http://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_MultivariateNormal&feed=atom&action=historyAP Statistics Curriculum 2007 MultivariateNormal - Revision history2024-03-28T22:49:56ZRevision history for this page on the wikiMediaWiki 1.15.1http://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_MultivariateNormal&diff=11704&oldid=prevIvoDinov: /* Bivariate (2D) case */2012-07-22T00:08:02Z<p><span class="autocomment">Bivariate (2D) case</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>===Bivariate (2D) case===</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>===Bivariate (2D) case===</div></td></tr>
<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;">: See the SOCR Bivariate Normal Distribution [[SOCR_BivariateNormal_JS_Activity| Activity]] and corresponding [http://socr.ucla.edu/htmls/HTML5/BivariateNormal/ Webapp].</ins></div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>In 2-dimensions, the nonsingular bi-variate Normal distribution with (<math>k=rank(\Sigma) = 2</math>), the probability density function of a (bivariate) vector (X,Y) is</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>In 2-dimensions, the nonsingular bi-variate Normal distribution with (<math>k=rank(\Sigma) = 2</math>), the probability density function of a (bivariate) vector (X,Y) is</div></td></tr>
<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>
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</table>IvoDinovhttp://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_MultivariateNormal&diff=11012&oldid=prevIvoDinov at 15:42, 1 June 20112011-06-01T15:42:21Z<p></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>: <math>Z = \begin{cases} 0,& |X| \leq 1.33,\\</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>: <math>Z = \begin{cases} 0,& |X| \leq 1.33,\\</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>2X,& |X| > 1.33.\end{cases}</math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>2X,& |X| > 1.33.\end{cases}</math></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;">===Applications===</ins></div></td></tr>
<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;">[[SOCR_EduMaterials_Activities_2D_PointSegmentation_EM_Mixture| This SOCR activity demonstrates the use of 2D Gaussian distribution, expectation maximization and mixture modeling for classification of points (objects) in 2D]].</ins></div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>===[[EBook_Problems_MultivariateNormal|Problems]]===</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>===[[EBook_Problems_MultivariateNormal|Problems]]===</div></td></tr>
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</table>IvoDinovhttp://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_MultivariateNormal&diff=10765&oldid=prevJayZzz: /* EBook - Multivariate Normal Distribution */2011-01-25T02:56:28Z<p><span class="autocomment">EBook - Multivariate Normal Distribution</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>-X,& |X| \leq 1.33.\end{cases}</math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>-X,& |X| \leq 1.33.\end{cases}</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline">Than</del>, both X and Y are individually Normally distributed; however, the pair (X,Y) is '''not''' jointly bivariate Normal distributed (of course, the constant c=1.33 is not special, any other non-trivial constant also works).</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">Then</ins>, both X and Y are individually Normally distributed; however, the pair (X,Y) is '''not''' jointly bivariate Normal distributed (of course, the constant c=1.33 is not special, any other non-trivial constant also works).</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Furthermore, as X and Y are not independent, the sum Z = X+Y is not guaranteed to be a (univariate) Normal variable. In this case, it's clear that Z is not Normal:</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Furthermore, as X and Y are not independent, the sum Z = X+Y is not guaranteed to be a (univariate) Normal variable. In this case, it's clear that Z is not Normal:</div></td></tr>
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</table>JayZzzhttp://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_MultivariateNormal&diff=10671&oldid=prevIvoDinov: /* Two normally distributed random variables need not be jointly bivariate normal */2010-12-14T06:34:52Z<p><span class="autocomment">Two normally distributed random variables need not be jointly bivariate normal</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Than, both X and Y are individually Normally distributed; however, the pair (X,Y) is '''not''' jointly bivariate Normal distributed (of course, the constant c=1.33 is not special, any other non-trivial constant also works).</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Than, both X and Y are individually Normally distributed; however, the pair (X,Y) is '''not''' jointly bivariate Normal distributed (of course, the constant c=1.33 is not special, any other non-trivial constant also works).</div></td></tr>
<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;"></ins></div></td></tr>
<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;">Furthermore, as X and Y are not independent, the sum Z = X+Y is not guaranteed to be a (univariate) Normal variable. In this case, it's clear that Z is not Normal:</ins></div></td></tr>
<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;">: <math>Z = \begin{cases} 0,& |X| \leq 1.33,\\</ins></div></td></tr>
<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;">2X,& |X| > 1.33.\end{cases}</math></ins></div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>===[[EBook_Problems_MultivariateNormal|Problems]]===</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>===[[EBook_Problems_MultivariateNormal|Problems]]===</div></td></tr>
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</table>IvoDinovhttp://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_MultivariateNormal&diff=10670&oldid=prevIvoDinov: /* Two normally distributed random variables need not be jointly bivariate normal */2010-12-14T06:18:20Z<p><span class="autocomment">Two normally distributed random variables need not be jointly bivariate normal</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The fact that two random variables ''X'' and ''Y'' both have a normal distribution does not imply that the pair (''X'',&nbsp;''Y'') has a joint normal distribution. A simple example is provided below:</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The fact that two random variables ''X'' and ''Y'' both have a normal distribution does not imply that the pair (''X'',&nbsp;''Y'') has a joint normal distribution. A simple example is provided below:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>: Let X ~ N(0,1).</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>: Let X ~ N(0,1).</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>: Let <math>Y = \begin{cases} X,& |X| > 1.<del class="diffchange diffchange-inline">54</del>,\\</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>: Let <math>Y = \begin{cases} X,& |X| > 1.<ins class="diffchange diffchange-inline">33</ins>,\\</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>-X,& |X| \leq 1.<del class="diffchange diffchange-inline">54</del>.\end{cases}</math></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>-X,& |X| \leq 1.<ins class="diffchange diffchange-inline">33</ins>.\end{cases}</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>Than, both X and Y are individually Normally distributed; however, the pair (X,Y) is '''not''' jointly bivariate Normal distributed (of course, the constant c=1.<del class="diffchange diffchange-inline">54 </del>is not special, any other non-trivial constant also works).</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>Than, both X and Y are individually Normally distributed; however, the pair (X,Y) is '''not''' jointly bivariate Normal distributed (of course, the constant c=1.<ins class="diffchange diffchange-inline">33 </ins>is not special, any other non-trivial constant also works).</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>===[[EBook_Problems_MultivariateNormal|Problems]]===</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>===[[EBook_Problems_MultivariateNormal|Problems]]===</div></td></tr>
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</table>IvoDinovhttp://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_MultivariateNormal&diff=10669&oldid=prevIvoDinov: /* Two normally distributed random variables need not be jointly bivariate normal */2010-12-14T05:59:49Z<p><span class="autocomment">Two normally distributed random variables need not be jointly bivariate normal</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>-X,& |X| \leq 1.54.\end{cases}</math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>-X,& |X| \leq 1.54.\end{cases}</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>Than, both X and Y are individually Normally distributed; however, the pair (X,Y) is '''not''' jointly bivariate Normal distributed.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>Than, both X and Y are individually Normally distributed; however, the pair (X,Y) is '''not''' jointly bivariate Normal distributed <ins class="diffchange diffchange-inline">(of course, the constant c=1.54 is not special, any other non-trivial constant also works)</ins>.</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>===[[EBook_Problems_MultivariateNormal|Problems]]===</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>===[[EBook_Problems_MultivariateNormal|Problems]]===</div></td></tr>
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</table>IvoDinovhttp://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_MultivariateNormal&diff=10668&oldid=prevIvoDinov: /* Two normally distributed random variables need not be jointly bivariate normal */2010-12-14T05:57:28Z<p><span class="autocomment">Two normally distributed random variables need not be jointly bivariate normal</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>====Two normally distributed random variables need not be jointly bivariate normal====</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>====Two normally distributed random variables need not be jointly bivariate normal====</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The fact that two random variables ''X'' and ''Y'' both have a normal distribution does not imply that the pair (''X'',&nbsp;''Y'') has a joint normal distribution. A simple example is provided below:</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The fact that two random variables ''X'' and ''Y'' both have a normal distribution does not imply that the pair (''X'',&nbsp;''Y'') has a joint normal distribution. A simple example is provided below:</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>: Let <del class="diffchange diffchange-inline"><math></del>X <del class="diffchange diffchange-inline">\sim </del>N(0,1)<del class="diffchange diffchange-inline"></math></del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>: Let X <ins class="diffchange diffchange-inline">~ </ins>N(0,1)<ins class="diffchange diffchange-inline">.</ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>: Let <math>Y = \begin{cases} X,& |X| > 1.54,\\</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>: Let <math>Y = \begin{cases} X,& |X| > 1.54,\\</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>-X,& |X| \leq 1.54.\end{cases}</math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>-X,& |X| \leq 1.54.\end{cases}</math></div></td></tr>
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</table>IvoDinovhttp://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_MultivariateNormal&diff=10667&oldid=prevIvoDinov: /* Two normally distributed random variables need not be jointly bivariate normal */2010-12-14T05:56:15Z<p><span class="autocomment">Two normally distributed random variables need not be jointly bivariate normal</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The fact that two random variables ''X'' and ''Y'' both have a normal distribution does not imply that the pair (''X'',&nbsp;''Y'') has a joint normal distribution. A simple example is provided below:</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The fact that two random variables ''X'' and ''Y'' both have a normal distribution does not imply that the pair (''X'',&nbsp;''Y'') has a joint normal distribution. A simple example is provided below:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>: Let <math>X \sim N(0,1)</math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>: Let <math>X \sim N(0,1)</math></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>: Let <math>Y = \begin{cases}X,& |X| > 1.54,\\</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>: Let <math>Y = \begin{cases} X,& |X| > 1.54,\\</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline"> −X</del>, & |X| \<del class="diffchange diffchange-inline">le </del>1.54.\end{cases}</math></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">-X</ins>,& |X| \<ins class="diffchange diffchange-inline">leq </ins>1.54.\end{cases}</math></div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Than, both X and Y are individually Normally distributed; however, the pair (X,Y) is '''not''' jointly bivariate Normal distributed.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Than, both X and Y are individually Normally distributed; however, the pair (X,Y) is '''not''' jointly bivariate Normal distributed.</div></td></tr>
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</table>IvoDinovhttp://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_MultivariateNormal&diff=10666&oldid=prevIvoDinov: /* Two normally distributed random variables need not be jointly bivariate normal */2010-12-14T05:52:56Z<p><span class="autocomment">Two normally distributed random variables need not be jointly bivariate normal</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>====Two normally distributed random variables need not be jointly bivariate normal====</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>====Two normally distributed random variables need not be jointly bivariate normal====</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>The fact that two random variables ''X'' and ''Y'' both have a normal distribution does not imply that the pair (''X'',&nbsp;''Y'') has a joint normal distribution. A simple example is <del class="diffchange diffchange-inline">one in which </del>X <del class="diffchange diffchange-inline">has a normal distribution with expected value </del>0 <del class="diffchange diffchange-inline">and variance 1</del>, <del class="diffchange diffchange-inline">and ''</del>Y<del class="diffchange diffchange-inline">''&nbsp;</del>=<del class="diffchange diffchange-inline">&nbsp;''</del>X<del class="diffchange diffchange-inline">'' if </del>|<del class="diffchange diffchange-inline">''</del>X<del class="diffchange diffchange-inline">''</del>|<del class="diffchange diffchange-inline">&nbsp;</del>>&<del class="diffchange diffchange-inline">nbsp;''c'' and ''Y''&nbsp;=&nbsp;−''X'' if </del>|<del class="diffchange diffchange-inline">''</del>X<del class="diffchange diffchange-inline">''</del>|<del class="diffchange diffchange-inline">&nbsp;<&nbsp;''c'', where ''c'' is about </del>1.54. <del class="diffchange diffchange-inline"> </del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>The fact that two random variables ''X'' and ''Y'' both have a normal distribution does not imply that the pair (''X'',&nbsp;''Y'') has a joint normal distribution. A simple example is <ins class="diffchange diffchange-inline">provided below:</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">: Let <math></ins>X <ins class="diffchange diffchange-inline">\sim N(</ins>0,<ins class="diffchange diffchange-inline">1)</math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">: Let <math></ins>Y = <ins class="diffchange diffchange-inline">\begin{cases}</ins>X<ins class="diffchange diffchange-inline">,& </ins>|X| > <ins class="diffchange diffchange-inline">1.54,\\</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline"> −X, </ins>& |X| <ins class="diffchange diffchange-inline">\le </ins>1.54.<ins class="diffchange diffchange-inline">\end{cases}</math></ins></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;">Than, both X and Y are individually Normally distributed; however, the pair (X,Y) is '''not''' jointly bivariate Normal distributed.</ins></div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>===[[EBook_Problems_MultivariateNormal|Problems]]===</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>===[[EBook_Problems_MultivariateNormal|Problems]]===</div></td></tr>
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</table>IvoDinovhttp://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_MultivariateNormal&diff=10665&oldid=prevIvoDinov: /* Definition */2010-12-14T05:39:16Z<p><span class="autocomment">Definition</span></p>
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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>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div> f_X(x) = \frac{1}{ (2\pi)^{k/2}|\Sigma|^{1/2} }</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div> f_X(x) = \frac{1}{ (2\pi)^{k/2}|\Sigma|^{1/2} }</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div> \exp\!\Big( {-\tfrac{1}{2}}(x-\mu)'\Sigma^{-1}(x-\mu) \Big)<del class="diffchange diffchange-inline">,</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> \exp\!\Big( {-\tfrac{1}{2}}(x-\mu)'\Sigma^{-1}(x-\mu) \Big)</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div> </math></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </math><ins class="diffchange diffchange-inline">, </ins>where |Σ| is the determinant of Σ, and where (2π)<sup>''k''/2</sup>|Σ|<sup>1/2</sup> = |2πΣ|<sup>1/2</sup>. This formulation reduces to the density of the univariate normal distribution if Σ is a scalar (i.e., a 1×1&nbsp;matrix).</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>where |Σ| is the determinant of Σ, and where (2π)<sup>''k''/2</sup>|Σ|<sup>1/2</sup> = |2πΣ|<sup>1/2</sup>. This formulation reduces to the density of the univariate normal distribution if Σ is a scalar (i.e., a 1×1&nbsp;matrix).</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>If the variance-covariance matrix is singular, the corresponding distribution has no density. An example of this case is the distribution of the vector of residual-errors in the ordinary least squares regression. Note also that the ''X''<sub>''i''</sub> are in general ''not'' independent; they can be seen as the result of applying the matrix ''A'' to a collection of independent Gaussian variables ''Z''.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>If the variance-covariance matrix is singular, the corresponding distribution has no density. An example of this case is the distribution of the vector of residual-errors in the ordinary least squares regression. Note also that the ''X''<sub>''i''</sub> are in general ''not'' independent; they can be seen as the result of applying the matrix ''A'' to a collection of independent Gaussian variables ''Z''.</div></td></tr>
<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;"></del></div></td><td colspan="2"> </td></tr>
<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;"><center>[[Image:SOCR_EBook_Dinov_RV_Normal_013108_Fig14.jpg|500px]]</center></del></div></td><td colspan="2"> </td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>===Bivariate (2D) case===</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>===Bivariate (2D) case===</div></td></tr>
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