Theory of Stochastic Signals/Two-Dimensional Gaussian Random Variables - Revision history

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	:$${\rm (I)}\hspace{0.8cm}\frac {1}{(1 - \rho_{\xi \eta}^2) \cdot \sigma_\xi^2} = \frac {4}{3} \left[ 1 - \frac {1}{2}\cdot {\sin (\alpha) }\right ] = 0.862 ,$$		:$${\rm (I)}\hspace{0.8cm}\frac {1}{(1 - \rho_{\xi \eta}^2) \cdot \sigma_\xi^2} = \frac {4}{3} \left[ 1 - \frac {1}{2}\cdot {\sin (\alpha) }\right ] = 0.862 ,$$
	:$${\rm (II)}\hspace{0.68cm}\frac {1}{(1 - \rho_{\xi \eta}^2) \cdot \sigma_\eta^2} = \frac {4}{3} \left[ 1 + \frac {1}{2}\cdot {\sin (\alpha) }\right ] = 1.805 ,\hspace{0.28cm}\frac {\rm (I)}{\rm (II)}: \frac		:$${\rm (II)}\hspace{0.68cm}\frac {1}{(1 - \rho_{\xi \eta}^2) \cdot \sigma_\eta^2} = \frac {4}{3} \left[ 1 + \frac {1}{2}\cdot {\sin (\alpha) }\right ] = 1.805 ,\hspace{0.28cm}\frac {\rm (I)}{\rm (II)}: \frac{\sigma_\eta}{\sigma_\xi} = \sqrt{\frac{0.862}{1.805} }= 0.691,$$
	{\sigma_\eta}{\sigma_\xi} = \sqrt{\frac{0.862}{1.805} }= 0.691,$$
	:$${\rm (III)}\hspace{0.54cm}\frac {\rho_{\xi \eta} }{(1 - \rho_{\xi \eta}^2) \cdot \sigma_\xi\cdot \sigma_\eta}= \frac {\rho_{\xi \eta} }{(1 - \rho_{\xi \eta}^2) \cdot \sigma_\xi^2 \cdot 0.691}=\frac {2}{3}\cdot \cos( \alpha) = 0.471.$$		:$${\rm (III)}\hspace{0.54cm}\frac {\rho_{\xi \eta} }{(1 - \rho_{\xi \eta}^2) \cdot \sigma_\xi\cdot \sigma_\eta}= \frac {\rho_{\xi \eta} }{(1 - \rho_{\xi \eta}^2) \cdot \sigma_\xi^2 \cdot 0.691}=\frac {2}{3}\cdot \cos( \alpha) = 0.471.$$

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	{{Display}}		{{Display}}
			[[de:Stochastische_Signaltheorie/Zweidimensionale_Gaußsche_Zufallsgrößen]]

Hwang at 13:44, 21 December 2022

2022-12-21T13:44:30Z

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	All previous statements of the fourth main chapter  "Random Variables with Statistical Dependence"  apply in general.		All previous statements of the fourth main chapter  "Random Variables with Statistical Dependence"  apply in general.

	For the special case  '''Gaussian random variables'''  – the name goes back to the scientist  [https://en.wikipedia.org/wiki/Carl_Friedrich_Gauss Carl Friedrich Gauss]  – we can further note:		For the special case  »'''Gaussian random variables'''«  – the name goes back to the scientist  [https://en.wikipedia.org/wiki/Carl_Friedrich_Gauss $\text{Carl Friedrich Gauss}$]  – we can further note:
	*The joint probability density function of a two-dimensional Gaussian random variable  $(x, y)$  with mean values  $m_x = 0$,  $m_y = 0$  and correlation coefficient  $ρ_{xy}$  is:		*The joint probability density function of a two-dimensional Gaussian random variable  $(x, y)$  with mean values  $m_x = 0$,  $m_y = 0$  and correlation coefficient  $ρ_{xy}$  is:
	: $$f_{xy}(x, y)=\frac{\rm 1}{\rm 2\it\pi \cdot \sigma_x \cdot \sigma_y \sqrt{\rm 1-\rho_{\it xy}^2}}\cdot\exp\Bigg[-\frac{\rm 1}{\rm 2\cdot (1- \it\rho_{xy}^{\rm 2} {\rm)}}\cdot(\frac {\it x^{\rm 2}}{\sigma_x^{\rm 2}}+\frac {\it y^{\rm 2}}{\sigma_y^{\rm 2}}-\rm 2\it\rho_{xy}\cdot\frac{x \cdot y}{\sigma_x \cdot \sigma_y}\rm ) \rm \Bigg].$$		: $$f_{xy}(x, y)=\frac{\rm 1}{\rm 2\it\pi \cdot \sigma_x \cdot \sigma_y \sqrt{\rm 1-\rho_{\it xy}^2}}\cdot\exp\Bigg[-\frac{\rm 1}{\rm 2\cdot (1- \it\rho_{xy}^{\rm 2} {\rm)}}\cdot(\frac {\it x^{\rm 2}}{\sigma_x^{\rm 2}}+\frac {\it y^{\rm 2}}{\sigma_y^{\rm 2}}-\rm 2\it\rho_{xy}\cdot\frac{x \cdot y}{\sigma_x \cdot \sigma_y}\rm ) \rm \Bigg].$$
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	:$$ρ_{xy} = 0.8.$$		:$$ρ_{xy} = 0.8.$$

	As in the  [[Theory_of_Stochastic_Signals/Two-Dimensional_Random_Variables#Correlation_coefficient\|previous examples]],  the random variable is more extended in  $x$  direction than in  $y$  direction:   $σ_x = 2 \cdot σ_y$.		As in the  [[Theory_of_Stochastic_Signals/Two-Dimensional_Random_Variables#Correlation_coefficient\|$\text{previous examples}$]],  the random variable is more extended in  $x$  direction than in  $y$  direction:   $σ_x = 2 \cdot σ_y$.
	<br clear=all>		<br clear=all>
	These representations can be interpreted as follows:		These representations can be interpreted as follows:
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	More information on this topic with signal examples is provided in the first part  "Gaussian random variables without statistical bindings"  of the  (German language)  learning video		More information on this topic with signal examples is provided in the first part  "Gaussian random variables without statistical bindings"  of the  (German language)  learning video
	::[[Gaußsche_2D-Zufallsgrößen_(Lernvideo)\|Gaußsche 2D-Zufallsgrößen]]   ⇒   "Two-dimensional Gaussian random variables".		::[[Gaußsche_2D-Zufallsgrößen_(Lernvideo)\|"Gaußsche 2D-Zufallsgrößen"]]   ⇒   "Two-dimensional Gaussian random variables".
	<br clear=all>		<br clear=all>
	{{GraueBox\|TEXT=		{{GraueBox\|TEXT=
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	More information on this topic is provided in the  (German language)  learning video		More information on this topic is provided in the  (German language)  learning video
	::[[Gaußsche_2D-Zufallsgrößen_(Lernvideo)\|Gaußsche 2D-Zufallsgrößen]]   ⇒   "Two-dimensional Gaussian random variables".		::[[Gaußsche_2D-Zufallsgrößen_(Lernvideo)\|"Gaußsche 2D-Zufallsgrößen"]]   ⇒   "Two-dimensional Gaussian random variables".

	*Part 1:   Gaussian random variables without statistical bindings,		*Part 1:   Gaussian random variables without statistical bindings,

Guenter at 15:05, 10 April 2022

2022-04-10T15:05:02Z

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	[[File:EN_Sto_T_4_2_S3_neu1.png\|right\|frame\|Contour lines of the 2D–PDF at correlated quantities]]		[[File:EN_Sto_T_4_2_S3_neu1.png\|right\|frame\|Contour lines of the 2D–PDF at correlated quantities]]
	*The ellipse major axis is dashed in dark blue.		*The ellipse major axis is dashed in dark blue.
	*The correlation line or  "~~regession~~ line"   $K(x)$  is drawn in solid red.		*The correlation line or  "regression line"   $(RL)$  is drawn in solid red.

Guenter at 16:24, 31 March 2022

2022-03-31T16:24:19Z

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	:$$f_{xy}(x, y) = {\rm const.} \hspace{0.5cm} \rightarrow \hspace{0.5cm} \frac{x^{\rm 2} }{\sigma_{x}^{\rm 2}}+\frac{y^{\rm 2} }{\sigma_{y}^{\rm 2} }-{\rm 2}\cdot\rho_{xy}\cdot\frac{x\cdot y}{\sigma_x\cdot \sigma_y}={\rm const.}$$		:$$f_{xy}(x, y) = {\rm const.} \hspace{0.5cm} \rightarrow \hspace{0.5cm} \frac{x^{\rm 2} }{\sigma_{x}^{\rm 2}}+\frac{y^{\rm 2} }{\sigma_{y}^{\rm 2} }-{\rm 2}\cdot\rho_{xy}\cdot\frac{x\cdot y}{\sigma_x\cdot \sigma_y}={\rm const.}$$
	The following graph shows in lighter blue two contour lines for different parameter sets,  each with  $ρ_{xy} ≠ 0$.		The following graph shows in lighter blue two contour lines for different parameter sets,  each with  $ρ_{xy} ≠ 0$.
	[[File:~~P_ID408__Sto_T_4_2_S3_neu~~.png\|right\|frame\|Contour lines of the 2D–PDF at correlated quantities]]		[[File:EN_Sto_T_4_2_S3_neu1.png\|right\|frame\|Contour lines of the 2D–PDF at correlated quantities]]
	*The ellipse major axis is dashed in dark blue.		*The ellipse major axis is dashed in dark blue.
	*The correlation line or  "regession line"   $K(x)$  is drawn in solid red.		*The correlation line or  "regession line"   $K(x)$  is drawn in solid red.

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

2022-02-15T15:45:53Z

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

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	: $$f_{xy}(x, y)=\frac{\rm 1}{\rm 2\it\pi \cdot \sigma_x \cdot \sigma_y \sqrt{\rm 1-\rho_{\it xy}^2}}\cdot\exp\Bigg[-\frac{\rm 1}{\rm 2\cdot (1- \it\rho_{xy}^{\rm 2} {\rm)}}\cdot(\frac {\it x^{\rm 2}}{\sigma_x^{\rm 2}}+\frac {\it y^{\rm 2}}{\sigma_y^{\rm 2}}-\rm 2\it\rho_{xy}\cdot\frac{x \cdot y}{\sigma_x \cdot \sigma_y}\rm ) \rm \Bigg].$$		: $$f_{xy}(x, y)=\frac{\rm 1}{\rm 2\it\pi \cdot \sigma_x \cdot \sigma_y \sqrt{\rm 1-\rho_{\it xy}^2}}\cdot\exp\Bigg[-\frac{\rm 1}{\rm 2\cdot (1- \it\rho_{xy}^{\rm 2} {\rm)}}\cdot(\frac {\it x^{\rm 2}}{\sigma_x^{\rm 2}}+\frac {\it y^{\rm 2}}{\sigma_y^{\rm 2}}-\rm 2\it\rho_{xy}\cdot\frac{x \cdot y}{\sigma_x \cdot \sigma_y}\rm ) \rm \Bigg].$$
	*Replacing  $x$  by  $(x - m_x)$  and  $y$  by  $(y- m_y)$,  we obtain the more general PDF of a two-dimensional Gaussian random variable with mean.		*Replacing  $x$  by  $(x - m_x)$  and  $y$  by  $(y- m_y)$,  we obtain the more general PDF of a two-dimensional Gaussian random variable with mean.
	*The two marginal probability density functions $f_{x}(x)$  and $f_{y}(y)$  of a two-dimensional Gaussian random variable are also Gaussian with ~~rms values~~  $σ_x$  and $σ_y$, resp.		*The two marginal probability density functions $f_{x}(x)$  and $f_{y}(y)$  of a two-dimensional Gaussian random variable are also Gaussian with standard deviations  $σ_x$  and $σ_y$, resp.
	*For uncorrelated components  $x$  and  $y$  in the above equation  $ρ_{xy} = 0$  must be substituted,  and then the result is obtained:		*For uncorrelated components  $x$  and  $y$  in the above equation  $ρ_{xy} = 0$  must be substituted,  and then the result is obtained:
	:$$f_{xy}(x,y)=\frac{1}{\sqrt{2\pi}\cdot\sigma_{x}} \cdot\rm e^{-\it {x^{\rm 2}}\hspace{-0.08cm}/{\rm (}{\rm 2\it\sigma_{x}^{\rm 2}} {\rm )}} \cdot\frac{1}{\sqrt{2\pi}\cdot\sigma_{\it y}}\cdot e^{-\it {y^{\rm 2}}\hspace{-0.08cm}/{\rm (}{\rm 2\it\sigma_{y}^{\rm 2}} {\rm )}} = \it f_{x} \rm ( \it x \rm ) \cdot \it f_{y} \rm ( \it y \rm ) .$$		:$$f_{xy}(x,y)=\frac{1}{\sqrt{2\pi}\cdot\sigma_{x}} \cdot\rm e^{-\it {x^{\rm 2}}\hspace{-0.08cm}/{\rm (}{\rm 2\it\sigma_{x}^{\rm 2}} {\rm )}} \cdot\frac{1}{\sqrt{2\pi}\cdot\sigma_{\it y}}\cdot e^{-\it {y^{\rm 2}}\hspace{-0.08cm}/{\rm (}{\rm 2\it\sigma_{y}^{\rm 2}} {\rm )}} = \it f_{x} \rm ( \it x \rm ) \cdot \it f_{y} \rm ( \it y \rm ) .$$

Guenter at 14:05, 28 January 2022

2022-01-28T14:05:18Z

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	{{GraueBox\|TEXT=		{{GraueBox\|TEXT=
	[[File:P_ID2911__Sto_T_4_2_S2_unten.png \|right\|frame\| Screen capture of the video "2D Gaussian random variables"]]		[[File:P_ID2911__Sto_T_4_2_S2_unten.png \|right\|frame\| Screen capture of the video "2D Gaussian random variables"]]
	$\text{Example 2:}$		$\text{Example 2:}$
			<br><br><br>
	*The graphic shows a snapshot of the first part video  "Gaussian random variables without statistical bindings".		*The graphic shows a snapshot of the first part video  "Gaussian random variables without statistical bindings".

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			Notes:
	*If the coordinate system were also rotated by  $β =45^\circ$,  there would be uncorrelated components because of  $σ_x = σ_y$  and because of  $\sin(β) = \cos(β) = 1/\sqrt{2}$  for the new correlation coefficient   ⇒   $ρ_{ξη} = 0$ .		*If the coordinate system were also rotated by  $β =45^\circ$,  there would be uncorrelated components because of  $σ_x = σ_y$  and because of  $\sin(β) = \cos(β) = 1/\sqrt{2}$  for the new correlation coefficient   ⇒   $ρ_{ξη} = 0$ .

Guenter at 13:07, 28 January 2022

2022-01-28T13:07:16Z

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	[[Aufgaben:Exercise_4.4:_Two-dimensional_Gaussian_PDF\|Exercise 4.4: Two-dimensional Gaussian PDF]]		[[Aufgaben:Exercise_4.4:_Two-dimensional_Gaussian_PDF\|Exercise 4.4: Two-dimensional Gaussian PDF]]

	[[Aufgaben:Exercise_4.4Z:~~_Contour_Lines_of_the_2D_PDF~~\|Exercise 4.4Z: Contour Lines of the 2D PDF]]		[[Aufgaben:Exercise_4.4Z:_Contour_Lines_of_the_"2D-PDF"\|Exercise 4.4Z: Contour Lines of the "2D-PDF"]]

	[[Aufgaben:Exercise_4.5:_Two-dimensional_Examination_Evaluation\|Exercise 4.5: Two-dimensional Examination Evaluation]]		[[Aufgaben:Exercise_4.5:_Two-dimensional_Examination_Evaluation\|Exercise 4.5: Two-dimensional Examination Evaluation]]

Guenter at 13:04, 28 January 2022

2022-01-28T13:04:57Z

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	==Exercises for the chapter==		==Exercises for the chapter==
	<br>		<br>
	[[Aufgaben:Exercise_4.4:~~_Gaussian_2D_PDF~~\|Exercise 4.4: Gaussian 2D PDF]]		[[Aufgaben:Exercise_4.4:_Two-dimensional_Gaussian_PDF\|Exercise 4.4: Two-dimensional Gaussian PDF]]

	[[Aufgaben:Exercise_4.4Z:_Contour_Lines_of_the_2D_PDF\|Exercise 4.4Z: Contour Lines of the 2D PDF]]		[[Aufgaben:Exercise_4.4Z:_Contour_Lines_of_the_2D_PDF\|Exercise 4.4Z: Contour Lines of the 2D PDF]]

Guenter at 13:01, 28 January 2022

2022-01-28T13:01:53Z

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	==Rotation of the coordinate system==		==Rotation of the coordinate system==
	<br>		<br>
	For some tasks it is advantageous to rotate the coordinate system, as indicated in the following graphic:		For some tasks it is advantageous to rotate the coordinate system,  as indicated in the following graphic:

	[[File:EN_Sto_T_4_2_S4.png \|frame\| To rotate the coordinate system \| right]]		[[File:EN_Sto_T_4_2_S4.png \|frame\| To rotate the coordinate system \| right]]
	*The  $(ξ, η)$ coordinate system is rotated with respect to the original  $(x, y)$ system by the angle  $β$~~ ~~ .		*The  $(ξ, η)$  coordinate system is rotated with respect to the original  $(x, y)$  system by the angle  $β$.
	*In contrast  $α$  denotes the angle between the ellipse major axis and the  $x$ axis.		*In contrast,  $α$  denotes the angle between the ellipse major axis and the  $x$–axis.


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	If  $(x, y)$  is a 2D Gaussian random variable, then the random variable  $(ξ, η)$  is also Gaussian distributed.		If  $(x, y)$  is a Gaussian random variable,  then the random variable  $(ξ, η)$  is also Gaussian distributed.

	Substituting the above equations into the 2D PDF $f_{xy}(x, y)$  and comparing the coefficients, we obtain the following governing equations for  $σ_x$,  $σ_y$  and  $ρ_{xy}$  respectively  $σ_ξ,  σ_η$  and  $ρ_{ξη}$:		Substituting the above equations into the 2D–PDF $f_{xy}(x, y)$  and comparing the coefficients,  we obtain the following governing equations for  $σ_x$,  $σ_y$  and  $ρ_{xy}$  respectively  $σ_ξ,  σ_η$   and  $ρ_{ξη}$:

	:$$\frac {1}{(1 - \rho_{\xi \eta}^2) \cdot \sigma_\xi^2} = \frac {1}{(1 - \rho_{xy}^2) } \left[ \frac {\cos^2 (\beta)}{\sigma_{x}^2 } + \frac {\sin^2 (\beta)}{\sigma_{y}^2 } - 2 \rho_{xy} \cdot \frac {\sin (\beta) \cdot \cos (\beta)}{\sigma_{x} \cdot \sigma_{y}}\right ] ,$$		:$$\frac {1}{(1 - \rho_{\xi \eta}^2) \cdot \sigma_\xi^2} = \frac {1}{(1 - \rho_{xy}^2) } \left[ \frac {\cos^2 (\beta)}{\sigma_{x}^2 } + \frac {\sin^2 (\beta)}{\sigma_{y}^2 } - 2 \rho_{xy} \cdot \frac {\sin (\beta) \cdot \cos (\beta)}{\sigma_{x} \cdot \sigma_{y}}\right ] ,$$
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	:$$\frac {\rho_{\xi \eta}}{(1 - \rho_{\xi \eta}^2) \cdot \sigma_\xi\cdot \sigma_\eta}= \frac {1}{(1 - \rho_{xy}^2) } \left[ \frac {\sin (\beta) \cdot \cos (\beta)}{\sigma_{x}^2 } - \frac {\sin (\beta) \cdot \cos (\beta)}{\sigma_{y}^2 } + \frac {\rho_{xy}}{\sigma_{x} \cdot \sigma_{y}} \cdot ( \cos^2( \beta) -\sin^2( \beta)) \right ] .$$		:$$\frac {\rho_{\xi \eta}}{(1 - \rho_{\xi \eta}^2) \cdot \sigma_\xi\cdot \sigma_\eta}= \frac {1}{(1 - \rho_{xy}^2) } \left[ \frac {\sin (\beta) \cdot \cos (\beta)}{\sigma_{x}^2 } - \frac {\sin (\beta) \cdot \cos (\beta)}{\sigma_{y}^2 } + \frac {\rho_{xy}}{\sigma_{x} \cdot \sigma_{y}} \cdot ( \cos^2( \beta) -\sin^2( \beta)) \right ] .$$

	With these three equations ~~the~~ in each case three parameters of the two coordinate systems can be converted directly, which is possible however only in special cases without substantial computational expenditure   Following an example with justifiable computational expenditure.		With these three equations,  in each case three parameters of the two coordinate systems can be converted directly,  which is possible however only in special cases without substantial computational expenditure.  Following an example with justifiable computational expenditure.



	~~[[File: EN_Sto_T_4_2_S4.png\|right\|frame]]~~
	{{GraueBox\|TEXT=		{{GraueBox\|TEXT=
	$\text{Example 3:}$  We consider a Gaussian 2D PDF with the following properties:		[[File: EN_Sto_T_4_2_S4.png\|right\|frame\|To rotate the coordinate system]]
			$\text{Example 3:}$  We consider a two-dimensional Gaussian PDF with the following properties:

			#The variances of the two components are equal:   $σ_x^2 = σ_y^2 = 1$.
			#The correlation coefficient between  $x$  and  $y$  is  $ρ_{xy} = 0.5$.
			#The angle of the ellipse major axis with respect to the  $x$–axis is thus  $α = 45^\circ$.


	*~~The variances of~~ the ~~two components are equal:~~   $~~σ_x^2~~ = ~~σ_y~~^~~2 = 1~~$.		*If the coordinate system were also rotated by  $β =45^\circ$,  there would be uncorrelated components because of  $σ_x = σ_y$  and because of  $\sin(β) = \cos(β) = 1/\sqrt{2}$  for the new correlation coefficient   ⇒   $ρ_{ξη} = 0$ .
	*The correlation coefficient between  ~~$x$  and~~  $y$  is  $ρ_{xy} ~~= 0.5~~$.
	*The angle of the ~~ellipse major axis with respect to~~  ~~$x$~~&~~ndash~~;~~axis is thus~~  $α = ~~45^\circ~~$.

			*The two standard deviations  – related to the new coordinate system –  would then result according to the first two equations to  $σ_ξ = \sqrt{1.5}$  and  $σ_η = \sqrt{0.5}$.

	If the coordinate system were also rotated by  $β =45^\circ$  , there would be uncorrelated components because of  $σ_x = σ_y$  and because of  $\sin(β) = \cos(β) = 1/\sqrt{2}$  for the new correlation coefficient  $ρ_{ξη} = 0$   ⇒   .

	~~The two standard deviations - related to the new coordinate system - would then result according to~~ the ~~first two equations to~~  $~~σ_ξ~~ = ~~\sqrt{1.5}~~$  ~~and~~  $~~σ_η~~ = ~~\sqrt{0.5}~~$.		However,  the above sketch is not based on  $β = α$  but on  $β = α/2$.

	~~However,~~ the ~~above sketch is not based on  $β = α$  but on  $β = α/2$.~~		With the parameters and equations
			# $σ_x = σ_y = 1$,  $ρ_{xy} = 0.5$,
	*With  $σ_x = σ_y = 1$,  $ρ_{xy} = 0.5$,		#$α = 45^\circ$,  $\sin(β) - \cos(β) = \sin(2β)/2 = \sin(α)/2$,  and
	*as well as  $α = 45^\circ$,  $\sin(β) - \cos(β) = \sin(2β)/2 = \sin(α)/2$  and		#$\cos^2(β) - \sin^2(β) = \cos(2β)= \cos(α)$
	*$\cos^2(β) - \sin^2(β) = \cos(2β)= \cos(α)$


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	:$${\rm (III)}\hspace{0.54cm}\frac {\rho_{\xi \eta} }{(1 - \rho_{\xi \eta}^2) \cdot \sigma_\xi\cdot \sigma_\eta}= \frac {\rho_{\xi \eta} }{(1 - \rho_{\xi \eta}^2) \cdot \sigma_\xi^2 \cdot 0.691}=\frac {2}{3}\cdot \cos( \alpha) = 0.471.$$		:$${\rm (III)}\hspace{0.54cm}\frac {\rho_{\xi \eta} }{(1 - \rho_{\xi \eta}^2) \cdot \sigma_\xi\cdot \sigma_\eta}= \frac {\rho_{\xi \eta} }{(1 - \rho_{\xi \eta}^2) \cdot \sigma_\xi^2 \cdot 0.691}=\frac {2}{3}\cdot \cos( \alpha) = 0.471.$$

	Dividing now the equation  $\rm (III)$  by the equation  $\rm (I)$, we get:		Dividing now the equation  $\rm (III)$  by the equation  $\rm (I)$,  we get:
	:$$ \frac {\rho_{\xi \eta} }{0.691}=\frac {0.471}{0.862}\hspace{0.5cm}\Rightarrow\hspace{0.5cm}{\rho_{\xi \eta} }= 0.378.$$		:$$ \frac {\rho_{\xi \eta} }{0.691}=\frac {0.471}{0.862}\hspace{0.5cm}\Rightarrow\hspace{0.5cm}{\rho_{\xi \eta} }= 0.378.$$

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	[[Aufgaben:Exercise_4.4:_Two-dimensional_Gaussian_PDF\|Exercise 4.4: Two-dimensional Gaussian PDF]]		[[Aufgaben:Exercise_4.4:_Two-dimensional_Gaussian_PDF\|Exercise 4.4: Two-dimensional Gaussian PDF]]

	[[Aufgaben:Exercise_4.4Z:~~_Contour_Lines_of_the_2D_PDF~~\|Exercise 4.4Z: Contour Lines of the 2D PDF]]		[[Aufgaben:Exercise_4.4Z:_Contour_Lines_of_the_"2D-PDF"\|Exercise 4.4Z: Contour Lines of the "2D-PDF"]]

	[[Aufgaben:Exercise_4.5:_Two-dimensional_Examination_Evaluation\|Exercise 4.5: Two-dimensional Examination Evaluation]]		[[Aufgaben:Exercise_4.5:_Two-dimensional_Examination_Evaluation\|Exercise 4.5: Two-dimensional Examination Evaluation]]