https://cfd-online.com/W/index.php?title=Introduction_to_turbulence/Statistical_analysis/Probability&feed=atom&action=historyIntroduction to turbulence/Statistical analysis/Probability - Revision history2024-03-28T18:40:58ZRevision history for this page on the wikiMediaWiki 1.16.5https://cfd-online.com/W/index.php?title=Introduction_to_turbulence/Statistical_analysis/Probability&diff=8190&oldid=prevAyyoubzadeh: /* Skewness and kurtosis */2007-08-31T16:30:28Z<p><span class="autocomment">Skewness and kurtosis</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Both these are easy to remember if you note the <math>S</math> and <math>K</math> must be dimensionless.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Both these are easy to remember if you note the <math>S</math> and <math>K</math> must be dimensionless.</div></td></tr>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>The pdf's in <font color="orange">Figure 2.4</font> can be distinguished by means of their skewness and kurtosis. The random variable shown in (b) has a higher kurtosis than that in (a). Thus the kurtosis can be used as an indication of the tails of a pdf, a higher kurtosis indicating that relatively larger excursions from the mean are more probable. The skewness of (a) and (b) are zero, whereas those for (c) and (d) are non-zero. Thus, as its name implies, a non-zero skewness indicates a skewed or asymmetric pdf, which in turn means that larger excursions in one direction are more probable <del class="diffchange diffchange-inline">tan </del>in the other. For a Gaussian pdf, the skewness is zero and then kurtosis is equal to three. The flatness factor, defined as <math>( K-3 )</math>, is sometimes used to indicate deviations from Gaussian behavior.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>The pdf's in <font color="orange">Figure 2.4</font> can be distinguished by means of their skewness and kurtosis. The random variable shown in (b) has a higher kurtosis than that in (a). Thus the kurtosis can be used as an indication of the tails of a pdf, a higher kurtosis indicating that relatively larger excursions from the mean are more probable. The skewness of (a) and (b) are zero, whereas those for (c) and (d) are non-zero. Thus, as its name implies, a non-zero skewness indicates a skewed or asymmetric pdf, which in turn means that larger excursions in one direction are more probable <ins class="diffchange diffchange-inline">than </ins>in the other. For a Gaussian pdf, the skewness is zero and then kurtosis is equal to three. The flatness factor, defined as <math>( K-3 )</math>, is sometimes used to indicate deviations from Gaussian behavior.</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>'''Exercise:''' Prove that the kurtosis of a Gaussian distributed random variable is 3.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>'''Exercise:''' Prove that the kurtosis of a Gaussian distributed random variable is 3.</div></td></tr>
</table>Ayyoubzadehhttps://cfd-online.com/W/index.php?title=Introduction_to_turbulence/Statistical_analysis/Probability&diff=8189&oldid=prevAyyoubzadeh: /* Gaussian (or normal) distributions */2007-08-31T16:25:48Z<p><span class="autocomment">Gaussian (or normal) distributions</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The Gaussian distribution is unusual in that it is completely determined by its first two moments, <math>X</math> and <math> \sigma </math>. This is ''not'' typical of most turbulence distributions. Nonetheless, it is sometimes useful to approximate turbulence as being Gaussian, often because of the absence of simple alternatives.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The Gaussian distribution is unusual in that it is completely determined by its first two moments, <math>X</math> and <math> \sigma </math>. This is ''not'' typical of most turbulence distributions. Nonetheless, it is sometimes useful to approximate turbulence as being Gaussian, often because of the absence of simple alternatives.</div></td></tr>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>It is <del class="diffchange diffchange-inline">straightvorward </del>to show by integrating by parts that all the even central moments above the second are given by the following recursive relationship,</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>It is <ins class="diffchange diffchange-inline">straightforward </ins>to show by integrating by parts that all the even central moments above the second are given by the following recursive relationship,</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:<math> </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:<math> </div></td></tr>
</table>Ayyoubzadehhttps://cfd-online.com/W/index.php?title=Introduction_to_turbulence/Statistical_analysis/Probability&diff=7986&oldid=prevJola at 12:40, 21 June 20072007-06-21T12:40:57Z<p></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>| [[Statistical analysis in turbulence|Up to statistical analysis]] | [[Ensemble average in turbulence|Back to ensemble average]] | [[Multivariate random variables|Forward to multivariate random variables]]</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>| [[Statistical analysis in turbulence|Up to statistical analysis]] | [[Ensemble average in turbulence|Back to ensemble average]] | [[Multivariate random variables|Forward to multivariate random variables]]</div></td></tr>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">{{Chapter navigation|Ensemble average|Multivariate random variables}}</del></div></td><td colspan="2"> </td></tr>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline">[[Category: Turbulence]]</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">{{Chapter navigation|Ensemble average|Multivariate random variables}}</ins></div></td></tr>
</table>Jolahttps://cfd-online.com/W/index.php?title=Introduction_to_turbulence/Statistical_analysis/Probability&diff=7978&oldid=prevJola at 12:34, 21 June 20072007-06-21T12:34:00Z<p></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[Category: Turbulence]]</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[Category: Turbulence]]</div></td></tr>
</table>Jolahttps://cfd-online.com/W/index.php?title=Introduction_to_turbulence/Statistical_analysis/Probability&diff=7974&oldid=prevJola at 12:29, 21 June 20072007-06-21T12:29:10Z<p></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>== The histogram and probability density function ==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>== The histogram and probability density function ==</div></td></tr>
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<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">{{Chapter navigation|Ensemble average|Mutivariate random variables}}</ins></div></td></tr>
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</table>Jolahttps://cfd-online.com/W/index.php?title=Introduction_to_turbulence/Statistical_analysis/Probability&diff=7972&oldid=prevJola: Probability in turbulence moved to Introduction to turbulence/Statistical analysis/Probability2007-06-21T12:27:26Z<p><a href="/Wiki/Probability_in_turbulence" class="mw-redirect" title="Probability in turbulence">Probability in turbulence</a> moved to <a href="/Wiki/Introduction_to_turbulence/Statistical_analysis/Probability" title="Introduction to turbulence/Statistical analysis/Probability">Introduction to turbulence/Statistical analysis/Probability</a></p>
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</tr></table>Jolahttps://cfd-online.com/W/index.php?title=Introduction_to_turbulence/Statistical_analysis/Probability&diff=7833&oldid=prevJola at 11:34, 18 June 20072007-06-18T11:34:14Z<p></p>
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</table>Jolahttps://cfd-online.com/W/index.php?title=Introduction_to_turbulence/Statistical_analysis/Probability&diff=7825&oldid=prevJola at 18:09, 17 June 20072007-06-17T18:09:18Z<p></p>
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</table>Jolahttps://cfd-online.com/W/index.php?title=Introduction_to_turbulence/Statistical_analysis/Probability&diff=7822&oldid=prevJola at 18:07, 17 June 20072007-06-17T18:07:38Z<p></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>'''Exercise:''' Prove that the kurtosis of a Gaussian distributed random variable is 3.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>'''Exercise:''' Prove that the kurtosis of a Gaussian distributed random variable is 3.</div></td></tr>
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</table>Jolahttps://cfd-online.com/W/index.php?title=Introduction_to_turbulence/Statistical_analysis/Probability&diff=7812&oldid=prevJola: Probability moved to Probability in turbulence2007-06-17T15:11:57Z<p><a href="/W/index.php?title=Probability&action=edit&redlink=1" class="new" title="Probability (page does not exist)">Probability</a> moved to <a href="/Wiki/Probability_in_turbulence" class="mw-redirect" title="Probability in turbulence">Probability in turbulence</a></p>
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