Aufgaben:Exercise 4.4: Two-dimensional Gaussian PDF - Revision history

Guenter at 16:37, 9 January 2024

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Guenter at 16:15, 9 January 2024

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Guenter at 16:02, 9 January 2024

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Guenter at 16:45, 23 February 2022

2022-02-23T16:45:23Z

Guenter at 16:23, 23 February 2022

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Guenter at 15:45, 23 February 2022

2022-02-23T15:45:52Z

Guenter: Guenter moved page Aufgaben:Exercise 4.4: Gaussian 2D PDF to Aufgaben:Exercise 4.4: Two-dimensional Gaussian PDF

2022-01-24T09:05:59Z

Guenter moved page Exercise 4.4: Gaussian 2D PDF to Exercise 4.4: Two-dimensional Gaussian PDF

Noah at 21:20, 23 January 2022

2022-01-23T21:20:49Z

Noah at 21:20, 23 January 2022

2022-01-23T21:20:37Z

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Noah at 17:21, 23 January 2022

2022-01-23T17:21:49Z

@@ Line 5: / Line 5: @@
 [[File:P_ID261__Sto_A_4_4_neu.png|right|frame|Table:&nbsp; Gaussian error functions]]
 We consider two-dimensional random variables,&nbsp; where both components are always assumed to be mean-free.
-*The two-dimensional probability density function&nbsp; $\text{(2D-PDF)}$&nbsp; of the random variable&nbsp; $(u, v)$&nbsp; is:
+*The&nbsp; &raquo;two-dimensional probability density function&laquo;&nbsp; of the random variable&nbsp; $(u, v)$&nbsp; is:
 :$$f_{uv}(u, v)={1}/{\pi} \cdot {\rm e}^{-(2u^{\rm 2} \hspace{0.05cm}+ \hspace{0.05cm}v^{\rm 2}\hspace{-0.05cm}/\rm 2)}.$$
-*The following parameters are known for another Gaussian two-dimensional random variable&nbsp; $(x, y)$:
+*For another Gaussian two-dimensional random variable&nbsp; $(x, y)$&nbsp; the following parameters are known:
 :$$\sigma_x= 0.5, \hspace{0.5cm}\sigma_y = 1,\hspace{0.5cm}\rho_{xy} = 1. $$
-*The values of the cumulative distribution function&nbsp; ${\rm \phi}(x)$&nbsp; and the complementary function&nbsp; ${\rm Q}(x) = 1- {\rm \phi}(x)$&nbsp; can be found in the adjacent table.
+*In the adjacent table can be found
@@ Line 17: / Line 19: @@
 Hints:
-*The exercise belongs to the chapter&nbsp; [[Theory_of_Stochastic_Signals/Two-Dimensional_Gaussian_Random_Variables|Two-dimensional Gaussian Random Variables]].
+*The exercise belongs to the chapter&nbsp; [[Theory_of_Stochastic_Signals/Two-Dimensional_Gaussian_Random_Variables|&raquo;Two-dimensional Gaussian Random Variables&laquo;]].
-*Reference is also made to the chapter&nbsp; [[Theory_of_Stochastic_Signals/Gaussian_Distributed_Random_Variables|Gaussian distributed random variables]].
-*More information on this topic is provided in the&nbsp; (German language)&nbsp;learning video&nbsp; [[Gaußsche_2D-Zufallsgrößen_(Lernvideo)|"Gaußsche 2D-Zufallsgrößen"]]:
+*Reference is also made to the chapter&nbsp; [[Theory_of_Stochastic_Signals/Gaussian_Distributed_Random_Variables|&raquo;Gaussian distributed random variables&laquo;]].
@@ Line 40: / Line 40: @@
-{Calculate the probability that&nbsp; $u$&nbsp; is less than&nbsp; $1$&nbsp;.
+{Calculate the probability that&nbsp; $u$&nbsp; is less than&nbsp; $1$.
 |type="{}"}
 ${\rm Pr}(u < 1)\ = \ $ { 0.9772 3% }
-{Calculate the probability that the random variable&nbsp; $u$&nbsp; is less than&nbsp; $1$&nbsp; and at the same time the random variable&nbsp; $v$&nbsp; is greater than&nbsp; $1$&nbsp; .
+{Calculate the probability that the random variable&nbsp; $u$&nbsp; is less than&nbsp; $1$&nbsp; and at the same time the random variable&nbsp; $v$&nbsp; is greater than&nbsp; $1$.
 |type="{}"}
 ${\rm Pr}\big[(u < 1) ∩ (υ > 1)\big]\ = \ $ { 0.1551 3% }
@@ Line 52: / Line 52: @@
 {Which of the statements are true for the two-dimensional random variable&nbsp; $(x, y)$?
 |type="[]"}
-+ The joint PDF $f_{xy}(x, y)$&nbsp; is always zero outside the straight line&nbsp; $y = 2x$.
++ The joint probability density function&nbsp; $f_{xy}(x, y)$&nbsp; is always zero outside the straight line&nbsp; $y = 2x$.
 - For all pairs of values on the straight line&nbsp; $y = 2x$&nbsp; holds: &nbsp;$f_{xy}(x, y)= 0.5$.
 + With respect to the edge PDFs:&nbsp; $f_{x}(x) = f_{u}(u)$&nbsp; and $f_{y}(y) = f_{v}(v)$ holds.

@@ Line 8: / Line 8: @@
 :$$f_{uv}(u, v)={1}/{\pi} \cdot {\rm e}^{-(2u^{\rm 2} \hspace{0.05cm}+ \hspace{0.05cm}v^{\rm 2}\hspace{-0.05cm}/\rm 2)}.$$
-*The following parameters are known from another two-dimensional random variable&nbsp; $(x, y)$&nbsp; which is also Gaussian:
+*The following parameters are known for another Gaussian two-dimensional random variable&nbsp; $(x, y)$:
 :$$\sigma_x= 0.5, \hspace{0.5cm}\sigma_y = 1,\hspace{0.5cm}\rho_{xy} = 1. $$
-*The values of the Gaussian error integral&nbsp; ${\rm \phi}(x)$&nbsp; and the complementary function&nbsp; ${\rm Q}(x) = 1- {\rm \phi}(x)$&nbsp; can be found in the adjacent table.
+*The values of the cumulative distribution function&nbsp; ${\rm \phi}(x)$&nbsp; and the complementary function&nbsp; ${\rm Q}(x) = 1- {\rm \phi}(x)$&nbsp; can be found in the adjacent table.

@@ Line 5: / Line 5: @@
 [[File:P_ID261__Sto_A_4_4_neu.png|right|frame|Table:&nbsp; Gaussian error functions]]
 We consider two-dimensional random variables,&nbsp; where both components are always assumed to be mean-free.
-*The 2D&ndash;PDF of the random variable&nbsp; $(u, v)$&nbsp; is:
+*The two-dimensional probability density function&nbsp; $\text{(2D-PDF)}$&nbsp; of the random variable&nbsp; $(u, v)$&nbsp; is:
 :$$f_{uv}(u, v)={1}/{\pi} \cdot {\rm e}^{-(2u^{\rm 2} \hspace{0.05cm}+ \hspace{0.05cm}v^{\rm 2}\hspace{-0.05cm}/\rm 2)}.$$

@@ Line 109: / Line 109: @@
 The sketch on the right illustrates the given constellation:
-*The PDF height lines&nbsp; (blue)&nbsp; are stretched ellipses due to&nbsp; $\sigma_v > \sigma_u$&nbsp; in vertical direction.
+*The PDF contour lines&nbsp; (blue)&nbsp; are stretched ellipses due to&nbsp; $\sigma_v > \sigma_u$&nbsp; in vertical direction.
 *Drawn in red&nbsp; (shading)&nbsp; is the area whose probability should be calculated in this subtask.

@@ Line 73: / Line 73: @@
 {{ML-Kopf}}
 '''(1)'''&nbsp; <u>Both statements are true</u>:
-*Comparing the given 2D PDF with the general 2D PDF.
+*Comparing the given 2D&ndash;PDF with the general 2D&ndash;PDF
-$$f_{uv}(u,v) = \frac{\rm 1}{{\rm 2}\it\pi \cdot \sigma_u \cdot \sigma_v \cdot \sqrt{{\rm 1}-\it \rho_{\it uv}^{\rm 2}}} \cdot \rm exp\left[\frac{\rm 1}{2\cdot (\rm 1-\it \rho_{uv}^{\rm 2}{\rm )}}(\frac{\it u^{\rm 2}}{\it\sigma_u^{\rm 2}} + \frac{\it v^{\rm 2}}{\it\sigma_v^{\rm 2}} - \rm 2\it\rho_{uv}\frac{\it u\cdot \it v}{\sigma_u\cdot \sigma_v}\rm )\right],$$
+:$$f_{uv}(u,v) = \frac{\rm 1}{{\rm 2}\it\pi \cdot \sigma_u \cdot \sigma_v \cdot \sqrt{{\rm 1}-\it \rho_{\it uv}^{\rm 2}}} \cdot \rm exp\left[\frac{\rm 1}{2\cdot (\rm 1-\it \rho_{uv}^{\rm 2}{\rm )}}(\frac{\it u^{\rm 2}}{\it\sigma_u^{\rm 2}} + \frac{\it v^{\rm 2}}{\it\sigma_v^{\rm 2}} - \rm 2\it\rho_{uv}\frac{\it u\cdot \it v}{\sigma_u\cdot \sigma_v}\rm )\right],$$
-:so it can be seen that no term with&nbsp; $u \cdot v$&nbsp; occurs in the exponent, which is only possible with&nbsp; $\rho_{uv} = 0$&nbsp; m&ouml;gible.
+:so it can be seen that no term with&nbsp; $u \cdot v$&nbsp; occurs in the exponent,&nbsp; which is only possible with&nbsp; $\rho_{uv} = 0$.
 *But this means that&nbsp; $u$&nbsp; and&nbsp; $v$&nbsp; are uncorrelated.
-*For Gaussian random variables, however, statistical independence always follows from uncorrelatedness.
+*For Gaussian random variables,&nbsp; however,&nbsp; statistical independence always follows from uncorrelatedness.
 '''(2)'''&nbsp; With statistical independence holds:
-:$$f_{uv}(u, v) = f_u(u)\cdot f_v(v), \hspace{0.5cm}
+:$$f_{uv}(u, v) = f_u(u)\cdot f_v(v) $$
-f_u(u)=\frac{{\rm e}^{-{\it u^{\rm 2}}/{(2\sigma_u^{\rm 2})}}}{\sqrt{\rm 2\pi}\cdot\sigma_u} , \hspace{0.5cm} \it f_v{\rm (}v{\rm )}=\frac{{\rm e}^{-{\it v^{\rm 2}}/{{\rm (}{\rm 2}\sigma_v^{\rm 2}{\rm )}}}}{\sqrt{\rm 2\pi}\cdot\sigma_v}.$$
+:$$f_u(u)=\frac{{\rm e}^{-{\it u^{\rm 2}}/{(2\sigma_u^{\rm 2})}}}{\sqrt{\rm 2\pi}\cdot\sigma_u} , $$
-*By comparing coefficients, we get&nbsp; $\sigma_u = 0.5$&nbsp; and&nbsp; $\sigma_v = 1$.
+*By comparing coefficients,&nbsp; we get&nbsp; $\sigma_u = 0.5$&nbsp; and&nbsp; $\sigma_v = 1$.
 *The quotient is thus&nbsp; $\sigma_u/\sigma_v\hspace{0.15cm}\underline{=0.5}$.
-[[File:P_ID265__Sto_A_4_4_d.png|right|frame|Probability: &nbsp; $\rm Pr\big[(\it u < \rm 1) \cap (\it v > \rm 1)\big]$]]
+'''(3)'''&nbsp; Because&nbsp; $u$&nbsp; is a continuous random variable:
 :$$\rm Pr(\it u < \rm 1) = \rm Pr(\it u \le \rm 1) =\it F_u\rm (1). $$
@@ Line 99: / Line 99: @@
+[[File:P_ID265__Sto_A_4_4_d.png|right|frame|$\rm Pr\big[(\it u < \rm 1) \cap (\it v > \rm 1)\big]$]]
 '''(4)'''&nbsp; Due to the statistical independence between&nbsp; $u$&nbsp; and&nbsp; $v$&nbsp; holds:
 :$$\rm Pr\big[(\it u < \rm 1) \cap (\it v > \rm 1)\big] = \rm Pr(\it u < \rm 1)\cdot \rm Pr(\it v > \rm 1).$$
@@ Line 108: / Line 108: @@
 :$$\Rightarrow \hspace{0.3cm} \rm Pr\big[(\it u < \rm 1) \cap (\it v > \rm 1)\big] = \rm 0.9772\cdot \rm 0.1587 \hspace{0.15cm}\underline{ = \rm 0.1551}.$$
-The sketch illustrates the given constellation:
+The sketch on the right illustrates the given constellation:
-*The PDF (blue) height lines are stretched ellipses due to&nbsp; $\sigma_v > \sigma_u$&nbsp; in vertical direction.
+*The PDF height lines&nbsp; (blue)&nbsp; are stretched ellipses due to&nbsp; $\sigma_v > \sigma_u$&nbsp; in vertical direction.
-*Drawn in red shading is the area whose probability should be calculated in this subtask.
+*Drawn in red&nbsp; (shading)&nbsp; is the area whose probability should be calculated in this subtask.
-[[File:P_ID266__Sto_A_4_4_e.png|right|frame|2D Dirac "wall" on the correlation line]]
+[[File:P_ID266__Sto_A_4_4_e.png|right|frame|"Dirac wall"&nbsp; on the correlation line]]
-'''(5)'''&nbsp; Correct are <u>the first and the third suggested solutions</u>:
+'''(5)'''&nbsp; Correct are&nbsp; <u>the first and the third suggested solutions</u>:
 *Because&nbsp; $\rho_{xy} = 1$&nbsp; there is a deterministic correlation between&nbsp; $x$&nbsp; and&nbsp; $y$
-:&#8658; &nbsp; All values lie on the straight line&nbsp; $y =K(x) \cdot x$.
+:&#8658; &nbsp; All values lie on the straight line&nbsp; $y =K \cdot x$.
 *Because of the standard deviations&nbsp; $\sigma_x = 0.5$&nbsp; and&nbsp; $\sigma_y = 1$&nbsp; it holds&nbsp; $K = 2$.
-*On this straight line&nbsp; $y = 2x$&nbsp; all PDF values are infinitely large.
+*On this straight line&nbsp; $y = 2x$ &nbsp; &rArr; &nbsp; all PDF values are infinitely large.
-*This means: &nbsp; The 2D PDF is here a "Dirac wall".
+*This means: &nbsp; The joint PDF is here a&nbsp; "Dirac wall".
-*As you can see from the sketch, the PDF values are distributed evenly on the straight line $y = 2x$&nbsp.
+*As you can see from the sketch,&nbsp; the PDF values are Gaussian distributed on the straight line $y = 2x$.
-*The straight line&nbsp; $y = 2x$&nbsp; also represents the correlation line.
+*The two marginal probability densities are also Gaussian functions,&nbsp; each with zero mean.
-*The two marginal probability densities are also Gaussian functions, each with zero mean.
+*Because of&nbsp; $\sigma_x = \sigma_u$&nbsp; and&nbsp; $\sigma_y = \sigma_v$&nbsp; also holds:
 :$$f_x(x) = f_u(u), \hspace{0.5cm}f_y(y) = f_v(v).$$
 [[File:P_ID274__Sto_A_4_4_g.png|right|frame|Probability calculation for the Dirac wall]]
-'''(6)'''&nbsp; Since the PDF of the random variable is&ouml;&aerospace;e&nbsp; $x$&nbsp; identical to the PDF&nbsp; $f_u(u)$, it also results in exactly the same probability as calculated in the subtask&nbsp; '''(3)'''&nbsp; :
+'''(6)'''&nbsp; Since the PDF of the random variable&nbsp; $x$&nbsp; is identical to the PDF&nbsp; $f_u(u)$,&nbsp; it also results in exactly the same probability as calculated in the subtask&nbsp; '''(3)''':
 :$$\rm Pr(\it x < \rm 1) \hspace{0.15cm}\underline{ = \rm 0.9772}.$$
@@ Line 136: / Line 135: @@
 '''(7)'''&nbsp; The random event&nbsp; $y > 1$&nbsp; is identical to the event&nbsp; $x > 0.5$.&nbsp;
 *Thus, the wanted probability is equal to:
-:$$\rm Pr \big[(\it x > \rm 0.5) \cap (\it x < \rm 1)\big] = \it F_x \rm( 1) - \it F_x\rm (0.5).  $$
+:$${\rm Pr}\big[(x < 1) ∩ (y > 1)\big] = \rm Pr \big[ (\it x < \rm 1)\cap (\it x > \rm 0.5) \big] = \it F_x \rm( 1) - \it F_x\rm (0.5).  $$
-*With the standard deviation&nbsp; $\sigma_x = 0.5$&nbsp; follows further:
+*With the standard deviation&nbsp; $\sigma_x = 0.5$&nbsp; it follows further:
 :$$\rm Pr \big[(\it x > \rm 0.5) \cap (\it x < \rm 1)\big] = \rm \phi(\rm 2) - \phi(1)=\rm 0.9772- \rm 0.8413\hspace{0.15cm}\underline{=\rm 0.1359}.$$
 {{ML-Fuß}}

@@ Line 3: / Line 3: @@
 }}
-[[File:P_ID261__Sto_A_4_4_neu.png|right|frame|table: Gaussian error functions]]
+[[File:P_ID261__Sto_A_4_4_neu.png|right|frame|Table:&nbsp; Gaussian error functions]]
-We consider two-dimensional random variables, where both components are always assumed to be mean-free.
+We consider two-dimensional random variables,&nbsp; where both components are always assumed to be mean-free.
-*The 2D PDF of the random variable&nbsp; $(u, v)$&nbsp; is:
+*The 2D&ndash;PDF of the random variable&nbsp; $(u, v)$&nbsp; is:
 :$$f_{uv}(u, v)={1}/{\pi} \cdot {\rm e}^{-(2u^{\rm 2} \hspace{0.05cm}+ \hspace{0.05cm}v^{\rm 2}\hspace{-0.05cm}/\rm 2)}.$$
-*The following parameters are known from the 2D random variable $(x, y)$&nbsp; which is also Gaussian:
+*The following parameters are known from another two-dimensional random variable&nbsp; $(x, y)$&nbsp; which is also Gaussian:
 :$$\sigma_x= 0.5, \hspace{0.5cm}\sigma_y = 1,\hspace{0.5cm}\rho_{xy} = 1. $$
-The values of the Gaussian error integral&nbsp; ${\rm \phi}(x)$&nbsp; and the complementary function&nbsp; ${\rm Q}(x) = 1- {\rm \phi}(x)$&nbsp; can be found in the adjacent table.
+*The values of the Gaussian error integral&nbsp; ${\rm \phi}(x)$&nbsp; and the complementary function&nbsp; ${\rm Q}(x) = 1- {\rm \phi}(x)$&nbsp; can be found in the adjacent table.
@@ Line 22: / Line 18: @@
 Hints:
 *The exercise belongs to the chapter&nbsp; [[Theory_of_Stochastic_Signals/Two-Dimensional_Gaussian_Random_Variables|Two-dimensional Gaussian Random Variables]].
-*Reference is also made to the chapter&nbsp; [[Theory_of_Stochastic_Signals/Gaussian_Distributed_Random_Variables|Gaussian distributed random variables]]
+*Reference is also made to the chapter&nbsp; [[Theory_of_Stochastic_Signals/Gaussian_Distributed_Random_Variables|Gaussian distributed random variables]].
+*More information on this topic is provided in the&nbsp; (German language)&nbsp;learning video&nbsp; [[Gaußsche_2D-Zufallsgrößen_(Lernvideo)|"Gaußsche 2D-Zufallsgrößen"]]:
 ::Part 1: &nbsp; Gaussian random variables without statistical bindings,
 ::Part 2: &nbsp; Gaussian random variables with statistical bindings.
@@ Line 34: / Line 29: @@
 <quiz display=simple>
-{Which of the statements are true with respect to 2D random variable&nbsp; $(u, v)$&nbsp;?
+{Which of the statements are true with respect to the two-dimensional random variable&nbsp; $(u, v)$&nbsp;?
 |type="[]"}
 + The random variables&nbsp; $u$&nbsp; and&nbsp; $v$&nbsp; are uncorrelated.
@@ Line 55: / Line 50: @@
-{Which of the statements are true for the 2D&ndash;random variable&nbsp; $(x, y)$&nbsp;?
+{Which of the statements are true for the two-dimensional random variable&nbsp; $(x, y)$?
 |type="[]"}
-+ The 2D PDF $f_{xy}(x, y)$&nbsp; is always zero outside the straight line&nbsp; $y = 2x$&nbsp;.
++ The joint PDF $f_{xy}(x, y)$&nbsp; is always zero outside the straight line&nbsp; $y = 2x$.
-- For all pairs of values on the straight line $y = 2x$&nbsp; $f_{xy}(x, y)= 0.5$.
+- For all pairs of values on the straight line&nbsp; $y = 2x$&nbsp; holds: &nbsp;$f_{xy}(x, y)= 0.5$.
-+ With respect to the edge PDF, $f_{x}(x) = f_{u}(u)$&nbsp; and $f_{y}(y) = f_{v}(v)$ holds.
++ With respect to the edge PDFs:&nbsp; $f_{x}(x) = f_{u}(u)$&nbsp; and $f_{y}(y) = f_{v}(v)$ holds.
-{Calculate the probability that&nbsp; $x$&nbsp; is less than&nbsp; $1$&nbsp;.
+{Calculate the probability that&nbsp; $x$&nbsp; is less than&nbsp; $1$.
 |type="{}"}
 ${\rm Pr}(x < 1)\ = \ $ { 0.9772 3% }
-{Now calculate the probability that the random variable&nbsp; $x$&nbsp; is smaller than&nbsp; $1$&nbsp; and at the same time the random variable&nbsp; $y$&nbsp; is larger than&nbsp; $1$&nbsp; .
+{Now calculate the probability that the random variable&nbsp; $x$&nbsp; is less than&nbsp; $1$&nbsp; and at the same time the random variable&nbsp; $y$&nbsp; is greater than&nbsp; $1$.
 |type="{}"}
 ${\rm Pr}\big[(x < 1) ∩ (y > 1)\big]\ = \ $ { 0.1359 3% }

@@ Line 1: / Line 1: @@
-{{quiz-Header|Buchseite=Stochastische Signaltheorie/Zweidimensionale Gaußsche Zufallsgrößen
+{{quiz-Header|Buchseite=Theory_of_Stochastic_Signals/Two-Dimensional_Gaussian_Random_Variablesn
 }}
-[[File:P_ID261__Sto_A_4_4_neu.png|right|frame|Tabelle: Gaußsche Fehlerfunktionen]]
+[[File:P_ID261__Sto_A_4_4_neu.png|right|frame|table: Gaussian error functions]]
-Wir betrachten zweidimensionale Zufallsgr&ouml;&szlig;en, wobei beide Komponenten stets als mittelwertfrei vorausgesetzt werden.
+We consider two-dimensional random variables, where both components are always assumed to be mean-free.
-*Die 2D-WDF der Zufallsgr&ouml;&szlig;e&nbsp; $(u, v)$&nbsp; lautet:
+*The 2D PDF of the random variable&nbsp; $(u, v)$&nbsp; is:
 :$$f_{uv}(u, v)={1}/{\pi} \cdot {\rm e}^{-(2u^{\rm 2} \hspace{0.05cm}+ \hspace{0.05cm}v^{\rm 2}\hspace{-0.05cm}/\rm 2)}.$$
-*Von der ebenfalls Gau&szlig;schen 2D-Zufallsgr&ouml;&szlig;e&nbsp; $(x, y)$&nbsp; sind die folgenden Parameter bekannt:
+*The following parameters are known from the 2D random variable $(x, y)$&nbsp; which is also Gaussian:
 :$$\sigma_x= 0.5, \hspace{0.5cm}\sigma_y = 1,\hspace{0.5cm}\rho_{xy} = 1. $$
@@ Line 20: / Line 20: @@
-''Hinweise:''
+Hints:
-*Die Aufgabe gehört zum  Kapitel&nbsp; [[Theory_of_Stochastic_Signals/Zweidimensionale_Gaußsche_Zufallsgrößen|Zweidimensionale Gaußsche Zufallsgrößen]].
+*The exercise belongs to the chapter&nbsp; [[Theory_of_Stochastic_Signals/Two-Dimensional_Gaussian_Random_Variables|Two-dimensional Gaussian Random Variables]].
-*Bezug genommen wird auch auf das Kapitel&nbsp; [[Theory_of_Stochastic_Signals/Gaußverteilte_Zufallsgrößen|Gaußverteilte Zufallsgrößen]]
+*Reference is also made to the chapter&nbsp; [[Theory_of_Stochastic_Signals/Gaussian_Distributed_Random_Variables|Gaussian distributed random variables]]
-*Weitere Informationen zu dieser Thematik liefert das Lernvideo&nbsp; [[Gaußsche_2D-Zufallsgrößen_(Lernvideo)|Gaußsche 2D-Zufallsgrößen]]:
+*More information on this topic is provided in the learning video&nbsp; [[Gaußsche_2D-Zufallsgrößen_(Lernvideo)|Gaussian 2D random variables]]:
-::Teil 1: &nbsp; Gaußsche Zufallsgrößen ohne statistische Bindungen,
+::Part 1: &nbsp; Gaussian random variables without statistical bindings,
-::Teil 2: &nbsp; Gaußsche Zufallsgrößen mit statistischen Bindungen.
+::Part 2: &nbsp; Gaussian random variables with statistical bindings.