + B \cdot E) \cdot  \sigma^{\rm 2 }
+:$$\mu_{xy } = (A \cdot D + B \cdot E) \cdot \sigma^{\rm 2 }
-\hspace{0.3cm}\Rightarrow\hspace{0.3cm}
+\hspace{0.3cm}\rightarrow\hspace{0.3cm}
-\rho_{xy } = \frac{\rho_{xy } }{\sigma_x \cdot \sigma_y} =  \frac {A \cdot D + B \cdot E}{\sqrt{(A^{\rm 2}+B^{\rm 2})(D^{\rm 2}+E^{\rm 2} ) } }. $$}}
+\rho_{xy } = \frac{\rho_{xy } }{\sigma_x \cdot \sigma_y} = \frac {A \cdot D + B \cdot E}{\sqrt{(A^{\rm 2}+B^{\rm 2})(D^{\rm 2}+E^{\rm 2} ) } }. $$}}
-Wir schließen nun Zwei Sonderfälle aus:
+We now exclude two special cases:
-*$A = B = 0$&nbsp;  &rArr; &nbsp; $x ≡ 0$&nbsp; sowie
+*$A = B = 0$&nbsp; &rArr; &nbsp; $x ≡ 0$&nbsp; as well as.
-*$D = E = 0$&nbsp;  &rArr; &nbsp; $y ≡ 0$.
+*$D = E = 0$&nbsp; &rArr; &nbsp; $y ≡ 0$.
-Dann liefert die obige Gleichung stets eindeutige Werte für den Korrelationskoeffizienten im Bereich &nbsp;$–1 ≤ ρ_{xy} ≤ +1$.
+Then the above equation always yields unique values for the correlation coefficient in the range &nbsp;$-1 ≤ ρ_{xy} ≤ +1$.
 {{GraueBox|TEXT=
-$\text{Beispiel 1:}$&nbsp;
+$\text{Example 1:}$&nbsp;
-Setzen wir&nbsp;  $A = E = 0$,&nbsp; so ergibt sich der Korrelationskoeffizient&nbsp; $ρ_{xy} = 0$.&nbsp; Dieses Ergebnis ist einsichtig:
+If we set&nbsp; $A = E = 0$,&nbsp; we get the correlation coefficient&nbsp; $ρ_{xy} = 0$.&nbsp; This result is insightful:
-*Nun hängt&nbsp; $x$&nbsp; nur noch von&nbsp; $v$&nbsp; und&nbsp; $y$&nbsp; ausschließlich von&nbsp; $u$&nbsp; ab.
+*Now&nbsp; $x$&nbsp; depends only on&nbsp; $v$&nbsp; and&nbsp; $y$&nbsp; depends exclusively on&nbsp; $u$&nbsp; .
-*Da aber&nbsp; $u$&nbsp; und&nbsp; $v$&nbsp; als statistisch unabhängig angenommen wurden, bestehen auch keine Beziehungen zwischen&nbsp; $x$&nbsp; und&nbsp; $y$.
+*But since&nbsp; $u$&nbsp; and&nbsp; $v$&nbsp; were assumed to be statistically independent, there are also no relationships between&nbsp; $x$&nbsp; and&nbsp; $y$.
-*Ebenso ergibt sich &nbsp;$ρ_{xy} = 0$&nbsp;  für die Kombination&nbsp; $B = D = 0$.}}
+*Similarly, &nbsp;$ρ_{xy} = 0$&nbsp; results for the combination&nbsp; $B = D = 0$.}}
 {{GraueBox|TEXT=
-$\text{Beispiel 2:}$&nbsp; Die Konstellation &nbsp;$B = E = 0$&nbsp; führt dazu, dass sowohl&nbsp; $x$&nbsp; als auch&nbsp; $y$&nbsp; nur noch von&nbsp; $u$&nbsp; abhängen.&nbsp; Dann ergibt sich für den Korrelationskoeffizienten:
+$\text{Example 2:}$&nbsp; The constellation &nbsp;$B = E = 0$&nbsp; leads to the fact that both&nbsp; $x$&nbsp; and&nbsp; $y$&nbsp; depend only on&nbsp; $u$&nbsp; Then the correlation coefficient is obtained:
-:$$\rho_{xy } =  \frac {A \cdot D }{\sqrt{A^{\rm 2}\cdot D^{\rm 2} } } = \frac {A \cdot D }{\vert A\vert \cdot \vert D\vert } =\pm 1. $$
+:$$\rho_{xy } = \frac {A \cdot D }{\sqrt{A^{\rm 2}\cdot D^{\rm 2} } } = \frac {A \cdot D }{\vert A\vert \cdot D\vert } =\pm 1. $$
 ==Erzeugung korrelierter Zufallsgrößen==
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