View source for Applets:Two-dimensional Gaussian Random Variables

{{LntAppletLinkEnDe|gauss_en|gauss}}

==Applet Description==
<br>
The applet illustrates the properties of two-dimensional Gaussian random variables&nbsp; $XY\hspace{-0.1cm}$, characterized by the standard deviations (rms)&nbsp; $\sigma_X$&nbsp; and&nbsp; $\sigma_Y$&nbsp; of their two components, and the correlation coefficient&nbsp; $\rho_{XY}$&nbsp;between them. The components are assumed to be zero mean:&nbsp; $m_X = m_Y = 0$.

The applet shows
* the two-dimensional probability density function &nbsp; &rArr; &nbsp; $\rm 2D\hspace{-0.1cm}-\hspace{-0.1cm}PDF$&nbsp; $f_{XY}(x, \hspace{0.1cm}y)$&nbsp; in three-dimensional representation as well as in the form of contour lines,
* the corresponding marginal probability density function&nbsp; &rArr; &nbsp; $\rm 1D\hspace{-0.1cm}-\hspace{-0.1cm}PDF$&nbsp; $f_{X}(x)$&nbsp; of the random variable&nbsp; $X$&nbsp; as a blue curve; likewise&nbsp; $f_{Y}(y)$&nbsp; for the second random variable, 
* the two-dimensional distribution function&nbsp; &rArr; &nbsp; $\rm 2D\hspace{-0.1cm}-\hspace{-0.1cm}CDF$&nbsp; $F_{XY}(x, \hspace{0.1cm}y)$&nbsp; as a 3D plot,
* the distribution function&nbsp; &rArr; &nbsp; $\rm 1D\hspace{-0.1cm}-\hspace{-0.1cm}CDF$&nbsp; $F_{X}(x)$&nbsp; of the random variable&nbsp; $X$; also&nbsp; $F_{Y}(y)$&nbsp; as a red curve.


The applet uses the framework &nbsp;[https://en.wikipedia.org/wiki/Plotly "Plot.ly"]

==Theoretical Background==
<br>  

===Joint probability density function &nbsp; &rArr; &nbsp; 2D&ndash;PDF===

We consider two continuous value random variables&nbsp; $X$&nbsp; and&nbsp; $Y\hspace{-0.1cm}$, between which statistical dependencies may exist. To describe the interrelationships between these variables, it is convenient to combine the two components into a&nbsp; '''two-dimensional random variable'''&nbsp; $XY =(X, Y)$&nbsp; . Then holds: 

{{BlaueBox|TEXT=  
$\text{Definition:}$&nbsp; 
The &nbsp;'''joint probability density function'''&nbsp; is the probability density function (PDF) of the two-dimensional random variable&nbsp; $XY$&nbsp; at location&nbsp; $(x, y)$: 
:$$f_{XY}(x, \hspace{0.1cm}y) = \lim_{\left.{\delta x\rightarrow 0 \atop {\delta y\rightarrow 0} }\right. }\frac{ {\rm Pr}\big [ (x - {\rm \Delta} x/{\rm 2} \le X \le x + {\rm \Delta} x/{\rm 2}) \cap (y - {\rm \Delta} y/{\rm 2} \le Y \le y +{\rm \Delta}y/{\rm 2}) \big]  }{ {\rm \Delta} \ x\cdot{\rm \Delta} y}.$$

*The joint probability density function, or in short&nbsp; $\rm 2D\hspace{-0.1cm}-\hspace{-0.1cm}PDF$&nbsp; is an extension of the one-dimensional PDF.
*$∩$&nbsp; denotes the logical AND operation.
*$X$&nbsp; and&nbsp; $Y$ denote the two random variables, and&nbsp; $x \in X$&nbsp; and &nbsp; $y \in Y$ indicate realizations thereof.
*The nomenclature used for this applet thus differs slightly from the description in the [[Theory_of_Stochastic_Signals/Two-Dimensional_Random_Variables#Joint_probability_density_function|"Theory section"]].}}


Using this 2D–PDF&nbsp; $f_{XY}(x, y)$&nbsp; statistical dependencies within the two-dimensional random variable &nbsp;$XY$&nbsp; are also fully captured in contrast to the two one-dimensional density functions &nbsp; ⇒ &nbsp; '''marginal probability density functions''': 
:$$f_{X}(x) = \int _{-\infty}^{+\infty} f_{XY}(x,y) \,\,{\rm d}y ,$$
:$$f_{Y}(y) = \int_{-\infty}^{+\infty} f_{XY}(x,y) \,\,{\rm d}x .$$

These two marginal density functions&nbsp; $f_X(x)$&nbsp; and&nbsp; $f_Y(y)$ 
*provide only statistical information about the individual components&nbsp; $X$&nbsp; and&nbsp; $Y$, respectively, 
*but not about the bindings between them.


As a quantitative measure of the linear statistical bindings&nbsp; &rArr; &nbsp; '''correlation'''&nbsp; one uses.
* the&nbsp; '''covariance'''&nbsp; $\mu_{XY}$, which is equal to the first-order common linear moment for mean-free components:
:$$\mu_{XY} = {\rm E}\big[X \cdot Y\big] = \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} X \cdot Y \cdot f_{XY}(x,y) \,{\rm d}x \, {\rm d}y ,$$  
*the&nbsp; '''correlation coefficient'''&nbsp; after normalization to the two rms values &nbsp;$σ_X$&nbsp; and&nbsp;$σ_Y$&nbsp; of the two components: 
:$$\rho_{XY}=\frac{\mu_{XY} }{\sigma_X \cdot \sigma_Y}.$$

{{BlaueBox|TEXT=  
$\text{Properties of correlation coefficient:}$&nbsp; 
*Because of normalization, $-1 \le ρ_{XY} ≤ +1$ always holds&nbsp;. 
*If the two random variables &nbsp;$X$&nbsp; and &nbsp;$Y$ are uncorrelated, then &nbsp;$ρ_{XY} = 0$. 
*For strict linear dependence between &nbsp;$X$&nbsp; and &nbsp;$Y$, &nbsp;$ρ_{XY}= ±1$ &nbsp; &rArr; &nbsp; complete correlation.
*A positive correlation coefficient means that when &nbsp;$X$ is larger, on statistical average, &nbsp;$Y$&nbsp; is also larger than when &nbsp;$X$ is smaller. 
*In contrast, a negative correlation coefficient expresses that &nbsp;$Y$&nbsp; becomes smaller on average as &nbsp;$X$&nbsp; increases}}.  
<br><br>

===2D&ndash;PDF for Gaussian random variables=== 

For the special case&nbsp; '''Gaussian random variables'''&nbsp; - the name goes back to the scientist&nbsp; [https://en.wikipedia.org/wiki/Carl_Friedrich_Gauss "Carl Friedrich Gauss"]&nbsp; - we can further note: 
*The joint PDF of a Gaussian 2D random variable&nbsp; $XY$&nbsp; with means&nbsp; $m_X = 0$&nbsp; and&nbsp; $m_Y = 0$&nbsp; and the correlation coefficient&nbsp; $ρ = ρ_{XY}$&nbsp; is: 
: $$f_{XY}(x, y)=\frac{\rm 1}{\rm 2\it\pi \cdot \sigma_X \cdot \sigma_Y \cdot \sqrt{\rm 1-\rho^2}}\ \cdot\ \exp\Bigg[-\frac{\rm 1}{\rm 2 \cdot (1- \it\rho^{\rm 2} {\rm)}}\cdot(\frac {\it x^{\rm 2}}{\sigma_X^{\rm 2}}+\frac {\it y^{\rm 2}}{\sigma_Y^{\rm 2}}-\rm 2\it\rho\cdot\frac{x \cdot y}{\sigma_x \cdot \sigma_Y}\rm ) \rm \Bigg]\hspace{0.8cm}{\rm with}\hspace{0.5cm}-1 \le \rho \le +1.$$
*Replacing&nbsp; $x$&nbsp; by&nbsp; $(x - m_X)$&nbsp; and&nbsp; $y$&nbsp; by&nbsp; $(y- m_Y)$, we obtain the more general PDF of a two-dimensional Gaussian random variable with mean. 
*The marginal probability density functions&nbsp; $f_{X}(x)$&nbsp; and&nbsp; $f_{Y}(y)$&nbsp; of a 2D Gaussian random variable are also Gaussian with the standard deviations&nbsp; $σ_X$&nbsp; and&nbsp; $σ_Y$, respectively.
*For uncorrelated components&nbsp; $X$&nbsp; and&nbsp; $Y$, in the above equation&nbsp; $ρ = 0$&nbsp; must be substituted, and then the result is obtained: 
:$$f_{XY}(x,y)=\frac{1}{\sqrt{2\pi}\cdot\sigma_{X}} \cdot\rm e^{-\it {x^{\rm 2}}\hspace{-0.08cm}/{\rm (}{\rm 2\hspace{0.05cm}\it\sigma_{X}^{\rm 2}} {\rm )}} \cdot\frac{1}{\sqrt{2\pi}\cdot\sigma_{\it Y}}\cdot e^{-\it {y^{\rm 2}}\hspace{-0.08cm}/{\rm (}{\rm 2\hspace{0.05cm}\it\sigma_{Y}^{\rm 2}} {\rm )}} = \it f_{X} \rm ( \it x \rm ) \cdot \it f_{Y} \rm ( \it y \rm ) .$$

{{BlaueBox|TEXT=  
$\text{Conclusion:}$&nbsp; In the special case of a 2D random variable with Gaussian PDF&nbsp; $f_{XY}(x, y)$&nbsp; it also follows directly from &nbsp;''uncorrelatedness''&nbsp; the&nbsp; ''statistical independence:''
:$$f_{XY}(x,y)= f_{X}(x) \cdot f_{Y}(y) . $$

Please note:
*For no other PDF can the&nbsp; ''uncorrelatedness''&nbsp; be used to infer&nbsp; ''statistical independence''&nbsp; . 
*But one can always &nbsp; ⇒ &nbsp; infer&nbsp; ''uncorrelatedness'' from&nbsp; ''statistical independence''&nbsp; for any 2D-PDF&nbsp; $f_{XY}(x, y)$&nbsp; because: 
*If two random variables&nbsp; $X$&nbsp; and&nbsp; $Y$&nbsp; are completely (statistically) independent of each other, then of course there are no ''linear''&nbsp; dependencies between them &nbsp; <br>⇒ &nbsp; they are then also uncorrelated&nbsp; &rArr; &nbsp; $ρ = 0$. }}
<br><br>
===Contour lines for uncorrelated random variables===

[[File:Sto_App_Bild2.png |frame| Contour lines of 2D-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; $Y$ are uncorrelated&nbsp; $(ρ_{XY} = 0)$, 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: 
*'''Circles'''&nbsp; (if&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. 
<br clear=all>
===Regression line===

As &nbsp;'''regression line'''&nbsp; is called the straight line &nbsp;$y = K(x)$&nbsp; in the &nbsp;$(x, y)$&ndash;plane through the "center" $(m_X, m_Y)$. This has the following properties:  
[[File:Sto_App_Bild1a.png|frame| Gaussian 2D PDF (approximation with $N$ measurement points) and <br>correlation line &nbsp;$y = K(x)$]]

*The mean square error from this straight line - viewed in &nbsp;$y$&ndash;direction and averaged over all &nbsp;$N$&nbsp; measurement points - is minimal: 
:$$\overline{\varepsilon_y^{\rm 2} }=\frac{\rm 1}{N} \cdot \sum_{\nu=\rm 1}^{N}\; \;\big [y_\nu - K(x_{\nu})\big ]^{\rm 2}={\rm minimum}.$$
*The correlation straight line can be interpreted as a kind of "statistical symmetry axis". The equation of the straight line in the general case is: 
:$$y=K(x)=\frac{\sigma_Y}{\sigma_X}\cdot\rho_{XY}\cdot(x - m_X)+m_Y.$$

*The angle that the correlation line makes to the &nbsp;$x$&ndash;axis is: 
:$$\theta={\rm arctan}(\frac{\sigma_{Y} }{\sigma_{X} }\cdot \rho_{XY}).$$



===Contour lines for correlated random variables===

For correlated components&nbsp; $(ρ_{XY} ≠ 0)$&nbsp; the contour lines of the PDF are (almost) always elliptic, so also for the special case&nbsp; $σ_X = σ_Y$. 

<u>Exception:</u>&nbsp; $ρ_{XY}=\pm 1$ &nbsp; &rArr; &nbsp; "Dirac-wall"; see&nbsp; [[Aufgaben:Exercise_4.4:_Two-dimensional_Gaussian_PDF|"Exercise 4.4"]]&nbsp; in the book "Stochastic Signal Theory", subtask &nbsp;''(5)''.
[[File:Sto_App_Bild3.png|right|frame|height lines of the two dimensional PDF with correlated quantities]]
Here, the determining equation of the PDF height lines is: 

:$$f_{XY}(x, y) = {\rm const.} \hspace{0.5cm} \rightarrow \hspace{0.5cm} \frac{x^{\rm 2} }{\sigma_{X}^{\rm 2}}+\frac{y^{\rm 2} }{\sigma_{Y}^{\rm 2} }-{\rm 2}\cdot\rho_{XY}\cdot\frac{x\cdot y}{\sigma_X\cdot \sigma_Y}={\rm const.}$$
The graph shows a contour line in lighter blue for each of two different sets of parameters. 

*The ellipse major axis is dashed in dark blue. 
*The&nbsp; [[Theory_of_Stochastic_Signals/Two-Dimensional_Random_Variables#Regression_line|"regression line"]]&nbsp; $K(x)$&nbsp; is drawn in red throughout. 


Based on this plot, the following statements are possible: 
*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 angle of inclination&nbsp; $α$&nbsp; of the ellipse major axis (dashed straight line) with respect to the&nbsp; $x$&ndash;axis also depends on&nbsp; $σ_X$,&nbsp; $σ_Y$&nbsp; and&nbsp; $ρ_{XY}$&nbsp; : 
:$$\alpha = {1}/{2} \cdot {\rm arctan } \big ( 2 \cdot \rho_{XY} \cdot \frac {\sigma_X \cdot \sigma_Y}{\sigma_X^2 - \sigma_Y^2} \big ).$$
*The (red) correlation line&nbsp; $y = K(x)$&nbsp; of a Gaussian 2D-random variable always lies below the (blue dashed) ellipse major axis. 
* $K(x)$&nbsp; can be geometrically constructed from the intersection of the contour lines and their vertical tangents, as indicated in the sketch in green color.  
<br><br>
===Two dimensional cumulative distribution function &nbsp; &rArr; &nbsp; 2D&ndash;CDF===

{{BlaueBox|TEXT=  
$\text{Definition:}$&nbsp; The&nbsp; '''2D cumulative distribution function'''&nbsp; like the 2D-CDF, is merely a useful extension of the&nbsp; [[Theory_of_Stochastic_Signals/Cumulative_Distribution_Function#CDF_for_continuous-valued_random_variables|"one-dimensional distribution function"]]&nbsp; (PDF): 
:$$F_{XY}(x,y) = {\rm Pr}\big [(X \le x) \cap (Y \le y) \big ] .$$}}


The following similarities and differences between the "1D&ndash;CDF" and the" 2D&ndash;CDF" emerge:
*The functional relationship between "2D&ndash;PDF" and "2D&ndash;CDF" is given by the integration as in the one-dimensional case, but now in two dimensions. For continuous random variables, the following holds: 
:$$F_{XY}(x,y)=\int_{-\infty}^{y} \int_{-\infty}^{x} f_{XY}(\xi,\eta) \,\,{\rm d}\xi \,\, {\rm d}\eta .$$
*Inversely, the probability density function can be given from the cumulative distribution function by partial differentiation to&nbsp; $x$&nbsp; and&nbsp; $y$&nbsp; : 
:$$f_{XY}(x,y)=\frac{{\rm d}^{\rm 2} F_{XY}(\xi,\eta)}{{\rm d} \xi \,\, {\rm d} \eta}\Bigg|_{\left.{x=\xi \atop {y=\eta}}\right.}.$$
*In terms of the cumulative distribution function&nbsp; $F_{XY}(x, y)$&nbsp; the following limits apply:
:$$F_{XY}(-\infty,\ -\infty) = 0,\hspace{0.5cm}F_{XY}(x,\ +\infty)=F_{X}(x ),\hspace{0.5cm}
F_{XY}(+\infty,\ y)=F_{Y}(y ) ,\hspace{0.5cm}F_{XY}(+\infty,\ +\infty) = 1.$$ 
*In the limiting case $($infinitely large&nbsp; $x$&nbsp; and&nbsp; $y)$&nbsp; thus the value&nbsp; $1$ is obtained for the "2D&ndash;CDF". From this we obtain the&nbsp; '''normalization condition'''&nbsp; for the two-dimensional probability density function: 
:$$\int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} f_{XY}(x,y) \,\,{\rm d}x \,\,{\rm d}y=1 . $$

{{BlaueBox|TEXT=
$\text{Conclusion:}$&nbsp; Note the significant difference between one-dimensional and two-dimensional random variables: 
*For one-dimensional random variables, the area under the PDF always yields $1$. 
*For two-dimensional random variables, the PDF volume always equals $1$.}}
<br><br>

==Exercises==
<br>
*Select the number&nbsp; $(1,\ 2$, ... $)$&nbsp; of the task to be processed.&nbsp; The number "0" corresponds to a "Reset":&nbsp; Setting as at the program start.
*A task description is displayed.&nbsp; Parameter values are adjusted.&nbsp; Solution after pressing "Sample solution".&nbsp; 
*In the task description, we use &nbsp;$\rho$&nbsp; instead of &nbsp;$\rho_{XY}$.
*For the one-dimensional Gaussian PDF holds:&nbsp; $f_{X}(x) = \sqrt{1/(2\pi \cdot \sigma_X^2)} \cdot {\rm e}^{-x^2/(2 \hspace{0.05cm}\cdot \hspace{0.05cm} \sigma_X^2)}$.


{{BlueBox|TEXT=
'''(1)'''&nbsp; Get familiar with the program using the default &nbsp;$(\sigma_X=1, \ \sigma_Y=0.5, \ \rho = 0.7)$.&nbsp; Interpret the graphs for &nbsp;$\rm PDF$&nbsp; and&nbsp; $\rm CDF$.}}

*&nbsp;$\rm PDF$&nbsp; is a ridge with the maximum at&nbsp; $x = 0, \ y = 0$.&nbsp; The ridge is slightly twisted with respect to the &nbsp;$x$&ndash;axis.
*&nbsp;$\rm CDF$&nbsp; is obtained from &nbsp;$\rm PDF$&nbsp; by continuous integration in both directions.&nbsp; The maximum $($near &nbsp;$1)$&nbsp; occurs at &nbsp;$x=3, \ y=3$.


{{BlueBox|TEXT=
'''(2)'''&nbsp; The new setting is &nbsp;$\sigma_X= \sigma_Y=1, \ \rho = 0$.&nbsp; What are the values for &nbsp;$f_{XY}(0,\ 0)$&nbsp; and &nbsp;$F_{XY}(0,\ 0)$?&nbsp; Interpret the results}}

*&nbsp;The PDF maximum is&nbsp; $f_{XY}(0,\ 0) = 1/(2\pi)= 0.1592$, because of &nbsp;$\sigma_X= \sigma_Y = 1, \ \rho = 0$.&nbsp; The contour lines are circles.
*&nbsp;For the CDF value:&nbsp; $F_{XY}(0,\ 0) = [{\rm Pr}(X \le 0)] \cdot [{\rm Pr}(Y \le 0)] = 0.25$.&nbsp; Minor deviation due to numerical integration.


{{BlueBox|TEXT=
'''(3)'''&nbsp; The settings of&nbsp; $(2)$&nbsp; continue to apply.&nbsp; What are the values for &nbsp;$f_{XY}(0,\ 1)$&nbsp; and &nbsp;$F_{XY}(0,\ 1)$?&nbsp; Interpret the results.}}

*&nbsp;It holds&nbsp; $f_{XY}(0,\ 1) = f_{X}(0) \cdot f_{Y}(1) = [ \sqrt{1/(2\pi)}] \cdot [\sqrt{1/(2\pi)} \cdot {\rm e}^{-0.5}] = 1/(2\pi) \cdot {\rm e}^{-0.5} = 0.0965$.
*&nbsp;The program returns&nbsp; $F_{XY}(0,\ 1) = [{\rm Pr}(X \le 0)] \cdot [{\rm Pr}(Y \le 1)] = 0.4187$, i.e. a larger value than in&nbsp; $(2)$,&nbsp; since it integrates over a wider range.


{{BlueBox|TEXT=
'''(4)'''&nbsp; The settings are kept.&nbsp; What values are obtained for &nbsp;$f_{XY}(1,\ 0)$&nbsp; and &nbsp;$F_{XY}(1,\ 0)$?&nbsp; Interpret the results}}

*&nbsp;Due to rotational symmetry, same results as in&nbsp; $(3)$.


{{BlueBox|TEXT=
'''(5)'''&nbsp; Is the statement true:&nbsp;"Elliptic contour lines exist only for &nbsp;$\rho \ne 0$".&nbsp; Interpret the&nbsp; $\rm 2D\hspace{-0.1cm}-\hspace{-0.1cm}PDF$&nbsp; and&nbsp; $\rm 2D\hspace{-0.1cm}-\hspace{-0.1cm}CDF$&nbsp; for &nbsp;$\sigma_X=1, \ \sigma_Y=0.5$&nbsp; and&nbsp; $\rho = 0$.}}

*&nbsp;No!&nbsp; Also, for&nbsp; $\ \rho = 0$&nbsp; the contour lines are elliptical&nbsp; (not circular)&nbsp; if &nbsp;$\sigma_X \ne \sigma_Y$.
*&nbsp;For&nbsp;$\sigma_X \gg \sigma_Y$&nbsp; the&nbsp; $\rm 2D\hspace{-0.1cm}-\hspace{-0.1cm}PDF$&nbsp; has the shape of an elongated ridge parallel to&nbsp; $x$&ndash;axis, for&nbsp;$\sigma_X \ll \sigma_Y$&nbsp; parallel to&nbsp; $y$&ndash;axis.
*&nbsp;For&nbsp;$\sigma_X \gg \sigma_Y$&nbsp; the slope of&nbsp; $\rm 2D\hspace{-0.1cm}-\hspace{-0.1cm}CDF$&nbsp; in the direction of the &nbsp;$y$&ndash;axis is much steeper than in the direction of the &nbsp;$x$&ndash;axis.


{{BlueBox|TEXT=
'''(6)'''&nbsp; Starting from&nbsp; $\sigma_X=\sigma_Y=1\ \rho = 0.7$&nbsp; vary the correlation coefficient&nbsp; $\rho$.&nbsp; What is the slope angle &nbsp;$\alpha$&nbsp; of the ellipse main axis?}}

*&nbsp;For&nbsp; $\rho > 0$:&nbsp; &nbsp;$\alpha = 45^\circ$. &nbsp; &nbsp; For&nbsp; $\rho < 0$:&nbsp; &nbsp;$\alpha = -45^\circ$.&nbsp; For&nbsp; $\rho = 0$:&nbsp; The contour lines are circular and thus there are no ellipses main axis.


{{BlueBox|TEXT=
'''(7)'''&nbsp; Starting from&nbsp; $\sigma_X=\sigma_Y=1\ \rho = 0.7$&nbsp; vary the correlation coefficient&nbsp; $\rho$.&nbsp; What is the slope angle &nbsp;$\theta$&nbsp; of the correlation line&nbsp; $K(x)$?}}

*&nbsp;For&nbsp; $\sigma_X=\sigma_Y$:&nbsp;  &nbsp;$\theta={\rm arctan}\ (\rho)$.&nbsp; The slope increases with increasing&nbsp; $\rho > 0$.&nbsp; In all cases, &nbsp;$\theta < \alpha = 45^\circ$ holds. For&nbsp; $\rho = 0.7$&nbsp; this gives &nbsp;$\theta = 35^\circ$.


{{BlueBox|TEXT=
'''(8)'''&nbsp; Starting from&nbsp; $\sigma_X=\sigma_Y=0.75, \ \rho = 0.7$&nbsp; vary the parameters&nbsp; $\sigma_Y$&nbsp; and&nbsp; $\rho $.&nbsp; What statements hold for the angles &nbsp;$\alpha$&nbsp; and&nbsp; $\theta$?}}

*&nbsp;For&nbsp; $\sigma_Y<\sigma_X$: &nbsp; $\alpha < 45^\circ$. &nbsp; &nbsp; For&nbsp; $\sigma_Y>\sigma_X$: &nbsp;  $\alpha > 45^\circ$. &nbsp;For all settings:&nbsp; '''The correlation line is below the ellipse main axis'''.


{{BlueBox|TEXT=
'''(9)'''&nbsp; Assume&nbsp; $\sigma_X= 1, \ \sigma_Y=0.75, \ \rho = 0.7$.&nbsp; Vary&nbsp; $\rho$.&nbsp; How to construct the correlation line from the contour lines?}}

*&nbsp;The correlation line intersects all contour lines at that points where the tangent line is perpendicular to the contour line.


{{BlueBox|TEXT=
'''(10)'''&nbsp; Now let be&nbsp; $\sigma_X= \sigma_Y=1, \ \rho = 0.95$.&nbsp; Interpret the&nbsp; $\rm 2D\hspace{-0.1cm}-\hspace{-0.1cm}PDF$.&nbsp; Which statements are true for the limiting case&nbsp; $\rho \to 1$&nbsp;?}}

*&nbsp;The&nbsp; $\rm 2D\hspace{-0.1cm}-\hspace{-0.1cm}WDF$&nbsp; only has components near the ellipse main axis.&nbsp; The correlation line is just below:&nbsp; $\alpha = 45^\circ, \ \theta = 43.5^\circ$.
*&nbsp;In the limiting case&nbsp; $\rho \to 1$&nbsp; it holds&nbsp; $\theta = \alpha = 45^\circ$.&nbsp; Outside the correlation line, the&nbsp; $\rm 2D\hspace{-0.1cm}-\hspace{-0.1cm}PDF$&nbsp; would have no shares.&nbsp; That is:
*&nbsp;Along the correlation line, there would be a&nbsp; "Dirac wall" &nbsp; &rArr; &nbsp; All values are infinitely large, nevertheless Gaussian weighted around the mean.

 






==Applet Manual==
<br>
[[File:Anleitung_2D-Gauss.png|left|500px|frame|Screen shot from the German version]]
<br><br>
&nbsp; &nbsp; '''(A)''' &nbsp; &nbsp; Parameter input via slider:&nbsp; $\sigma_X$, &nbsp;$\sigma_Y$ and&nbsp; $\rho$. 

&nbsp; &nbsp; '''(B)''' &nbsp; &nbsp; Selection:&nbsp; Representation of PDF or CDF.

&nbsp; &nbsp; '''(C)''' &nbsp; &nbsp; Reset:&nbsp; Setting as at program start.

&nbsp; &nbsp; '''(D)''' &nbsp; &nbsp; Display contour lines instead of one-dimensional PDF.

&nbsp; &nbsp; '''(E)''' &nbsp; &nbsp; Display range for two-dimensional PDF.

&nbsp; &nbsp; '''(F)''' &nbsp; &nbsp; Manipulation of the three-dimensional graph (zoom, rotate, ...)

&nbsp; &nbsp; '''(G)''' &nbsp; &nbsp; Display range for&nbsp; "one-dimensional PDF"&nbsp; or&nbsp; "contour lines".

&nbsp; &nbsp; '''(H)''' &nbsp; &nbsp; Manipulation of the two-dimensional graphics ("one-dimensional PDF")

&nbsp; &nbsp; '''( I )''' &nbsp; &nbsp; Area for exercises: Task selection.  

&nbsp; &nbsp; '''(J)''' &nbsp; &nbsp; Area for exercises: Task description

&nbsp; &nbsp; '''(K)''' &nbsp; &nbsp; Area for exercises: Show/hide solution

&nbsp; &nbsp; '''( L)''' &nbsp; &nbsp; Area for exercises: Output of the sample solution

<u>Note:</u> &nbsp; &nbsp;Value output of the graphics&nbsp; $($both 2D and 3D$)$&nbsp; via mouse control. 
<br clear=all>



==About the Authors==
<br>
This interactive calculation tool was designed and implemented at the&nbsp; [https://www.ei.tum.de/en/lnt/home/ Institute for Communications Engineering]&nbsp; at the&nbsp; [https://www.tum.de/en Technical University of Munich]. 
*The first version was created in 2003 by&nbsp; [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Studierende#Ji_Li_.28Bachelorarbeit_EI_2003.2C_Diplomarbeit_EI_2005.29|Ji Li]] &nbsp; as part of his diploma thesis with “FlashMX – Actionscript” (Supervisor: [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|Günter Söder]]). 
*In 2019 the program was redesigned by&nbsp; [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Studierende#Carolin_Mirschina_.28Ingenieurspraxis_Math_2019.2C_danach_Werkstudentin.29|Carolin Mirschina]]&nbsp; as part of her bachelor thesis&nbsp; (Supervisor: [[Biographies_and_Bibliographies/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29|Tasnád Kernetzky]] ) via "HTML5".
*Last revision and English version 2021 by&nbsp; [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Carolin_Mirschina_.28Ingenieurspraxis_Math_2019.2C_danach_Werkstudentin.29|Carolin Mirschina]]&nbsp; in the context of a working student activity.&nbsp; 


The conversion of this applet to HTML 5 was financially supported by&nbsp; [https://www.ei.tum.de/studium/studienzuschuesse/ "Studienzuschüsse"]&nbsp; (Faculty EI of the TU Munich).&nbsp; We thank.



==Once again: Open Applet in new Tab==

{{LntAppletLinkEnDe|gauss_en|gauss}}