https://wiki.socr.umich.edu/index.php?action=history&feed=atom&title=AP_Statistics_Curriculum_2007_Distrib_Dists 2026-07-21T12:52:52Z Revision history for this page on the wiki MediaWiki 1.31.6 https://wiki.socr.umich.edu/index.php?title=AP_Statistics_Curriculum_2007_Distrib_Dists&diff=16834&oldid=prev 2020-03-03T17:57:35Z

<p>Text replacement - &quot;{{translate|pageName=http://wiki.stat.ucla.edu/socr/&quot; to &quot;&quot;{{translate|pageName=http://wiki.socr.umich.edu/&quot;</p> <table class="diff diff-contentalign-left" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 17:57, 3 March 2020</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l269" >Line 269:</td> <td colspan="2" class="diff-lineno">Line 269:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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>* SOCR Home page: http://www.socr.ucla.edu</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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>* SOCR Home page: http://www.socr.ucla.edu</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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;"></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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;"></td></tr> <tr><td class='diff-marker'>−</td><td style="color: #222; 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>{{translate|pageName=http://wiki.<del class="diffchange diffchange-inline">stat</del>.<del class="diffchange diffchange-inline">ucla</del>.edu<del class="diffchange diffchange-inline">/socr</del>/index.php?title=AP_Statistics_Curriculum_2007_Distrib_Dists}}</div></td><td class='diff-marker'>+</td><td style="color: #222; 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><ins class="diffchange diffchange-inline">&quot;</ins>{{translate|pageName=http://wiki.<ins class="diffchange diffchange-inline">socr</ins>.<ins class="diffchange diffchange-inline">umich</ins>.edu/index.php?title=AP_Statistics_Curriculum_2007_Distrib_Dists}}</div></td></tr> </table>

Tdlee https://wiki.socr.umich.edu/index.php?title=AP_Statistics_Curriculum_2007_Distrib_Dists&diff=13004&oldid=prev 2014-08-01T13:27:08Z

<p>‎<span dir="auto"><span class="autocomment">Negative Multinomial Summary</span></span></p> <table class="diff diff-contentalign-left" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 13:27, 1 August 2014</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l149" >Line 149:</td> <td colspan="2" class="diff-lineno">Line 149:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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;"></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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;"></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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>====Negative Multinomial Summary====</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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>====Negative Multinomial Summary====</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color: #222; 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>* Probability Mass Function: &lt;math&gt; P(k_1, \cdots, k_m|k_0,\{p_1,\cdots,p_m\}) = \left (\sum_{i=0}^m{k_i}-1\right)!\frac{p_0^{k_0}}{(k_0-1)!} \prod_{i=1}^m{\frac{p_i^{k_i}}{k_i!}}&lt;/math&gt;<del class="diffchange diffchange-inline">, </del>or equivalently:</div></td><td class='diff-marker'>+</td><td style="color: #222; 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>* Probability Mass Function: &lt;math&gt; P(k_1, \cdots, k_m|k_0,\{p_1,\cdots,p_m\}) = \left (\sum_{i=0}^m{k_i}-1\right)!\frac{p_0^{k_0}}{(k_0-1)!} \prod_{i=1}^m{\frac{p_i^{k_i}}{k_i!}}<ins class="diffchange diffchange-inline">,</ins>&lt;/math&gt; or equivalently:</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color: #222; 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>: &lt;math&gt; P(k_1, \cdots, k_m|k_0,\{p_1,\cdots,p_m\}) = \Gamma\left(\sum_{i=1}^m{k_i}\right)\frac{p_0^{k_0}}{\Gamma(k_0)} \prod_{i=1}^m{\frac{p_i^{k_i}}{k_i!}}&lt;/math&gt;<del class="diffchange diffchange-inline">, </del>where &lt;math&gt;\Gamma(x)&lt;/math&gt; is the [http://en.wikipedia.org/wiki/Gamma_function Gamma function].</div></td><td class='diff-marker'>+</td><td style="color: #222; 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>: &lt;math&gt; P(k_1, \cdots, k_m|k_0,\{p_1,\cdots,p_m\}) = \Gamma\left(\sum_{i=1}^m{k_i}\right)\frac{p_0^{k_0}}{\Gamma(k_0)} \prod_{i=1}^m{\frac{p_i^{k_i}}{k_i!}}<ins class="diffchange diffchange-inline">,</ins>&lt;/math&gt; where &lt;math&gt;\Gamma(x)&lt;/math&gt; is the [http://en.wikipedia.org/wiki/Gamma_function Gamma function].</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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>* Mean (vector): &lt;math&gt;\mu=E(X_1,\cdots,X_m)= (\mu_1=E(X_1), \cdots, \mu_m=E(X_m)) = \left ( \frac{k_0p_1}{p_0}, \cdots, \frac{k_0p_m}{p_0} \right).&lt;/math&gt;</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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>* Mean (vector): &lt;math&gt;\mu=E(X_1,\cdots,X_m)= (\mu_1=E(X_1), \cdots, \mu_m=E(X_m)) = \left ( \frac{k_0p_1}{p_0}, \cdots, \frac{k_0p_m}{p_0} \right).&lt;/math&gt;</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color: #222; 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>* Variance-Covariance (matrix): &lt;math&gt;Cov(X_i,X_j)= \{cov[i,j]\}&lt;/math&gt;<del class="diffchange diffchange-inline">, </del>where &#160;</div></td><td class='diff-marker'>+</td><td style="color: #222; 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>* Variance-Covariance (matrix): &lt;math&gt;Cov(X_i,X_j)= \{cov[i,j]\}<ins class="diffchange diffchange-inline">,</ins>&lt;/math&gt; where &#160;</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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>: &lt;math&gt; cov[i,j] = \begin{cases} \frac{k_0 p_i p_j}{p_0^2},&amp; i\not= j,\\</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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>: &lt;math&gt; cov[i,j] = \begin{cases} \frac{k_0 p_i p_j}{p_0^2},&amp; i\not= j,\\</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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>\frac{k_0 p_i&#160; (p_i + p_0)}{p_0^2},&amp; i=j.\end{cases}.&lt;/math&gt;</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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>\frac{k_0 p_i&#160; (p_i + p_0)}{p_0^2},&amp; i=j.\end{cases}.&lt;/math&gt;</div></td></tr> </table>

Dinov https://wiki.socr.umich.edu/index.php?title=AP_Statistics_Curriculum_2007_Distrib_Dists&diff=13003&oldid=prev 2014-08-01T13:23:08Z

<p>‎<span dir="auto"><span class="autocomment">Negative Multinomial Distribution (NMD)</span></span></p> <table class="diff diff-contentalign-left" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 13:23, 1 August 2014</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l143" >Line 143:</td> <td colspan="2" class="diff-lineno">Line 143:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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>===Negative Multinomial Distribution (NMD)===</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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>===Negative Multinomial Distribution (NMD)===</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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 ''Negative Multinomial Distribution'' is a generalization of the two-parameter [[AP_Statistics_Curriculum_2007_Distrib_Dists#Negative_Binomial|Negative Binomial distribution]] (NB(r,p)) to &lt;math&gt;m\ge 1&lt;/math&gt; outcomes. Suppose we have an experiment that generates &lt;math&gt;m\ge 1&lt;/math&gt; possible outcomes, &lt;math&gt;\{X_0,\cdots,X_m\}&lt;/math&gt;, each occurring with probability &lt;math&gt;\{p_0,\cdots,p_m\}&lt;/math&gt;, respectively, where with &lt;math&gt;0&lt;p_i&lt;1&lt;/math&gt; and &lt;math&gt;\sum_{i=0}^m{p_i}=1&lt;/math&gt;. That is, &lt;math&gt;p_0 = 1-\sum_{i=1}^m{p_i}&lt;/math&gt;. If the experiment proceeds to generate independent outcomes until &lt;math&gt;\{X_0, X_1, \cdots, X_m\}&lt;/math&gt; occur exactly &lt;math&gt;\{k_0, k_1, \cdots, k_m\}&lt;/math&gt; times, then the distribution of the m-tuple &lt;math&gt;\{X_1, \cdots, X_m\}&lt;/math&gt; is Negative Multinomial with parameter vector &lt;math&gt;(k_0,\{p_1,\cdots,p_m\})&lt;/math&gt;. Notice that the degree-of-freedom here is actually m, not (m+1). That is why we only have a probability parameter vector of size m, not (m+1), as all probabilities add up to 1 (so this introduces one relation). Contrast this with the combinatorial interpretation of Negative Binomial (special case with m=1):</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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 ''Negative Multinomial Distribution'' is a generalization of the two-parameter [[AP_Statistics_Curriculum_2007_Distrib_Dists#Negative_Binomial|Negative Binomial distribution]] (NB(r,p)) to &lt;math&gt;m\ge 1&lt;/math&gt; outcomes. Suppose we have an experiment that generates &lt;math&gt;m\ge 1&lt;/math&gt; possible outcomes, &lt;math&gt;\{X_0,\cdots,X_m\}&lt;/math&gt;, each occurring with probability &lt;math&gt;\{p_0,\cdots,p_m\}&lt;/math&gt;, respectively, where with &lt;math&gt;0&lt;p_i&lt;1&lt;/math&gt; and &lt;math&gt;\sum_{i=0}^m{p_i}=1&lt;/math&gt;. That is, &lt;math&gt;p_0 = 1-\sum_{i=1}^m{p_i}&lt;/math&gt;. If the experiment proceeds to generate independent outcomes until &lt;math&gt;\{X_0, X_1, \cdots, X_m\}&lt;/math&gt; occur exactly &lt;math&gt;\{k_0, k_1, \cdots, k_m\}&lt;/math&gt; times, then the distribution of the m-tuple &lt;math&gt;\{X_1, \cdots, X_m\}&lt;/math&gt; is Negative Multinomial with parameter vector &lt;math&gt;(k_0,\{p_1,\cdots,p_m\})&lt;/math&gt;. Notice that the degree-of-freedom here is actually m, not (m+1). That is why we only have a probability parameter vector of size m, not (m+1), as all probabilities add up to 1 (so this introduces one relation). Contrast this with the combinatorial interpretation of Negative Binomial (special case with m=1):</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color: #222; 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>: &lt;math&gt;X \sim NegativeBinomial(NumberOfSuccesses=r,ProbOfSuccess=p)&lt;/math&gt;<del class="diffchange diffchange-inline">, </del></div></td><td class='diff-marker'>+</td><td style="color: #222; 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>: &lt;math&gt;X \sim NegativeBinomial(NumberOfSuccesses=r,ProbOfSuccess=p)<ins class="diffchange diffchange-inline">,</ins>&lt;/math&gt; &#160;</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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=Total # of experiments (n) to get r successes'' (and therefore n-r failures);</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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=Total # of experiments (n) to get r successes'' (and therefore n-r failures);</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color: #222; 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>: &lt;math&gt;X \sim Negative Multinomial(k_0,\{p_0,p_1\})&lt;/math&gt;<del class="diffchange diffchange-inline">,</del></div></td><td class='diff-marker'>+</td><td style="color: #222; 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>: &lt;math&gt;X \sim Negative Multinomial(k_0,\{p_0,p_1\})<ins class="diffchange diffchange-inline">,</ins>&lt;/math&gt;</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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=Total # of experiments (n) to get &lt;math&gt;k_0&lt;/math&gt; (default variable, &lt;math&gt;X_o&lt;/math&gt;) and &lt;math&gt;n-k_0&lt;/math&gt; outcomes of the other possible outcome (&lt;math&gt;X_1&lt;/math&gt;)''. &#160;</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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=Total # of experiments (n) to get &lt;math&gt;k_0&lt;/math&gt; (default variable, &lt;math&gt;X_o&lt;/math&gt;) and &lt;math&gt;n-k_0&lt;/math&gt; outcomes of the other possible outcome (&lt;math&gt;X_1&lt;/math&gt;)''. &#160;</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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;"></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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;"></td></tr> <tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l151" >Line 151:</td> <td colspan="2" class="diff-lineno">Line 151:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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>* Probability Mass Function: &lt;math&gt; P(k_1, \cdots, k_m|k_0,\{p_1,\cdots,p_m\}) = \left (\sum_{i=0}^m{k_i}-1\right)!\frac{p_0^{k_0}}{(k_0-1)!} \prod_{i=1}^m{\frac{p_i^{k_i}}{k_i!}}&lt;/math&gt;, or equivalently:</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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>* Probability Mass Function: &lt;math&gt; P(k_1, \cdots, k_m|k_0,\{p_1,\cdots,p_m\}) = \left (\sum_{i=0}^m{k_i}-1\right)!\frac{p_0^{k_0}}{(k_0-1)!} \prod_{i=1}^m{\frac{p_i^{k_i}}{k_i!}}&lt;/math&gt;, or equivalently:</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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>: &lt;math&gt; P(k_1, \cdots, k_m|k_0,\{p_1,\cdots,p_m\}) = \Gamma\left(\sum_{i=1}^m{k_i}\right)\frac{p_0^{k_0}}{\Gamma(k_0)} \prod_{i=1}^m{\frac{p_i^{k_i}}{k_i!}}&lt;/math&gt;, where &lt;math&gt;\Gamma(x)&lt;/math&gt; is the [http://en.wikipedia.org/wiki/Gamma_function Gamma function].</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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>: &lt;math&gt; P(k_1, \cdots, k_m|k_0,\{p_1,\cdots,p_m\}) = \Gamma\left(\sum_{i=1}^m{k_i}\right)\frac{p_0^{k_0}}{\Gamma(k_0)} \prod_{i=1}^m{\frac{p_i^{k_i}}{k_i!}}&lt;/math&gt;, where &lt;math&gt;\Gamma(x)&lt;/math&gt; is the [http://en.wikipedia.org/wiki/Gamma_function Gamma function].</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color: #222; 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>* Mean (vector): &lt;math&gt;\mu=E(X_1,\cdots,X_m)= (\mu_1=E(X_1), \cdots, \mu_m=E(X_m)) = \left ( \frac{k_0p_1}{p_0}, \cdots, \frac{k_0p_m}{p_0} \right)&lt;/math&gt;<del class="diffchange diffchange-inline">.</del></div></td><td class='diff-marker'>+</td><td style="color: #222; 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>* Mean (vector): &lt;math&gt;\mu=E(X_1,\cdots,X_m)= (\mu_1=E(X_1), \cdots, \mu_m=E(X_m)) = \left ( \frac{k_0p_1}{p_0}, \cdots, \frac{k_0p_m}{p_0} \right)<ins class="diffchange diffchange-inline">.</ins>&lt;/math&gt;</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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>* Variance-Covariance (matrix): &lt;math&gt;Cov(X_i,X_j)= \{cov[i,j]\}&lt;/math&gt;, where &#160;</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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>* Variance-Covariance (matrix): &lt;math&gt;Cov(X_i,X_j)= \{cov[i,j]\}&lt;/math&gt;, where &#160;</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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>: &lt;math&gt; cov[i,j] = \begin{cases} \frac{k_0 p_i p_j}{p_0^2},&amp; i\not= j,\\</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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>: &lt;math&gt; cov[i,j] = \begin{cases} \frac{k_0 p_i p_j}{p_0^2},&amp; i\not= j,\\</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color: #222; 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>\frac{k_0 p_i&#160; (p_i + p_0)}{p_0^2},&amp; i=j.\end{cases}&lt;/math&gt;<del class="diffchange diffchange-inline">.</del></div></td><td class='diff-marker'>+</td><td style="color: #222; 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>\frac{k_0 p_i&#160; (p_i + p_0)}{p_0^2},&amp; i=j.\end{cases}<ins class="diffchange diffchange-inline">.</ins>&lt;/math&gt;</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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;"></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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;"></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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>====Cancer Example====</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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>====Cancer Example====</div></td></tr> </table>

Dinov https://wiki.socr.umich.edu/index.php?title=AP_Statistics_Curriculum_2007_Distrib_Dists&diff=11673&oldid=prev 2012-06-23T19:41:31Z

<p>‎<span dir="auto"><span class="autocomment">Normal approximation to Negative Binomial distribution</span></span></p> <table class="diff diff-contentalign-left" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 19:41, 23 June 2012</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l110" >Line 110:</td> <td colspan="2" class="diff-lineno">Line 110:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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;"></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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;"></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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>====Normal approximation to Negative Binomial distribution====</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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>====Normal approximation to Negative Binomial distribution====</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color: #222; 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>The [[AP_Statistics_Curriculum_2007_Limits_CLT|central limit theorem]] provides the foundation for approximation of negative binomial distribution by [[AP_Statistics_Curriculum_2007_Normal_Std| Normal distribution]]. Each negative binomial random variable, \(V_k \sim NB(<del class="diffchange diffchange-inline">r</del>,p)\), may be expressed as a sum of '''k''' independent, identically distributed ([[AP_Statistics_Curriculum_2007_Distrib_Dists#Geometric|geometric]]) random variables \(\{X_i\}\), i.e., \( V_k = \sum_{i=1}^k{X_i}\), where [[AP_Statistics_Curriculum_2007_Distrib_Dists |\( X_i \sim Geometric(p)\)]]. In various scientific applications, given a large '''k''', the distribution of \(V_k\) is approximately normal with mean and variance given by \(\mu=k\frac{1}{p}\) and \(\sigma^2=k\frac{1-p}{p^2}\), as \(k \longrightarrow \infty\). Depending on the parameter '''p''', '''k''' may need to be rather large for the approximation to work well. Also, when using the normal approximation, we should remember to use the continuity correction, since the negative binomial and Normal distributions are discrete and continuous, respectively.</div></td><td class='diff-marker'>+</td><td style="color: #222; 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>The [[AP_Statistics_Curriculum_2007_Limits_CLT|central limit theorem]] provides the foundation for approximation of negative binomial distribution by [[AP_Statistics_Curriculum_2007_Normal_Std| Normal distribution]]. Each negative binomial random variable, \(V_k \sim NB(<ins class="diffchange diffchange-inline">k</ins>,p)\), may be expressed as a sum of '''k''' independent, identically distributed ([[AP_Statistics_Curriculum_2007_Distrib_Dists#Geometric|geometric]]) random variables \(\{X_i\}\), i.e., \( V_k = \sum_{i=1}^k{X_i}\), where [[AP_Statistics_Curriculum_2007_Distrib_Dists |\( X_i \sim Geometric(p)\)]]. In various scientific applications, given a large '''k''', the distribution of \(V_k\) is approximately normal with mean and variance given by \(\mu=k\frac{1}{p}\) and \(\sigma^2=k\frac{1-p}{p^2}\), as \(k \longrightarrow \infty\). Depending on the parameter '''p''', '''k''' may need to be rather large for the approximation to work well. Also, when using the normal approximation, we should remember to use the continuity correction, since the negative binomial and Normal distributions are discrete and continuous, respectively.</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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;"></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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;"></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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>In the above example, \(P(X\le 8)\), \(V_k \sim NegBin(k=r=6, p=0.3)\), the normal distribution approximation, \(N(\mu=\frac{k}{p}=20, \sigma=\sqrt{k\frac{1-p}{p^2}}=6.83)\), is shown it the following image and table:</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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>In the above example, \(P(X\le 8)\), \(V_k \sim NegBin(k=r=6, p=0.3)\), the normal distribution approximation, \(N(\mu=\frac{k}{p}=20, \sigma=\sqrt{k\frac{1-p}{p^2}}=6.83)\), is shown it the following image and table:</div></td></tr> </table>

IvoDinov https://wiki.socr.umich.edu/index.php?title=AP_Statistics_Curriculum_2007_Distrib_Dists&diff=11672&oldid=prev 2012-06-23T19:40:22Z

<p>‎<span dir="auto"><span class="autocomment">Normal approximation to Negative Binomial distribution</span></span></p> <table class="diff diff-contentalign-left" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 19:40, 23 June 2012</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l110" >Line 110:</td> <td colspan="2" class="diff-lineno">Line 110:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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;"></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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;"></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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>====Normal approximation to Negative Binomial distribution====</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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>====Normal approximation to Negative Binomial distribution====</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color: #222; 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>The [[AP_Statistics_Curriculum_2007_Limits_CLT|central limit theorem]] <del class="diffchange diffchange-inline">applies </del>provides the foundation for approximation of negative binomial distribution by [[AP_Statistics_Curriculum_2007_Normal_Std| Normal distribution]]. Each negative binomial random variable, \(V_k \sim NB(r,p)\), may be expressed as a sum of '''k''' independent, identically distributed ([[AP_Statistics_Curriculum_2007_Distrib_Dists#Geometric|geometric]]) random variables \(\{X_i\}\), i.e., \( V_k = \sum_{i=1}^k{X_i}\), where [[AP_Statistics_Curriculum_2007_Distrib_Dists |\( X_i \sim Geometric(p)\)]]. In various scientific applications, given a large '''k''', the distribution of \(V_k\) is approximately normal with mean and variance given by \(\mu=k\frac{1}{p}\) and \(\sigma^2=k\frac{1-p}{p^2}\), as \(k \longrightarrow \infty\). Depending on the parameter '''p''', '''k''' may need to be rather large for the approximation to work well. Also, when using the normal approximation, we should remember to use the continuity correction, since the negative binomial and Normal distributions are discrete and continuous, respectively.</div></td><td class='diff-marker'>+</td><td style="color: #222; 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>The [[AP_Statistics_Curriculum_2007_Limits_CLT|central limit theorem]] provides the foundation for approximation of negative binomial distribution by [[AP_Statistics_Curriculum_2007_Normal_Std| Normal distribution]]. Each negative binomial random variable, \(V_k \sim NB(r,p)\), may be expressed as a sum of '''k''' independent, identically distributed ([[AP_Statistics_Curriculum_2007_Distrib_Dists#Geometric|geometric]]) random variables \(\{X_i\}\), i.e., \( V_k = \sum_{i=1}^k{X_i}\), where [[AP_Statistics_Curriculum_2007_Distrib_Dists |\( X_i \sim Geometric(p)\)]]. In various scientific applications, given a large '''k''', the distribution of \(V_k\) is approximately normal with mean and variance given by \(\mu=k\frac{1}{p}\) and \(\sigma^2=k\frac{1-p}{p^2}\), as \(k \longrightarrow \infty\). Depending on the parameter '''p''', '''k''' may need to be rather large for the approximation to work well. Also, when using the normal approximation, we should remember to use the continuity correction, since the negative binomial and Normal distributions are discrete and continuous, respectively.</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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;"></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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;"></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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>In the above example, \(P(X\le 8)\), \(V_k \sim NegBin(k=r=6, p=0.3)\), the normal distribution approximation, \(N(\mu=\frac{k}{p}=20, \sigma=\sqrt{k\frac{1-p}{p^2}}=6.83)\), is shown it the following image and table:</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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>In the above example, \(P(X\le 8)\), \(V_k \sim NegBin(k=r=6, p=0.3)\), the normal distribution approximation, \(N(\mu=\frac{k}{p}=20, \sigma=\sqrt{k\frac{1-p}{p^2}}=6.83)\), is shown it the following image and table:</div></td></tr> </table>

IvoDinov https://wiki.socr.umich.edu/index.php?title=AP_Statistics_Curriculum_2007_Distrib_Dists&diff=11671&oldid=prev 2012-06-23T19:40:00Z

<p>‎<span dir="auto"><span class="autocomment">Normal approximation to Negative Binomial distribution</span></span></p> <table class="diff diff-contentalign-left" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 19:40, 23 June 2012</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l110" >Line 110:</td> <td colspan="2" class="diff-lineno">Line 110:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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;"></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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;"></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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>====Normal approximation to Negative Binomial distribution====</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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>====Normal approximation to Negative Binomial distribution====</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color: #222; 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>The central limit theorem applies provides the foundation for approximation of negative binomial distribution by [[AP_Statistics_Curriculum_2007_Normal_Std| Normal distribution]]. Each negative binomial random variable, \(V_k \sim NB(r,p)\), may be expressed as a sum of '''k''' independent, identically distributed ([[AP_Statistics_Curriculum_2007_Distrib_Dists#Geometric|geometric]]) random variables \(\{X_i\}\), i.e., \( V_k = \sum_{i=1}^k{X_i}\), where [[AP_Statistics_Curriculum_2007_Distrib_Dists |\( X_i \sim Geometric(p)\)]]. In various scientific applications, given a large '''k''', the distribution of \(V_k\) is approximately normal with mean and variance given by \(\mu=k\frac{1}{p}\) and \(\sigma^2=k\frac{1-p}{p^2}\), as \(k \longrightarrow \infty\). Depending on the parameter '''p''', '''k''' may need to be rather large for the approximation to work well. Also, when using the normal approximation, we should remember to use the continuity correction, since the negative binomial and Normal distributions are discrete and continuous, respectively.</div></td><td class='diff-marker'>+</td><td style="color: #222; 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>The <ins class="diffchange diffchange-inline">[[AP_Statistics_Curriculum_2007_Limits_CLT|</ins>central limit theorem<ins class="diffchange diffchange-inline">]] </ins>applies provides the foundation for approximation of negative binomial distribution by [[AP_Statistics_Curriculum_2007_Normal_Std| Normal distribution]]. Each negative binomial random variable, \(V_k \sim NB(r,p)\), may be expressed as a sum of '''k''' independent, identically distributed ([[AP_Statistics_Curriculum_2007_Distrib_Dists#Geometric|geometric]]) random variables \(\{X_i\}\), i.e., \( V_k = \sum_{i=1}^k{X_i}\), where [[AP_Statistics_Curriculum_2007_Distrib_Dists |\( X_i \sim Geometric(p)\)]]. In various scientific applications, given a large '''k''', the distribution of \(V_k\) is approximately normal with mean and variance given by \(\mu=k\frac{1}{p}\) and \(\sigma^2=k\frac{1-p}{p^2}\), as \(k \longrightarrow \infty\). Depending on the parameter '''p''', '''k''' may need to be rather large for the approximation to work well. Also, when using the normal approximation, we should remember to use the continuity correction, since the negative binomial and Normal distributions are discrete and continuous, respectively.</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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;"></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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;"></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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>In the above example, \(P(X\le 8)\), \(V_k \sim NegBin(k=r=6, p=0.3)\), the normal distribution approximation, \(N(\mu=\frac{k}{p}=20, \sigma=\sqrt{k\frac{1-p}{p^2}}=6.83)\), is shown it the following image and table:</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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>In the above example, \(P(X\le 8)\), \(V_k \sim NegBin(k=r=6, p=0.3)\), the normal distribution approximation, \(N(\mu=\frac{k}{p}=20, \sigma=\sqrt{k\frac{1-p}{p^2}}=6.83)\), is shown it the following image and table:</div></td></tr> </table>

IvoDinov https://wiki.socr.umich.edu/index.php?title=AP_Statistics_Curriculum_2007_Distrib_Dists&diff=11670&oldid=prev 2012-06-23T19:38:36Z

<p>‎<span dir="auto"><span class="autocomment">Normal approximation to Negative Binomial distribution</span></span></p> <table class="diff diff-contentalign-left" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 19:38, 23 June 2012</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l135" >Line 135:</td> <td colspan="2" class="diff-lineno">Line 135:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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>! colspan=3|Probability Areas</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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>! colspan=3|Probability Areas</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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>|-</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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>|-</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color: #222; 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>| <del class="diffchange diffchange-inline">&lt;</del>8|| .011292|| 0.039433</div></td><td class='diff-marker'>+</td><td style="color: #222; 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>| <ins class="diffchange diffchange-inline">\(\le </ins>8<ins class="diffchange diffchange-inline">\)</ins>|| .011292|| 0.039433</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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>|-</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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>|-</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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>| &gt;8|| .988708||0.960537</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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>| &gt;8|| .988708||0.960537</div></td></tr> </table>

IvoDinov https://wiki.socr.umich.edu/index.php?title=AP_Statistics_Curriculum_2007_Distrib_Dists&diff=11669&oldid=prev 2012-06-23T19:37:52Z

<p>‎<span dir="auto"><span class="autocomment">Normal approximation to Negative Binomial distribution</span></span></p> <table class="diff diff-contentalign-left" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 19:37, 23 June 2012</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l112" >Line 112:</td> <td colspan="2" class="diff-lineno">Line 112:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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 central limit theorem applies provides the foundation for approximation of negative binomial distribution by [[AP_Statistics_Curriculum_2007_Normal_Std| Normal distribution]]. Each negative binomial random variable, \(V_k \sim NB(r,p)\), may be expressed as a sum of '''k''' independent, identically distributed ([[AP_Statistics_Curriculum_2007_Distrib_Dists#Geometric|geometric]]) random variables \(\{X_i\}\), i.e., \( V_k = \sum_{i=1}^k{X_i}\), where [[AP_Statistics_Curriculum_2007_Distrib_Dists |\( X_i \sim Geometric(p)\)]]. In various scientific applications, given a large '''k''', the distribution of \(V_k\) is approximately normal with mean and variance given by \(\mu=k\frac{1}{p}\) and \(\sigma^2=k\frac{1-p}{p^2}\), as \(k \longrightarrow \infty\). Depending on the parameter '''p''', '''k''' may need to be rather large for the approximation to work well. Also, when using the normal approximation, we should remember to use the continuity correction, since the negative binomial and Normal distributions are discrete and continuous, respectively.</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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 central limit theorem applies provides the foundation for approximation of negative binomial distribution by [[AP_Statistics_Curriculum_2007_Normal_Std| Normal distribution]]. Each negative binomial random variable, \(V_k \sim NB(r,p)\), may be expressed as a sum of '''k''' independent, identically distributed ([[AP_Statistics_Curriculum_2007_Distrib_Dists#Geometric|geometric]]) random variables \(\{X_i\}\), i.e., \( V_k = \sum_{i=1}^k{X_i}\), where [[AP_Statistics_Curriculum_2007_Distrib_Dists |\( X_i \sim Geometric(p)\)]]. In various scientific applications, given a large '''k''', the distribution of \(V_k\) is approximately normal with mean and variance given by \(\mu=k\frac{1}{p}\) and \(\sigma^2=k\frac{1-p}{p^2}\), as \(k \longrightarrow \infty\). Depending on the parameter '''p''', '''k''' may need to be rather large for the approximation to work well. Also, when using the normal approximation, we should remember to use the continuity correction, since the negative binomial and Normal distributions are discrete and continuous, respectively.</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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;"></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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;"></td></tr> <tr><td class='diff-marker'>−</td><td style="color: #222; 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 the above example, \(V_k \sim NegBin(k=r=6, p=0.3)\), the normal distribution approximation, \(N(\mu=\frac{k}{p}=20, \sigma=\sqrt{k\frac{1-p}{p^2}}=6.83)\), is shown it the following image and table:</div></td><td class='diff-marker'>+</td><td style="color: #222; 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 the above example<ins class="diffchange diffchange-inline">, \(P(X\le 8)\)</ins>, \(V_k \sim NegBin(k=r=6, p=0.3)\), the normal distribution approximation, \(N(\mu=\frac{k}{p}=20, \sigma=\sqrt{k\frac{1-p}{p^2}}=6.83)\), is shown it the following image and table:</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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;"></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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;"></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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>&lt;center&gt;[[Image:SOCR_EBook_Dinov_RV_NegBinomial_013008_Fig4a.png|500px]]&lt;/center&gt;</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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>&lt;center&gt;[[Image:SOCR_EBook_Dinov_RV_NegBinomial_013008_Fig4a.png|500px]]&lt;/center&gt;</div></td></tr> </table>

IvoDinov https://wiki.socr.umich.edu/index.php?title=AP_Statistics_Curriculum_2007_Distrib_Dists&diff=11668&oldid=prev 2012-06-23T19:37:16Z

<p>‎<span dir="auto"><span class="autocomment">Normal approximation to Negative Binomial distribution</span></span></p> <table class="diff diff-contentalign-left" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 19:37, 23 June 2012</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l114" >Line 114:</td> <td colspan="2" class="diff-lineno">Line 114:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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>In the above example, \(V_k \sim NegBin(k=r=6, p=0.3)\), the normal distribution approximation, \(N(\mu=\frac{k}{p}=20, \sigma=\sqrt{k\frac{1-p}{p^2}}=6.83)\), is shown it the following image and table:</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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>In the above example, \(V_k \sim NegBin(k=r=6, p=0.3)\), the normal distribution approximation, \(N(\mu=\frac{k}{p}=20, \sigma=\sqrt{k\frac{1-p}{p^2}}=6.83)\), is shown it the following image and table:</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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;"></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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;"></td></tr> <tr><td class='diff-marker'>−</td><td style="color: #222; 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>&lt;center&gt;[[Image:<del class="diffchange diffchange-inline">EBook_Gamma_Fig2</del>.png|500px]]&lt;/center&gt;</div></td><td class='diff-marker'>+</td><td style="color: #222; 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>&lt;center&gt;[[Image:<ins class="diffchange diffchange-inline">SOCR_EBook_Dinov_RV_NegBinomial_013008_Fig4a</ins>.png|500px]]&lt;/center&gt;</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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;"></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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;"></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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 probabilities of the real [http://socr.ucla.edu/htmls/dist/NegativeBinomial_Distribution.html Negative Binomial] and [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html approximate Normal] distributions (on the range [2:4]) are not identical but are sufficiently close.</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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 probabilities of the real [http://socr.ucla.edu/htmls/dist/NegativeBinomial_Distribution.html Negative Binomial] and [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html approximate Normal] distributions (on the range [2:4]) are not identical but are sufficiently close.</div></td></tr> </table>

IvoDinov https://wiki.socr.umich.edu/index.php?title=AP_Statistics_Curriculum_2007_Distrib_Dists&diff=11666&oldid=prev 2012-06-23T19:36:09Z

<p>‎<span dir="auto"><span class="autocomment">Normal approximation to Negative Binomial distribution</span></span></p> <table class="diff diff-contentalign-left" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 19:36, 23 June 2012</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l116" >Line 116:</td> <td colspan="2" class="diff-lineno">Line 116:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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>&lt;center&gt;[[Image:EBook_Gamma_Fig2.png|500px]]&lt;/center&gt;</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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>&lt;center&gt;[[Image:EBook_Gamma_Fig2.png|500px]]&lt;/center&gt;</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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;"></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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;"></td></tr> <tr><td class='diff-marker'>−</td><td style="color: #222; 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>The probabilities of the [http://socr.ucla.edu/htmls/dist/<del class="diffchange diffchange-inline">Gamma_Distribution</del>.html <del class="diffchange diffchange-inline">real Gamma</del>] and [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html approximate Normal] distributions (on the range [2:4]) are not identical but are sufficiently close.</div></td><td class='diff-marker'>+</td><td style="color: #222; 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>The probabilities of the <ins class="diffchange diffchange-inline">real </ins>[http://socr.ucla.edu/htmls/dist/<ins class="diffchange diffchange-inline">NegativeBinomial_Distribution</ins>.html <ins class="diffchange diffchange-inline">Negative Binomial</ins>] and [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html approximate Normal] distributions (on the range [2:4]) are not identical but are sufficiently close.</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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;"></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; 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color: #222; 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>{| class=&quot;wikitable&quot; style=&quot;text-align:center; width:75%&quot; border=&quot;1&quot;</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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>|-</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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>|-</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color: #222; 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>! Summary|| [http://socr.ucla.edu/htmls/dist/<del class="diffchange diffchange-inline">Gamma_Distribution</del>.html \(NegativeBinomial(k=6,p=0.3)\) ] || [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html \(Normal(\mu=20, \sigma=6.83)\) ]</div></td><td class='diff-marker'>+</td><td style="color: #222; 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>! Summary|| [http://socr.ucla.edu/htmls/dist/<ins class="diffchange diffchange-inline">NegativeBinomial_Distribution</ins>.html \(NegativeBinomial(k=6,p=0.3)\) ] || [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html \(Normal(\mu=20, \sigma=6.83)\) ]</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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>|-</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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>|-</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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>| Mean||20.0||20.0</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; 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>| Mean||20.0||20.0</div></td></tr> </table>

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