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Theory of Stochastic Signals/Gaussian Distributed Random Variables - Revision history
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https://en.lntwww.lnt.ei.tum.de/index.php?title=Theory_of_Stochastic_Signals/Gaussian_Distributed_Random_Variables&diff=51272&oldid=prev
Guenter at 09:00, 22 December 2022
2022-12-22T09:00:58Z
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 11:00, 22 December 2022</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$x=\sum\limits_{i=\rm 1}^{\it I}x_i .$$</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$x=\sum\limits_{i=\rm 1}^{\it I}x_i .$$</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*According to the&nbsp; [https://en.wikipedia.org/wiki/Central_limit_theorem $\text{central limit theorem of statistics}$]&nbsp; this sum has a Gaussian PDF in the limiting case&nbsp; $(I → ∞)$&nbsp; as long as the individual components&nbsp; $x_i$&nbsp; have no statistical <del style="font-weight: bold; text-decoration: none;">ties '''KORREKTUR: ties or </del>bindings<del style="font-weight: bold; text-decoration: none;">?'''</del>.&nbsp; This holds&nbsp; (almost)&nbsp; for all density functions of the individual summands. </div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*According to the&nbsp; [https://en.wikipedia.org/wiki/Central_limit_theorem $\text{central limit theorem of statistics}$]&nbsp; this sum has a Gaussian PDF in the limiting case&nbsp; $(I → ∞)$&nbsp; as long as the individual components&nbsp; $x_i$&nbsp; have no statistical <ins style="font-weight: bold; text-decoration: none;"> </ins>bindings.&nbsp; This holds&nbsp; (almost)&nbsp; for all density functions of the individual summands. </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*Many&nbsp; "noise processes"&nbsp; fulfill exactly this condition,&nbsp; that is,&nbsp; they are additively composed of a large number of independent individual contributions,&nbsp; so that their pattern functions&nbsp; ("noise signals")&nbsp; exhibit a Gaussian amplitude distribution. </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*Many&nbsp; "noise processes"&nbsp; fulfill exactly this condition,&nbsp; that is,&nbsp; they are additively composed of a large number of independent individual contributions,&nbsp; so that their pattern functions&nbsp; ("noise signals")&nbsp; exhibit a Gaussian amplitude distribution. </div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*If one applies a Gaussian distributed signal to a linear filter for spectral shaping,&nbsp; the output signal is also Gaussian distributed. &nbsp; Only the distribution parameters such as mean and standard deviation change,&nbsp; as well as the internal statistical <del style="font-weight: bold; text-decoration: none;">ties '''KORREKTUR: ties or </del>bindings<del style="font-weight: bold; text-decoration: none;">?''' </del>of the samples.}}. </div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*If one applies a Gaussian distributed signal to a linear filter for spectral shaping,&nbsp; the output signal is also Gaussian distributed. &nbsp; Only the distribution parameters such as mean and standard deviation change,&nbsp; as well as the internal statistical bindings of the samples.}}. </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[File:P_ID68__Sto_T_3_5_S1_neu.png |right|frame|Gaussian distributed and uniformly distributed random signal]]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[File:P_ID68__Sto_T_3_5_S1_neu.png |right|frame|Gaussian distributed and uniformly distributed random signal]]</div></td></tr>
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Guenter
https://en.lntwww.lnt.ei.tum.de/index.php?title=Theory_of_Stochastic_Signals/Gaussian_Distributed_Random_Variables&diff=51198&oldid=prev
Hwang at 12:56, 21 December 2022
2022-12-21T12:56:49Z
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 14:56, 21 December 2022</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l164">Line 164:</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The Box and Muller method&nbsp; &ndash; hereafter abbreviated to&nbsp; "BM"&nbsp; &ndash; can be characterized as follows: </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The Box and Muller method&nbsp; &ndash; hereafter abbreviated to&nbsp; "BM"&nbsp; &ndash; can be characterized as follows: </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*The theoretical background for the validity of above generation rules is based on the regularities for&nbsp; [[Theory_of_Stochastic_Signals/Two-Dimensional_Random_Variables|$\text{two-dimensional random variables}$]]. </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*The theoretical background for the validity of above generation rules is based on the regularities for&nbsp; [[Theory_of_Stochastic_Signals/Two-Dimensional_Random_Variables|$\text{two-dimensional random variables}$]]. </div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*Obvious equations successively yield two Gaussian values without statistical bindings <del style="font-weight: bold; text-decoration: none;">'''KORREKTUR: ties or bindings?'''</del>.&nbsp; This fact can be used to reduce simulation time by generating a tuple&nbsp; $(x, \ y)$&nbsp; of Gaussian values at each function call.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*Obvious equations successively yield two Gaussian values without statistical bindings.&nbsp; This fact can be used to reduce simulation time by generating a tuple&nbsp; $(x, \ y)$&nbsp; of Gaussian values at each function call.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*A comparison of the computation times shows that&nbsp; &ndash; with the best possible implementation in each case&nbsp; &ndash; the BM method is superior to the addition method with&nbsp; $I =12$&nbsp; by&nbsp; (approximately)&nbsp; a factor of&nbsp; $3$.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*A comparison of the computation times shows that&nbsp; &ndash; with the best possible implementation in each case&nbsp; &ndash; the BM method is superior to the addition method with&nbsp; $I =12$&nbsp; by&nbsp; (approximately)&nbsp; a factor of&nbsp; $3$.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*The range of values is less limited in the BM method than in the addition method,&nbsp; so that even small probabilities are simulated more accurately.&nbsp; But even with the BM method,&nbsp; it is not possible to simulate arbitrarily small error probabilities.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*The range of values is less limited in the BM method than in the addition method,&nbsp; so that even small probabilities are simulated more accurately.&nbsp; But even with the BM method,&nbsp; it is not possible to simulate arbitrarily small error probabilities.</div></td></tr>
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Hwang
https://en.lntwww.lnt.ei.tum.de/index.php?title=Theory_of_Stochastic_Signals/Gaussian_Distributed_Random_Variables&diff=51195&oldid=prev
Hwang at 12:47, 21 December 2022
2022-12-21T12:47:27Z
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 14:47, 21 December 2022</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l14">Line 14:</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$x=\sum\limits_{i=\rm 1}^{\it I}x_i .$$</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$x=\sum\limits_{i=\rm 1}^{\it I}x_i .$$</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*According to the&nbsp; [https://en.wikipedia.org/wiki/Central_limit_theorem $\text{central limit theorem of statistics}$]&nbsp; this sum has a Gaussian PDF in the limiting case&nbsp; $(I → ∞)$&nbsp; as long as the individual components&nbsp; $x_i$&nbsp; have no statistical ties.&nbsp; This holds&nbsp; (almost)&nbsp; for all density functions of the individual summands. </div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*According to the&nbsp; [https://en.wikipedia.org/wiki/Central_limit_theorem $\text{central limit theorem of statistics}$]&nbsp; this sum has a Gaussian PDF in the limiting case&nbsp; $(I → ∞)$&nbsp; as long as the individual components&nbsp; $x_i$&nbsp; have no statistical ties <ins style="font-weight: bold; text-decoration: none;">'''KORREKTUR: ties or bindings?'''</ins>.&nbsp; This holds&nbsp; (almost)&nbsp; for all density functions of the individual summands. </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*Many&nbsp; "noise processes"&nbsp; fulfill exactly this condition,&nbsp; that is,&nbsp; they are additively composed of a large number of independent individual contributions,&nbsp; so that their pattern functions&nbsp; ("noise signals")&nbsp; exhibit a Gaussian amplitude distribution. </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*Many&nbsp; "noise processes"&nbsp; fulfill exactly this condition,&nbsp; that is,&nbsp; they are additively composed of a large number of independent individual contributions,&nbsp; so that their pattern functions&nbsp; ("noise signals")&nbsp; exhibit a Gaussian amplitude distribution. </div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*If one applies a Gaussian distributed signal to a linear filter for spectral shaping,&nbsp; the output signal is also Gaussian distributed. &nbsp; Only the distribution parameters such as mean and standard deviation change,&nbsp; as well as the internal statistical ties of the samples.}}. </div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*If one applies a Gaussian distributed signal to a linear filter for spectral shaping,&nbsp; the output signal is also Gaussian distributed. &nbsp; Only the distribution parameters such as mean and standard deviation change,&nbsp; as well as the internal statistical ties <ins style="font-weight: bold; text-decoration: none;">'''KORREKTUR: ties or bindings?''' </ins>of the samples.}}. </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[File:P_ID68__Sto_T_3_5_S1_neu.png |right|frame|Gaussian distributed and uniformly distributed random signal]]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[File:P_ID68__Sto_T_3_5_S1_neu.png |right|frame|Gaussian distributed and uniformly distributed random signal]]</div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l164">Line 164:</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The Box and Muller method&nbsp; &ndash; hereafter abbreviated to&nbsp; "BM"&nbsp; &ndash; can be characterized as follows: </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The Box and Muller method&nbsp; &ndash; hereafter abbreviated to&nbsp; "BM"&nbsp; &ndash; can be characterized as follows: </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*The theoretical background for the validity of above generation rules is based on the regularities for&nbsp; [[Theory_of_Stochastic_Signals/Two-Dimensional_Random_Variables|$\text{two-dimensional random variables}$]]. </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*The theoretical background for the validity of above generation rules is based on the regularities for&nbsp; [[Theory_of_Stochastic_Signals/Two-Dimensional_Random_Variables|$\text{two-dimensional random variables}$]]. </div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*Obvious equations successively yield two Gaussian values without statistical bindings '''KORREKTUR: ties?'''.&nbsp; This fact can be used to reduce simulation time by generating a tuple&nbsp; $(x, \ y)$&nbsp; of Gaussian values at each function call.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*Obvious equations successively yield two Gaussian values without statistical bindings '''KORREKTUR: ties <ins style="font-weight: bold; text-decoration: none;">or bindings</ins>?'''.&nbsp; This fact can be used to reduce simulation time by generating a tuple&nbsp; $(x, \ y)$&nbsp; of Gaussian values at each function call.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*A comparison of the computation times shows that&nbsp; &ndash; with the best possible implementation in each case&nbsp; &ndash; the BM method is superior to the addition method with&nbsp; $I =12$&nbsp; by&nbsp; (approximately)&nbsp; a factor of&nbsp; $3$.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*A comparison of the computation times shows that&nbsp; &ndash; with the best possible implementation in each case&nbsp; &ndash; the BM method is superior to the addition method with&nbsp; $I =12$&nbsp; by&nbsp; (approximately)&nbsp; a factor of&nbsp; $3$.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*The range of values is less limited in the BM method than in the addition method,&nbsp; so that even small probabilities are simulated more accurately.&nbsp; But even with the BM method,&nbsp; it is not possible to simulate arbitrarily small error probabilities.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*The range of values is less limited in the BM method than in the addition method,&nbsp; so that even small probabilities are simulated more accurately.&nbsp; But even with the BM method,&nbsp; it is not possible to simulate arbitrarily small error probabilities.</div></td></tr>
</table>
Hwang
https://en.lntwww.lnt.ei.tum.de/index.php?title=Theory_of_Stochastic_Signals/Gaussian_Distributed_Random_Variables&diff=51178&oldid=prev
Hwang at 19:56, 20 December 2022
2022-12-20T19:56:57Z
<p></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 21:56, 20 December 2022</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>$\text{Example 4:}$&nbsp; </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>$\text{Example 4:}$&nbsp; </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The sketch shows the PDF splitting for&nbsp; $J = 16$&nbsp; by the boundaries&nbsp; $I_{-7}$, ... , $ I_7$. </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The sketch shows the PDF splitting for&nbsp; $J = 16$&nbsp; by the boundaries&nbsp; $I_{-7}$, ... , $ I_7$. </div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*These interval boundaries were chosen so that each interval has the same area&nbsp; $p_j = 1/J = 1/16$&nbsp; <del style="font-weight: bold; text-decoration: none;">. </del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*These interval boundaries were chosen so that each interval has the same area&nbsp; $p_j = 1/J = 1/16$<ins style="font-weight: bold; text-decoration: none;">.</ins>&nbsp; <ins style="font-weight: bold; text-decoration: none;"> </ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*The characteristic value&nbsp; $C_j$&nbsp; of each interval lies exactly midway between&nbsp; $I_{j-1}$&nbsp; and&nbsp; $I_j$. </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*The characteristic value&nbsp; $C_j$&nbsp; of each interval lies exactly midway between&nbsp; $I_{j-1}$&nbsp; and&nbsp; $I_j$. </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
</table>
Hwang
https://en.lntwww.lnt.ei.tum.de/index.php?title=Theory_of_Stochastic_Signals/Gaussian_Distributed_Random_Variables&diff=51177&oldid=prev
Hwang at 19:55, 20 December 2022
2022-12-20T19:55:11Z
<p></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 21:55, 20 December 2022</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>However,&nbsp; a simulation documented in&nbsp; [ES96]<ref name='ES96'>Eck, P.; Söder, G.:&nbsp; Tabulated Inversion, a Fast Method for White Gaussian Noise Simulation.&nbsp; In:&nbsp; AEÜ Int. J. Electron. Commun. 50 (1996), pp. 41-48.</ref>&nbsp; over&nbsp; $10^{9}$&nbsp; samples has shown that the BM method approximates the Q function very well only up to error probabilities of&nbsp; $10^{-5}$&nbsp; but then the curve shape breaks off steeply. </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>However,&nbsp; a simulation documented in&nbsp; [ES96]<ref name='ES96'>Eck, P.; Söder, G.:&nbsp; Tabulated Inversion, a Fast Method for White Gaussian Noise Simulation.&nbsp; In:&nbsp; AEÜ Int. J. Electron. Commun. 50 (1996), pp. 41-48.</ref>&nbsp; over&nbsp; $10^{9}$&nbsp; samples has shown that the BM method approximates the Q function very well only up to error probabilities of&nbsp; $10^{-5}$&nbsp; but then the curve shape breaks off steeply. </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*The maximum occurring value of the root expression was not&nbsp; $6.55$,&nbsp; but due to the current random variables&nbsp; $u$&nbsp; and&nbsp; $v$&nbsp; only about&nbsp; $4.6$,&nbsp; which explains the abrupt drop from about&nbsp; $10^{-5}$&nbsp; on. </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*The maximum occurring value of the root expression was not&nbsp; $6.55$,&nbsp; but due to the current random variables&nbsp; $u$&nbsp; and&nbsp; $v$&nbsp; only about&nbsp; $4.6$,&nbsp; which explains the abrupt drop from about&nbsp; $10^{-5}$&nbsp; on. </div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*Of course,&nbsp; this method works much better with 64 bit arithmetic operations}}<del style="font-weight: bold; text-decoration: none;">.</del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*Of course,&nbsp; this method works much better with 64 bit arithmetic operations<ins style="font-weight: bold; text-decoration: none;">.</ins>}}</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Gaussian generation with the "Tabulated Inversion" method==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Gaussian generation with the "Tabulated Inversion" method==</div></td></tr>
</table>
Hwang
https://en.lntwww.lnt.ei.tum.de/index.php?title=Theory_of_Stochastic_Signals/Gaussian_Distributed_Random_Variables&diff=51176&oldid=prev
Hwang at 19:54, 20 December 2022
2022-12-20T19:54:42Z
<p></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 21:54, 20 December 2022</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>However,&nbsp; <del style="font-weight: bold; text-decoration: none;">an </del>in&nbsp; [ES96]<ref name='ES96'>Eck, P.; Söder, G.:&nbsp; Tabulated Inversion, a Fast Method for White Gaussian Noise Simulation.&nbsp; In:&nbsp; AEÜ Int. J. Electron. Commun. 50 (1996), pp. 41-48.</ref>&nbsp; <del style="font-weight: bold; text-decoration: none;">documented simulation </del>over&nbsp; $10^{9}$&nbsp; samples has shown that the BM method approximates the Q function very well only up to error probabilities of&nbsp; $10^{-5}$&nbsp; but then the curve shape breaks off steeply. </div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>However,&nbsp; <ins style="font-weight: bold; text-decoration: none;">a simulation documented </ins>in&nbsp; [ES96]<ref name='ES96'>Eck, P.; Söder, G.:&nbsp; Tabulated Inversion, a Fast Method for White Gaussian Noise Simulation.&nbsp; In:&nbsp; AEÜ Int. J. Electron. Commun. 50 (1996), pp. 41-48.</ref>&nbsp; over&nbsp; $10^{9}$&nbsp; samples has shown that the BM method approximates the Q function very well only up to error probabilities of&nbsp; $10^{-5}$&nbsp; but then the curve shape breaks off steeply. </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*The maximum occurring value of the root expression was not&nbsp; $6.55$,&nbsp; but due to the current random variables&nbsp; $u$&nbsp; and&nbsp; $v$&nbsp; only about&nbsp; $4.6$,&nbsp; which explains the abrupt drop from about&nbsp; $10^{-5}$&nbsp; on. </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*The maximum occurring value of the root expression was not&nbsp; $6.55$,&nbsp; but due to the current random variables&nbsp; $u$&nbsp; and&nbsp; $v$&nbsp; only about&nbsp; $4.6$,&nbsp; which explains the abrupt drop from about&nbsp; $10^{-5}$&nbsp; on. </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*Of course,&nbsp; this method works much better with 64 bit arithmetic operations}}.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*Of course,&nbsp; this method works much better with 64 bit arithmetic operations}}.</div></td></tr>
</table>
Hwang
https://en.lntwww.lnt.ei.tum.de/index.php?title=Theory_of_Stochastic_Signals/Gaussian_Distributed_Random_Variables&diff=51175&oldid=prev
Hwang at 19:48, 20 December 2022
2022-12-20T19:48:51Z
<p></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 21:48, 20 December 2022</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The Box and Muller method&nbsp; &ndash; hereafter abbreviated to&nbsp; "BM"&nbsp; &ndash; can be characterized as follows: </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The Box and Muller method&nbsp; &ndash; hereafter abbreviated to&nbsp; "BM"&nbsp; &ndash; can be characterized as follows: </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*The theoretical background for the validity of above generation rules is based on the regularities for&nbsp; [[Theory_of_Stochastic_Signals/Two-Dimensional_Random_Variables|$\text{two-dimensional random variables}$]]. </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*The theoretical background for the validity of above generation rules is based on the regularities for&nbsp; [[Theory_of_Stochastic_Signals/Two-Dimensional_Random_Variables|$\text{two-dimensional random variables}$]]. </div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*Obvious equations successively yield two Gaussian values without statistical bindings.&nbsp; This fact can be used to reduce simulation time by generating a tuple&nbsp; $(x, \ y)$&nbsp; of Gaussian values at each function call.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*Obvious equations successively yield two Gaussian values without statistical bindings <ins style="font-weight: bold; text-decoration: none;">'''KORREKTUR: ties?'''</ins>.&nbsp; This fact can be used to reduce simulation time by generating a tuple&nbsp; $(x, \ y)$&nbsp; of Gaussian values at each function call.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*A comparison of the computation times shows that&nbsp; &ndash; with the best possible implementation in each case&nbsp; &ndash; the BM method is superior to the addition method with&nbsp; $I =12$&nbsp; by&nbsp; (approximately)&nbsp; a factor of&nbsp; $3$.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*A comparison of the computation times shows that&nbsp; &ndash; with the best possible implementation in each case&nbsp; &ndash; the BM method is superior to the addition method with&nbsp; $I =12$&nbsp; by&nbsp; (approximately)&nbsp; a factor of&nbsp; $3$.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*The range of values is less limited in the BM method than in the addition method,&nbsp; so that even small probabilities are simulated more accurately.&nbsp; But even with the BM method,&nbsp; it is not possible to simulate arbitrarily small error probabilities.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*The range of values is less limited in the BM method than in the addition method,&nbsp; so that even small probabilities are simulated more accurately.&nbsp; But even with the BM method,&nbsp; it is not possible to simulate arbitrarily small error probabilities.</div></td></tr>
</table>
Hwang
https://en.lntwww.lnt.ei.tum.de/index.php?title=Theory_of_Stochastic_Signals/Gaussian_Distributed_Random_Variables&diff=51174&oldid=prev
Hwang at 19:45, 20 December 2022
2022-12-20T19:45:32Z
<p></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 21:45, 20 December 2022</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l158">Line 158:</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Gaussian generation with the Box/Muller method==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Gaussian generation with the Box/Muller method==</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><br></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><br></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>In this method,&nbsp; two statistically independent Gaussian distributed random variables&nbsp; $x$&nbsp; and&nbsp; $y$&nbsp; are generated&nbsp; (approximately)&nbsp; from the two&nbsp; $($between&nbsp; $0$&nbsp; and&nbsp; $1$&nbsp; <del style="font-weight: bold; text-decoration: none;">uniformlyly </del>distributed and statistically independent random variables&nbsp; $u$&nbsp; and&nbsp; $v)$&nbsp; by&nbsp; [[Theory_of_Stochastic_Signals/Exponentially_Distributed_Random_Variables#Transformation_of_random_variables|$\text{nonlinear transformation}$]]: </div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>In this method,&nbsp; two statistically independent Gaussian distributed random variables&nbsp; $x$&nbsp; and&nbsp; $y$&nbsp; are generated&nbsp; (approximately)&nbsp; from the two&nbsp; $($between&nbsp; $0$&nbsp; and&nbsp; $1$&nbsp; <ins style="font-weight: bold; text-decoration: none;">uniformly </ins>distributed and statistically independent random variables&nbsp; $u$&nbsp; and&nbsp; $v)$&nbsp; by&nbsp; [[Theory_of_Stochastic_Signals/Exponentially_Distributed_Random_Variables#Transformation_of_random_variables|$\text{nonlinear transformation}$]]: </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$x=m_x+\sigma_{x}\cdot \cos(2 \pi u)\cdot\sqrt{-2\cdot \ln(v)},$$</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$x=m_x+\sigma_{x}\cdot \cos(2 \pi u)\cdot\sqrt{-2\cdot \ln(v)},$$</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$y=m_y+\sigma_{y}\cdot \sin(2 \pi u)\cdot\sqrt{-2\cdot \ln(v)}.$$</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$y=m_y+\sigma_{y}\cdot \sin(2 \pi u)\cdot\sqrt{-2\cdot \ln(v)}.$$</div></td></tr>
</table>
Hwang
https://en.lntwww.lnt.ei.tum.de/index.php?title=Theory_of_Stochastic_Signals/Gaussian_Distributed_Random_Variables&diff=51173&oldid=prev
Hwang at 19:38, 20 December 2022
2022-12-20T19:38:23Z
<p></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 21:38, 20 December 2022</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[File:P_ID621__Sto_T_3_5_S3neu.png |right|frame| Complementary Gaussian error integral&nbsp; ${\rm Q}(x)$]]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[File:P_ID621__Sto_T_3_5_S3neu.png |right|frame| Complementary Gaussian error integral&nbsp; ${\rm Q}(x)$]]</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$${\rm Pr}(x > x_{\rm 0})={\rm Q}({x_{\rm 0} }/{\sigma}).$$</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$${\rm Pr}(x > x_{\rm 0})={\rm Q}({x_{\rm 0} }/{\sigma}).$$</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*Here,&nbsp; ${\rm Q}(x) = 1 - {\rm ϕ}(x)$&nbsp; denotes the complementary function to&nbsp; $ {\rm ϕ}(x)$.&nbsp; This function is called the&nbsp; &raquo;'''<del style="font-weight: bold; text-decoration: none;">Complementary </del>Gaussian error integral'''&laquo;&nbsp; and the following calculation rule applies: </div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*Here,&nbsp; ${\rm Q}(x) = 1 - {\rm ϕ}(x)$&nbsp; denotes the complementary function to&nbsp; $ {\rm ϕ}(x)$.&nbsp; This function is called the&nbsp; &raquo;'''<ins style="font-weight: bold; text-decoration: none;">complementary </ins>Gaussian error integral'''&laquo;&nbsp; and the following calculation rule applies: </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$\rm Q (\it x\rm ) = \rm 1- \phi (\it x)$$</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$\rm Q (\it x\rm ) = \rm 1- \phi (\it x)$$</div></td></tr>
</table>
Hwang