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

Hwang at 13:44, 21 December 2022

2022-12-21T13:44:30Z

Guenter at 15:05, 10 April 2022

2022-04-10T15:05:02Z

Guenter at 16:24, 31 March 2022

2022-03-31T16:24:19Z

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

2022-02-15T15:45:53Z

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

Guenter at 14:05, 28 January 2022

2022-01-28T14:05:18Z

Guenter at 13:07, 28 January 2022

2022-01-28T13:07:16Z

Guenter at 13:04, 28 January 2022

2022-01-28T13:04:57Z

Guenter at 13:01, 28 January 2022

2022-01-28T13:01:53Z

Guenter at 10:59, 28 January 2022

2022-01-28T10:59:12Z

Guenter at 10:07, 28 January 2022

2022-01-28T10:07:25Z

@@ Line 9: / Line 9: @@
 All previous statements of the fourth main chapter&nbsp; "Random Variables with Statistical Dependence"&nbsp; apply in general.
-For the special case&nbsp; '''Gaussian random variables'''&nbsp; &ndash; the name goes back to the scientist&nbsp; [https://en.wikipedia.org/wiki/Carl_Friedrich_Gauss Carl Friedrich Gauss]&nbsp; &ndash; we can further note:
+For the special case&nbsp; &raquo;'''Gaussian random variables'''&laquo;&nbsp; &ndash; the name goes back to the scientist&nbsp; [https://en.wikipedia.org/wiki/Carl_Friedrich_Gauss $\text{Carl Friedrich Gauss}$]&nbsp; &ndash; we can further note:
 *The joint probability density function of a two-dimensional Gaussian random variable&nbsp; $(x, y)$&nbsp; with mean values&nbsp; $m_x = 0$,&nbsp; $m_y = 0$&nbsp; and correlation coefficient&nbsp; $ρ_{xy}$&nbsp; is:
 : $$f_{xy}(x, y)=\frac{\rm 1}{\rm 2\it\pi \cdot \sigma_x \cdot \sigma_y \sqrt{\rm 1-\rho_{\it xy}^2}}\cdot\exp\Bigg[-\frac{\rm 1}{\rm 2\cdot (1- \it\rho_{xy}^{\rm 2} {\rm)}}\cdot(\frac {\it x^{\rm 2}}{\sigma_x^{\rm 2}}+\frac {\it y^{\rm 2}}{\sigma_y^{\rm 2}}-\rm 2\it\rho_{xy}\cdot\frac{x \cdot y}{\sigma_x \cdot \sigma_y}\rm ) \rm \Bigg].$$
@@ Line 40: / Line 40: @@
 :$$ρ_{xy} = 0.8.$$
-As in the&nbsp; [[Theory_of_Stochastic_Signals/Two-Dimensional_Random_Variables#Correlation_coefficient|previous examples]],&nbsp; the random variable is more extended in&nbsp; $x$&nbsp; direction than in&nbsp; $y$&nbsp; direction: &nbsp; $σ_x = 2 \cdot σ_y$.
+As in the&nbsp; [[Theory_of_Stochastic_Signals/Two-Dimensional_Random_Variables#Correlation_coefficient|$\text{previous examples}$]],&nbsp; the random variable is more extended in&nbsp; $x$&nbsp; direction than in&nbsp; $y$&nbsp; direction: &nbsp; $σ_x = 2 \cdot σ_y$.
 <br clear=all>
 These representations can be interpreted as follows:
@@ Line 64: / Line 64: @@
 More information on this topic with signal examples is provided in the first part&nbsp; "Gaussian random variables without statistical bindings"&nbsp; of the&nbsp; (German language)&nbsp; learning video
-::[[Gaußsche_2D-Zufallsgrößen_(Lernvideo)|Gaußsche 2D-Zufallsgrößen]] &nbsp; &rArr; &nbsp; "Two-dimensional Gaussian random variables".
+::[[Gaußsche_2D-Zufallsgrößen_(Lernvideo)|"Gaußsche 2D-Zufallsgrößen"]] &nbsp; &rArr; &nbsp; "Two-dimensional Gaussian random variables".
 <br clear=all>
 {{GraueBox|TEXT=
@@ Line 95: / Line 95: @@
 More information on this topic is provided in the&nbsp; (German language)&nbsp; learning video
-::[[Gaußsche_2D-Zufallsgrößen_(Lernvideo)|Gaußsche 2D-Zufallsgrößen]] &nbsp; &rArr; &nbsp; "Two-dimensional Gaussian random variables".
+::[[Gaußsche_2D-Zufallsgrößen_(Lernvideo)|"Gaußsche 2D-Zufallsgrößen"]] &nbsp; &rArr; &nbsp; "Two-dimensional Gaussian random variables".
 *Part 1: &nbsp; Gaussian random variables without statistical bindings,

@@ Line 83: / Line 83: @@
 [[File:EN_Sto_T_4_2_S3_neu1.png|right|frame|Contour lines of the 2D&ndash;PDF at correlated quantities]]
 *The ellipse major axis is dashed in dark blue.
-*The correlation line or&nbsp; "regession line" &nbsp; $K(x)$&nbsp; is drawn in solid red.
+*The correlation line or&nbsp; "regression line" &nbsp; $(RL)$&nbsp; is drawn in solid red.

@@ Line 81: / Line 81: @@
 :$$f_{xy}(x, y) = {\rm const.} \hspace{0.5cm} \rightarrow \hspace{0.5cm} \frac{x^{\rm 2} }{\sigma_{x}^{\rm 2}}+\frac{y^{\rm 2} }{\sigma_{y}^{\rm 2} }-{\rm 2}\cdot\rho_{xy}\cdot\frac{x\cdot y}{\sigma_x\cdot \sigma_y}={\rm const.}$$
 The following graph shows in lighter blue two contour lines for different parameter sets,&nbsp; each with&nbsp; $ρ_{xy} ≠ 0$.
-[[File:P_ID408__Sto_T_4_2_S3_neu.png|right|frame|Contour lines of the 2D&ndash;PDF at correlated quantities]]
+[[File:EN_Sto_T_4_2_S3_neu1.png|right|frame|Contour lines of the 2D&ndash;PDF at correlated quantities]]
 *The ellipse major axis is dashed in dark blue.
 *The correlation line or&nbsp; "regession line" &nbsp; $K(x)$&nbsp; is drawn in solid red.

@@ Line 13: / Line 13: @@
 : $$f_{xy}(x, y)=\frac{\rm 1}{\rm 2\it\pi \cdot \sigma_x \cdot \sigma_y \sqrt{\rm 1-\rho_{\it xy}^2}}\cdot\exp\Bigg[-\frac{\rm 1}{\rm 2\cdot (1- \it\rho_{xy}^{\rm 2} {\rm)}}\cdot(\frac {\it x^{\rm 2}}{\sigma_x^{\rm 2}}+\frac {\it y^{\rm 2}}{\sigma_y^{\rm 2}}-\rm 2\it\rho_{xy}\cdot\frac{x \cdot y}{\sigma_x \cdot \sigma_y}\rm ) \rm \Bigg].$$
 *Replacing&nbsp; $x$&nbsp; by&nbsp; $(x - m_x)$&nbsp; and&nbsp; $y$&nbsp; by&nbsp; $(y- m_y)$,&nbsp; we obtain the more general PDF of a two-dimensional Gaussian random variable with mean.
-*The two marginal probability density functions $f_{x}(x)$&nbsp; and $f_{y}(y)$&nbsp; of a two-dimensional Gaussian random variable are also Gaussian with rms values&nbsp; $σ_x$&nbsp; and $σ_y$, resp.
+*The two marginal probability density functions $f_{x}(x)$&nbsp; and $f_{y}(y)$&nbsp; of a two-dimensional Gaussian random variable are also Gaussian with standard deviations&nbsp; $σ_x$&nbsp; and $σ_y$, resp.
 *For uncorrelated components&nbsp; $x$&nbsp; and&nbsp; $y$&nbsp; in the above equation&nbsp; $ρ_{xy} = 0$&nbsp; must be substituted,&nbsp; and then the result is obtained:
 :$$f_{xy}(x,y)=\frac{1}{\sqrt{2\pi}\cdot\sigma_{x}} \cdot\rm e^{-\it {x^{\rm 2}}\hspace{-0.08cm}/{\rm (}{\rm 2\it\sigma_{x}^{\rm 2}} {\rm )}} \cdot\frac{1}{\sqrt{2\pi}\cdot\sigma_{\it y}}\cdot e^{-\it {y^{\rm 2}}\hspace{-0.08cm}/{\rm (}{\rm 2\it\sigma_{y}^{\rm 2}} {\rm )}} = \it f_{x} \rm ( \it x \rm ) \cdot \it f_{y} \rm ( \it y \rm ) .$$

@@ Line 68: / Line 68: @@
 {{GraueBox|TEXT=
 [[File:P_ID2911__Sto_T_4_2_S2_unten.png |right|frame| Screen capture of the video "2D Gaussian random variables"]]
 $\text{Example 2:}$&nbsp;
 *The graphic shows a snapshot of the first part video&nbsp; "Gaussian random variables without statistical bindings".
@@ Line 134: / Line 135: @@
 *If the coordinate system were also rotated by&nbsp; $β =45^\circ$,&nbsp; there would be uncorrelated components because of&nbsp; $σ_x = σ_y$&nbsp; and because of&nbsp; $\sin(β) = \cos(β) = 1/\sqrt{2}$&nbsp; for the new correlation coefficient &nbsp; &rArr; &nbsp;  $ρ_{ξη} = 0$ .

@@ Line 162: / Line 162: @@
 [[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]]

@@ Line 160: / Line 160: @@
 ==Exercises for the chapter==
 <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]]

@@ Line 101: / Line 101: @@
 ==Rotation of the coordinate system==
 <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,&nbsp; as indicated in the following graphic:
 [[File:EN_Sto_T_4_2_S4.png |frame| To rotate the coordinate system | right]]
-*The&nbsp; $(ξ, η)$ coordinate system is rotated with respect to the original&nbsp; $(x, y)$ system by the angle&nbsp; $β$&nbsp; .
+*The&nbsp; $(ξ, η)$&nbsp; coordinate system is rotated with respect to the original&nbsp; $(x, y)$&nbsp; system by the angle&nbsp; $β$.
-*In contrast&nbsp; $α$&nbsp; denotes the angle between the ellipse major axis and the&nbsp; $x$ axis.
+*In contrast,&nbsp; $α$&nbsp; denotes the angle between the ellipse major axis and the&nbsp; $x$&ndash;axis.
@@ Line 113: / Line 113: @@
-If&nbsp; $(x, y)$&nbsp; is a 2D Gaussian random variable, then the random variable&nbsp; $(ξ, η)$&nbsp; is also Gaussian distributed.
+If&nbsp; $(x, y)$&nbsp; is a Gaussian random variable,&nbsp; then the random variable&nbsp; $(ξ, η)$&nbsp; is also Gaussian distributed.
-Substituting the above equations into the 2D PDF $f_{xy}(x, y)$&nbsp; and comparing the coefficients, we obtain the following governing equations for&nbsp; $σ_x$,&nbsp; $σ_y$&nbsp; and&nbsp; $ρ_{xy}$&nbsp; respectively&nbsp; $σ_ξ,&nbsp; σ_η$&nbsp; and&nbsp; $ρ_{ξη}$:
+Substituting the above equations into the 2D&ndash;PDF $f_{xy}(x, y)$&nbsp; and comparing the coefficients,&nbsp; we obtain the following governing equations for&nbsp; $σ_x$,&nbsp; $σ_y$&nbsp; and&nbsp; $ρ_{xy}$&nbsp; respectively&nbsp; $σ_ξ,&nbsp; σ_η$ &nbsp; and&nbsp; $ρ_{ξη}$:
 :$$\frac {1}{(1 - \rho_{\xi \eta}^2) \cdot \sigma_\xi^2} = \frac {1}{(1 - \rho_{xy}^2) }  \left[ \frac {\cos^2 (\beta)}{\sigma_{x}^2 } + \frac {\sin^2 (\beta)}{\sigma_{y}^2 } - 2 \rho_{xy} \cdot \frac {\sin (\beta) \cdot \cos (\beta)}{\sigma_{x} \cdot \sigma_{y}}\right ] ,$$
@@ Line 121: / Line 121: @@
 :$$\frac {\rho_{\xi \eta}}{(1 - \rho_{\xi \eta}^2) \cdot \sigma_\xi\cdot \sigma_\eta}= \frac {1}{(1 - \rho_{xy}^2) }  \left[ \frac {\sin (\beta) \cdot \cos (\beta)}{\sigma_{x}^2 } - \frac {\sin (\beta) \cdot \cos (\beta)}{\sigma_{y}^2 } + \frac {\rho_{xy}}{\sigma_{x} \cdot \sigma_{y}} \cdot ( \cos^2( \beta) -\sin^2( \beta)) \right ] .$$
-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 &nbsp; Following an example with justifiable computational expenditure.
+With these three equations,&nbsp; in each case three parameters of the two coordinate systems can be converted directly,&nbsp; which is possible however only in special cases without substantial computational expenditure.&nbsp; Following an example with justifiable computational expenditure.
 {{GraueBox|TEXT=
-$\text{Example 3:}$&nbsp; 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]]
-*The variances of the two components are equal: &nbsp; $σ_x^2 = σ_y^2 = 1$.
+*If the coordinate system were also rotated by&nbsp; $β =45^\circ$,&nbsp; there would be uncorrelated components because of&nbsp; $σ_x = σ_y$&nbsp; and because of&nbsp; $\sin(β) = \cos(β) = 1/\sqrt{2}$&nbsp; for the new correlation coefficient &nbsp; &rArr; &nbsp;  $ρ_{ξη} = 0$ .
-The two standard deviations - related to the new coordinate system - would then result according to the first two equations to&nbsp; $σ_ξ = \sqrt{1.5}$&nbsp; and&nbsp; $σ_η = \sqrt{0.5}$.
+However,&nbsp; the above sketch is not based on&nbsp; $β = α$&nbsp; but on&nbsp; $β = α/2$.
-However, the above sketch is not based on&nbsp; $β = α$&nbsp; but on&nbsp; $β = α/2$.
+With the parameters and equations
+# $σ_x = σ_y = 1$,&nbsp; $ρ_{xy} = 0.5$,
-*With&nbsp; $σ_x = σ_y = 1$,&nbsp; $ρ_{xy} = 0.5$,
+#$α = 45^\circ$,&nbsp; $\sin(β) - \cos(β) = \sin(2β)/2 = \sin(α)/2$,&nbsp; and
-*as well as&nbsp; $α = 45^\circ$,&nbsp; $\sin(β) - \cos(β) = \sin(2β)/2 = \sin(α)/2$&nbsp; and
+#$\cos^2(β) - \sin^2(β) = \cos(2β)= \cos(α)$
@@ Line 151: / Line 154: @@
 :$${\rm (III)}\hspace{0.54cm}\frac {\rho_{\xi \eta} }{(1 - \rho_{\xi \eta}^2) \cdot \sigma_\xi\cdot \sigma_\eta}= \frac {\rho_{\xi \eta} }{(1 - \rho_{\xi \eta}^2) \cdot \sigma_\xi^2 \cdot 0.691}=\frac {2}{3}\cdot \cos( \alpha) = 0.471.$$
-Dividing now the equation&nbsp; $\rm (III)$&nbsp; by the equation&nbsp; $\rm (I)$, we get:
+Dividing now the equation&nbsp; $\rm (III)$&nbsp; by the equation&nbsp; $\rm (I)$,&nbsp; we get:
 :$$ \frac {\rho_{\xi \eta} }{0.691}=\frac {0.471}{0.862}\hspace{0.5cm}\Rightarrow\hspace{0.5cm}{\rho_{\xi \eta} }= 0.378.$$

@@ Line 63: / Line 63: @@
-More information on this topic with signal examples is provided in the first part "Gaussian random variables without statistical bindings" of the learning video (German).
+More information on this topic with signal examples is provided in the first part&nbsp; "Gaussian random variables without statistical bindings"&nbsp; of the&nbsp; (German language)&nbsp; learning video
-::[[Gaußsche_2D-Zufallsgrößen_(Lernvideo)|Gaussian 2D random variables]].
+::[[Gaußsche_2D-Zufallsgrößen_(Lernvideo)|Gaußsche 2D-Zufallsgrößen]] &nbsp; &rArr; &nbsp; "Two-dimensional Gaussian random variables".
 <br clear=all>
-[[File:P_ID2911__Sto_T_4_2_S2_unten.png |right|frame| Screen capture of tutorial video "Gaussian 2D random variables"]]
+{{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"]]
 $\text{Example 2:}$&nbsp;
-*More information on this topic with signal examples is provided in the first part "Gaussian random variables without statistical bindings" of the learning video (German).
+*The graphic shows a snapshot of the first part video&nbsp; "Gaussian random variables without statistical bindings".
-*The graphic shows a snapshot of this learning video.
+*The second video part covers&nbsp; "Gaussian random variables with statistical bindings"&nbsp; according to the following section.
 }}
 ==Contour lines for correlated random variables==
 <br>
-For correlated components&nbsp; $(ρ_{xy} ≠ 0)$&nbsp; the PDF contour lines are always elliptic, thus also for the special case&nbsp; $σ_x = σ_y$.&nbsp;
+For correlated components&nbsp; $(ρ_{xy} ≠ 0)$&nbsp; the PDF contour lines are always elliptic,&nbsp; thus also for the special case&nbsp; $σ_x = σ_y$:&nbsp;
 :$$f_{xy}(x, y) = {\rm const.} \hspace{0.5cm} \rightarrow \hspace{0.5cm} \frac{x^{\rm 2} }{\sigma_{x}^{\rm 2}}+\frac{y^{\rm 2} }{\sigma_{y}^{\rm 2} }-{\rm 2}\cdot\rho_{xy}\cdot\frac{x\cdot y}{\sigma_x\cdot \sigma_y}={\rm const.}$$
-The following graph shows in lighter blue two contour lines for different parameter sets, each with&nbsp; $ρ_{xy} ≠ 0$.
+The following graph shows in lighter blue two contour lines for different parameter sets,&nbsp; each with&nbsp; $ρ_{xy} ≠ 0$.
-[[File:P_ID408__Sto_T_4_2_S3_neu.png|right|frame|height lines of 2D PDF at correlated sizes]]
+[[File:P_ID408__Sto_T_4_2_S3_neu.png|right|frame|Contour lines of the 2D&ndash;PDF at correlated quantities]]
 *The ellipse major axis is dashed in dark blue.
-*The correlation line&nbsp; $K(x)$&nbsp; is drawn in red throughout.
+*The correlation line or&nbsp; "regession line" &nbsp; $K(x)$&nbsp; is drawn in solid red.
-Based on this plot, the following statements can be made:
+Based on this plot,&nbsp; the following statements can be made:
-*The ellipse shape depends not only on the correlation coefficient&nbsp; $ρ_{xy}$&nbsp; but also on the ratio of the two standard deviations&nbsp; $σ_x$&nbsp; and&nbsp; $σ_y$&nbsp; .
+*The ellipse shape depends not only on the correlation coefficient&nbsp; $ρ_{xy}$&nbsp; but also on the ratio of the two standard deviations&nbsp; $σ_x$&nbsp; and&nbsp; $σ_y$.
-*The angle of inclination&nbsp; $α$&nbsp; of the ellipse major axis&nbsp; (dashed straight line)&nbsp; with respect to&nbsp; $x$&ndash;axis also depends on&nbsp; $σ_x$,&nbsp; $σ_y$&nbsp; and&nbsp; $ρ_{xy}$&nbsp; :
+*The angle of inclination&nbsp; $α$&nbsp; of the ellipse major axis&nbsp; (blue dashed straight line)&nbsp; with respect to the&nbsp; $x$&ndash;axis also depends on&nbsp; $σ_x$,&nbsp; $σ_y$&nbsp; and&nbsp; $ρ_{xy}$:
 :$$\alpha = {1}/{2} \cdot {\rm arctan } \ ( 2 \cdot \rho_{xy} \cdot \frac {\sigma_x \cdot \sigma_y}{\sigma_x^2 - \sigma_y^2}).$$
-*The (red) correlation line&nbsp; $y = K(x)$&nbsp; of a Gaussian 2D random variable always lies below the (blue dashed) ellipse major axis.
+*The&nbsp; (red solid)&nbsp; correlation line&nbsp; $y = K(x)$&nbsp; of a Gaussian random variable always lies below the&nbsp; (blue dashed)&nbsp; ellipse major axis.
-* $K(x)$&nbsp; can also be constructed geometrically from the intersection of the contour lines and their vertical tangents, as indicated in green in the sketches above.
+* $K(x)$&nbsp; can also be constructed geometrically from the intersection of the contour lines and their vertical tangents,&nbsp; as indicated in green in the sketches above.
 *Part 1: &nbsp; Gaussian random variables without statistical bindings,
 *Part 2: &nbsp; Gaussian random variables with statistical bindings.

@@ Line 27: / Line 27: @@
 ::If two random variables&nbsp; $x$&nbsp; and&nbsp; $y$&nbsp; are completely&nbsp; (statistically)&nbsp; independent of each other,&nbsp; <br>then of course there are no&nbsp; "linear dependencies"&nbsp; between them &nbsp; ⇒ &nbsp; they are also uncorrelated. }}
-The interactive applet&nbsp; [[Applets:Two-dimensional_Gaussian_Random_Variables|PDF and CDF for 2D Gaussian random variables]]&nbsp; allows plotting the two-dimensional functions for arbitrary values of&nbsp; $σ_x, \ σ_y$&nbsp; and&nbsp; $ρ_{xy}$.
 {{GraueBox|TEXT=
@@ Line 43: / Line 44: @@
 These representations can be interpreted as follows:
 *The PDF here is comparable to a mountain ridge extending from the lower left to the upper right.
-*The maximum is at&nbsp; $m_x = 0$&nbsp; and&nbsp; $m_y = 0$.&nbsp; This means that the the two-Dimensional random variable is mean-free.
+*The maximum is at&nbsp; $m_x = 0$&nbsp; and&nbsp; $m_y = 0$.&nbsp; This means that the the two-dimensional random variable is mean-free.
 *The 2D&ndash;CDF as the integral in two directions over the 2D&ndash;PDF increases continuously from lower left to upper right from&nbsp; $0$&nbsp; to&nbsp; $1$. }}
@@ Line 51: / Line 52: @@
 ==Contour lines for uncorrelated random variables==
 <br>
-[[File:P_ID318__Sto_T_4_2_S2_ganz_neu.png |frame| Contour lines of 2D PDF with uncorrelated variables | right]]
+[[File:P_ID318__Sto_T_4_2_S2_ganz_neu.png |frame| Contour lines of 2D&ndash;PDF with uncorrelated variables | right]]
 From the conditional equation&nbsp; $f_{xy}(x, y) = \rm const.$&nbsp; the contour lines of the PDF can be calculated.
-If the components&nbsp; $x$&nbsp; and&nbsp; $y$&nbsp; are uncorrelated&nbsp; $(ρ_{xy} = 0)$, the equation obtained for the contour lines is:
+If the components&nbsp; $x$&nbsp; and&nbsp; $y$&nbsp; are uncorrelated&nbsp; $(ρ_{xy} = 0)$,&nbsp; the equation obtained for the contour lines is:
 :$$\frac{x^{\rm 2}}{\sigma_{x}^{\rm 2}}+\frac{y^{\rm 2}}{\sigma_{y}^{\rm 2}} =\rm const.$$
-In this case, the contour lines describe the following figures:
+In this case,&nbsp; the contour lines describe the following figures:
-*'''Circles'''&nbsp; $($falls&nbsp; $σ_x = σ_y$, &nbsp; green curve$)$, or
+*"Circles"&nbsp; $($for&nbsp; $σ_x = σ_y$, &nbsp; green curve$)$, or
-*'''Ellipses'''&nbsp; $($for&nbsp; $σ_x ≠ σ_y$, &nbsp; blue curve$)$ in alignment of the two axes.
+*"Ellipses"&nbsp; $($for&nbsp; $σ_x ≠ σ_y$, &nbsp; blue curve$)$ in alignment of the two axes.
 <br clear=all>
 [[File:P_ID2911__Sto_T_4_2_S2_unten.png |right|frame| Screen capture of tutorial video "Gaussian 2D random variables"]]