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

Hwang at 13:38, 21 December 2022

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Hwang at 13:38, 21 December 2022

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Hwang at 13:35, 21 December 2022

2022-12-21T13:35:47Z

Hwang at 13:35, 21 December 2022

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Hwang at 13:34, 21 December 2022

2022-12-21T13:34:27Z

Hwang at 13:31, 21 December 2022

2022-12-21T13:31:38Z

Hwang at 13:27, 21 December 2022

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Hwang at 13:17, 21 December 2022

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Hwang at 13:15, 21 December 2022

2022-12-21T13:15:26Z

Hwang at 13:08, 21 December 2022

2022-12-21T13:08:57Z

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 {{GraueBox|TEXT=
-$\text{Example 2:}$&nbsp; In the graph,&nbsp; the instantaneous values of a two-dimensional random variable are plotted as points in the&nbsp; $(x,\, y)$ plane.
+$\text{Example 2:}$&nbsp; In the graph,&nbsp; the instantaneous values of a two-dimensional random variable are plotted as points in the&nbsp; $(x,\, y)$&ndash;plane.
 *Ranges with many points,&nbsp; which accordingly appear dark,&nbsp; indicate large values of the two-dimensional PDF&nbsp; $f_{xy}(x,\, y)$.
 *In contrast,&nbsp; the random variable&nbsp; $(x,\, y)$&nbsp; has relatively few components in rather bright areas.
@@ Line 254: / Line 254: @@
 <br>
 {{BlaueBox|TEXT=
-$\text{Definition:}$&nbsp; The&nbsp; &raquo;'''regression line'''&laquo;&nbsp; &ndash; sometimes called&nbsp; "correlation line" &ndash;&nbsp; is the straight line&nbsp; $y = K(x)$&nbsp; in the&nbsp; $(x,  y)$ plane through the&nbsp; "midpoint"&nbsp; $(m_x, m_y)$.&nbsp;
+$\text{Definition:}$&nbsp; The&nbsp; &raquo;'''regression line'''&laquo;&nbsp; &ndash; sometimes called&nbsp; "correlation line" &ndash;&nbsp; is the straight line&nbsp; $y = K(x)$&nbsp; in the&nbsp; $(x,  y)$&ndash;plane through the&nbsp; "midpoint"&nbsp; $(m_x, m_y)$.&nbsp;
 [[File: EN_Sto_T_4_1_S7neu.png |frame|Two-dimensional Gaussian PDF with regression line&nbsp; $\rm (RL)$ ]]
 The regression line has the following properties:

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 <br>
 {{BlaueBox|TEXT=
-$\text{Definition:}$&nbsp; The&nbsp; &raquo;'''regression line'''&laquo;&nbsp; &ndash; sometimes called&nbsp; "correlation line" &ndash;&nbsp; is the straight line&nbsp; $y = K(x)$&nbsp; in the&nbsp; $(x,  y)$&ndash;plane through the&nbsp; "midpoint"&nbsp; $(m_x, m_y)$.&nbsp;
+$\text{Definition:}$&nbsp; The&nbsp; &raquo;'''regression line'''&laquo;&nbsp; &ndash; sometimes called&nbsp; "correlation line" &ndash;&nbsp; is the straight line&nbsp; $y = K(x)$&nbsp; in the&nbsp; $(x,  y)$ plane through the&nbsp; "midpoint"&nbsp; $(m_x, m_y)$.&nbsp;
 [[File: EN_Sto_T_4_1_S7neu.png |frame|Two-dimensional Gaussian PDF with regression line&nbsp; $\rm (RL)$ ]]
 The regression line has the following properties:

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 Unlike&nbsp; [[Theory_of_Stochastic_Signals/Two-Dimensional_Random_Variables#PDF_for_statistically_independent_components|$\text{Example 2}$]]&nbsp; with statistically independent components &nbsp; &rArr; &nbsp; $ρ_{xy} = 0$&nbsp; $($even though&nbsp; $σ_y < σ_x)$&nbsp; one recognizes that here
-*with larger&nbsp; $x$&ndash;value on statistical average,&nbsp; $y$&nbsp; is also larger
+*with larger&nbsp; $x$&ndash;value, on statistical average,&nbsp; $y$&nbsp; is also larger
 *than with a smaller&nbsp; $x$&ndash;value.}}

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 <br>One recognizes from this representation:
 #Integration over $f_{xy}(x, y)$&nbsp; parallel to the&nbsp; $x$&ndash;axis leads to the triangular marginal PDF&nbsp; $f_{y}(y)$,&nbsp; integration parallel to&nbsp; $y$&ndash;axis to the trapezoidal PDF $f_{x}(x)$.
-#From the joint PDF $f_{xy}(x, y)$&nbsp; it can already be guessed that for each&nbsp; $x$&ndash;value on statistical average a different&nbsp; $y$&ndash;value is to be expected.
+#From the joint PDF $f_{xy}(x, y)$&nbsp; it can already be guessed that for each&nbsp; $x$&ndash;value on statistical average, a different&nbsp; $y$&ndash;value is to be expected.
 #This means that the components&nbsp; $x$&nbsp; and&nbsp; $y$&nbsp; are statistically dependent on each other. }}
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 *If the two random variables&nbsp; $x$&nbsp; and&nbsp; $y$&nbsp; are uncorrelated,&nbsp; then&nbsp; $ρ_{xy} = 0$.
 *For strict linear dependence between&nbsp; $x$&nbsp; and&nbsp; $y$ &nbsp;  &rArr; &nbsp; $ρ_{xy}= ±1$ &nbsp; &rArr; &nbsp; complete correlation.
-*A positive correlation coefficient means that when&nbsp; $x$&nbsp; is larger,&nbsp; on statistical average&nbsp; $y$&nbsp; is also larger than when&nbsp; $x$&nbsp; is smaller.
+*A positive correlation coefficient means that when&nbsp; $x$&nbsp; is larger,&nbsp; on statistical average,&nbsp; $y$&nbsp; is also larger than when&nbsp; $x$&nbsp; is smaller.
 *In contrast,&nbsp; a negative correlation coefficient expresses that&nbsp; $y$&nbsp; becomes smaller on average as&nbsp; $x$&nbsp; increases.
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 Unlike&nbsp; [[Theory_of_Stochastic_Signals/Two-Dimensional_Random_Variables#PDF_for_statistically_independent_components|$\text{Example 2}$]]&nbsp; with statistically independent components &nbsp; &rArr; &nbsp; $ρ_{xy} = 0$&nbsp; $($even though&nbsp; $σ_y < σ_x)$&nbsp; one recognizes that here
-*with larger&nbsp; $x$&ndash;value on statistical average&nbsp; $y$&nbsp; is also larger
+*with larger&nbsp; $x$&ndash;value on statistical average,&nbsp; $y$&nbsp; is also larger
 *than with a smaller&nbsp; $x$&ndash;value.}}

@@ Line 246: / Line 246: @@
-Unlike the&nbsp; [[Theory_of_Stochastic_Signals/Two-Dimensional_Random_Variables#PDF_for_statistically_independent_components|$\text{Example 2}$]]&nbsp; with statistically independent components &nbsp; &rArr; &nbsp; $ρ_{xy} = 0$&nbsp; $($even though&nbsp; $σ_y < σ_x)$&nbsp; one recognizes that here
+Unlike&nbsp; [[Theory_of_Stochastic_Signals/Two-Dimensional_Random_Variables#PDF_for_statistically_independent_components|$\text{Example 2}$]]&nbsp; with statistically independent components &nbsp; &rArr; &nbsp; $ρ_{xy} = 0$&nbsp; $($even though&nbsp; $σ_y < σ_x)$&nbsp; one recognizes that here
 *with larger&nbsp; $x$&ndash;value on statistical average&nbsp; $y$&nbsp; is also larger
 *than with a smaller&nbsp; $x$&ndash;value.}}

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 ==Correlation coefficient==
 <br>
-With statistical independence of the two components&nbsp; $x$&nbsp; and&nbsp; $y$ &nbsp; the covariance&nbsp; $\mu_{xy} \equiv 0$.&nbsp; This case has already been considered in&nbsp; $\text{Example 2}$&nbsp; in the section&nbsp; [[Theory_of_Stochastic_Signals/Two-Dimensional_Random_Variables#PDF_and_CDF_for_statistically_independent_components|"PDF and CDF for statistically independent components"]].
+With statistical independence of the two components&nbsp; $x$&nbsp; and&nbsp; $y$ &nbsp; the covariance&nbsp; $\mu_{xy} \equiv 0$.&nbsp; This case has already been considered in&nbsp; $\text{Example 2}$&nbsp; in the section&nbsp; [[Theory_of_Stochastic_Signals/Two-Dimensional_Random_Variables#PDF_for_statistically_independent_components|"PDF for statistically independent components"]].
 *But the result&nbsp; $\mu_{xy} = 0$&nbsp; is also possible for statistically dependent components&nbsp; $x$&nbsp; and&nbsp; $y$&nbsp; namely when they are uncorrelated,&nbsp; i.e.&nbsp; "linearly independent".

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 ==PDF for statistically dependent components==
 <br>
-If there are statistical bindings between&nbsp; $x$&nbsp; and&nbsp; $y$,&nbsp; then different cuts parallel to&nbsp; $x$&ndash; and&nbsp; $y$&ndash;axis,&nbsp; resp.,&nbsp; yield different&nbsp; (non-shape equivalent)&nbsp; functions.&nbsp; In this case,&nbsp; of course,&nbsp; the joint PDF cannot be described as a product of the two&nbsp; (one-dimensional)&nbsp; marginal probability densities functionseither.
+If there are statistical bindings between&nbsp; $x$&nbsp; and&nbsp; $y$,&nbsp; then different cuts parallel to&nbsp; $x$&ndash; and&nbsp; $y$&ndash;axis,&nbsp; resp.,&nbsp; yield different&nbsp; (non-shape equivalent)&nbsp; functions.&nbsp; In this case,&nbsp; of course,&nbsp; the joint PDF cannot be described as a product of the two&nbsp; (one-dimensional)&nbsp; marginal probability densities functions either.
 {{GraueBox|TEXT=

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 $\text{Conclusion:}$&nbsp; Note the significant difference between one-dimensional and two-dimensional random variables:
 *For one-dimensional random variables,&nbsp; the area under the PDF always yields the value&nbsp; $1$.
-*For two-dimensional random variables,&nbsp; the PDF volume is always equal&nbsp; $1$.}}
+*For two-dimensional random variables,&nbsp; the PDF volume is always equal to&nbsp; $1$.}}
 ==PDF for statistically independent components==

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 In the right diagram,&nbsp; the maximum temperatures of the&nbsp; $31$ days in May 2002 of Munich&nbsp; (to the top)&nbsp; and the mountain&nbsp; "Zugspitze"&nbsp; (to the right)&nbsp; are contrasted.&nbsp; Both random variables are continuous in value:
-*Although the measurement points are about&nbsp; $\text{100 km}$&nbsp; apart,&nbsp; and on the Zugspitze,&nbsp; due to the different altitudes &nbsp;$($nearly&nbsp; $3000$&nbsp; versus&nbsp; $520$&nbsp; meters$)$&nbsp; is on average about&nbsp; $20$&nbsp; degrees colder than in Munich,&nbsp; one recognizes nevertheless a certain statistical dependence between the two random variables&nbsp; ${\it Θ}_{\rm M}$&nbsp; and&nbsp; ${\it Θ}_{\rm Z}$.
+*Although the measurement points are about&nbsp; $\text{100 km}$&nbsp; apart,&nbsp; and on the Zugspitze,&nbsp; it is on average about &nbsp; $20$&nbsp; degrees colder than in Munich due to the different altitudes &nbsp;$($nearly&nbsp; $3000$&nbsp; versus&nbsp; $520$&nbsp; meters$)$,&nbsp; one recognizes nevertheless a certain statistical dependence between the two random variables&nbsp; ${\it Θ}_{\rm M}$&nbsp; and&nbsp; ${\it Θ}_{\rm Z}$.
 *If it is warm in Munich,&nbsp; then pleasant temperatures are also more likely to be expected on the Zugspitze.&nbsp; However,&nbsp; the relationship is not deterministic:&nbsp; The coldest day in May 2002 was a different day in Munich than the coldest day on the Zugspitze. }}