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<div class="moz-cite-prefix">Hello Masahiro,<br>
<br>
Le 18/03/2018 à 17:31, fujimoto2005 a écrit :<br>
</div>
<blockquote cite="mid:1521390702445-0.post@n3.nabble.com"
type="cite">
<pre wrap="">Let x1, x2, .., xn be an random variable of n dimensional normal
distribution. Is there any function that gives the probability of {x1> =
k1}&{x2> = k2}&...&{xn> = kn}?</pre>
</blockquote>
<br>
I thougth at the first sight that it would be possible to get this
probability with some lexicographic sorting, but it's not the case.
I don't thing that we can avoid an explicit loop. Here is a possible
direct calculation, from a list of actual samples :<br>
<pre style="font-family:Monospaced;font-style:normal;font-size:12.0;"><span style="color:rgb(0,0,0);">ns</span> <span style="color:rgb(92,92,92);">=</span> <span style="color:rgb(188,143,143);">20</span><span style="color:rgb(0,0,0);">;</span> <span style="color:rgb(100,174,100);font-style:italic;">// number of samples</span>
<span style="color:rgb(0,0,0);">nd</span> <span style="color:rgb(92,92,92);">=</span> <span style="color:rgb(188,143,143);">3</span><span style="color:rgb(0,0,0);">;</span> <span style="color:rgb(100,174,100);font-style:italic;">// number of dimensions</span>
<span style="color:rgb(0,0,0);">k</span> <span style="color:rgb(92,92,92);">=</span> <span style="color:rgb(74,85,219);">[</span><span style="color:rgb(188,143,143);">5</span> <span style="color:rgb(188,143,143);">3</span> <span style="color:rgb(188,143,143);">7</span><span style="color:rgb(74,85,219);">]</span><span style="color:rgb(0,0,0);">;</span>
<span style="color:rgb(0,0,0);">m</span> <span style="color:rgb(92,92,92);">=</span> <span style="color:rgb(50,185,185);">grand</span><span style="color:rgb(74,85,219);">(</span><span style="color:rgb(0,0,0);">ns</span><span style="color:rgb(0,0,0);">,</span><span style="color:rgb(0,0,0);">nd</span><span style="color:rgb(0,0,0);">,</span><span style="color:rgb(188,143,143);">"</span><span style="color:rgb(188,143,143);">nor</span><span style="color:rgb(188,143,143);">"</span><span style="color:rgb(0,0,0);">,</span><span style="color:rgb(188,143,143);">8</span><span style="color:rgb(0,0,0);">,</span><span style="color:rgb(188,143,143);">3</span><span style="color:rgb(74,85,219);">)</span>
<span style="color:rgb(160,32,240);">for</span> <span style="color:rgb(0,0,0);">i</span> <span style="color:rgb(92,92,92);">=</span> <span style="color:rgb(188,143,143);">1</span><span style="color:rgb(255,170,0);">:</span><span style="color:rgb(0,0,0);">nd</span>
<span style="color:rgb(74,85,219);">[</span><span style="color:rgb(0,0,0);">?</span><span style="color:rgb(0,0,0);">,</span><span style="color:rgb(0,0,0);">r</span><span style="color:rgb(74,85,219);">]</span> <span style="color:rgb(92,92,92);">=</span> <span style="color:rgb(50,185,185);">gsort</span><span style="color:rgb(74,85,219);">(</span><span style="color:rgb(0,0,0);">m</span><span style="color:rgb(74,85,219);">(</span><span style="color:rgb(255,170,0);">:</span><span style="color:rgb(0,0,0);">,</span><span style="color:rgb(0,0,0);">i</span><span style="color:rgb(74,85,219);">)</span><span style="color:rgb(74,85,219);">)</span><span style="color:rgb(0,0,0);">;</span>
<span style="color:rgb(0,0,0);">m</span> <span style="color:rgb(92,92,92);">=</span> <span style="color:rgb(0,0,0);">m</span><span style="color:rgb(74,85,219);">(</span><span style="color:rgb(0,0,0);">r</span><span style="color:rgb(0,0,0);">,</span><span style="color:rgb(255,170,0);">:</span><span style="color:rgb(74,85,219);">)</span><span style="color:rgb(0,0,0);">;</span>
<span style="color:rgb(0,0,0);">m</span> <span style="color:rgb(92,92,92);">=</span> <span style="color:rgb(0,0,0);">m</span><span style="color:rgb(74,85,219);">(</span><span style="color:rgb(50,185,185);">find</span><span style="color:rgb(74,85,219);">(</span><span style="color:rgb(0,0,0);">m</span><span style="color:rgb(74,85,219);">(</span><span style="color:rgb(255,170,0);">:</span><span style="color:rgb(0,0,0);">,</span><span style="color:rgb(0,0,0);">i</span><span style="color:rgb(74,85,219);">)</span><span style="color:rgb(92,92,92);">>=</span><span style="color:rgb(0,0,0);">k</span><span style="color:rgb(74,85,219);">(</span><span style="color:rgb(0,0,0);">i</span><span style="color:rgb(74,85,219);">)</span><span style="color:rgb(74,85,219);">)</span><span style="color:rgb(0,0,0);">,</span><span style="color:rgb(255,170,0);">:</span><span style="color:rgb(74,85,219);">)</span>
<span style="color:rgb(160,32,240);">end</span>
<span style="color:rgb(0,0,0);">p</span> <span style="color:rgb(92,92,92);">=</span> <span style="color:rgb(50,185,185);">size</span><span style="color:rgb(74,85,219);">(</span><span style="color:rgb(0,0,0);">m</span><span style="color:rgb(0,0,0);">,</span><span style="color:rgb(188,143,143);">1</span><span style="color:rgb(74,85,219);">)</span><span style="color:rgb(92,92,92);">/</span><span style="color:rgb(0,0,0);">ns</span> <span style="color:rgb(100,174,100);font-style:italic;">// requested probability</span></pre>
Example of run :<br>
m = <br>
12.750144 7.1200267 5.900484 <br>
12.575508 5.410083 10.976422<br>
12.399993 3.7475677 10.092495<br>
12.220889 5.4940195 6.1479044<br>
12.158487 5.2062957 11.651957<br>
11.611694 6.6661928 5.9750098<br>
10.718146 2.2454739 11.737011<br>
10.343892 4.2714818 4.7199587<br>
9.9907016 7.7903253 5.9802778<br>
8.3135823 4.8094984 7.1769228<br>
8.0939865 9.2484944 13.215993<br>
8.0098652 3.0198012 7.5767533<br>
7.5354006 10.715856 8.985266 <br>
7.4971339 17.821625 5.5456382<br>
6.6502306 8.7791304 8.812858 <br>
6.5728805 12.299302 7.9823783<br>
6.2940806 10.376389 7.8221558<br>
5.2862072 6.0566186 11.102784<br>
<br>
m = <br>
7.4971339 17.821625 5.5456382<br>
6.5728805 12.299302 7.9823783<br>
7.5354006 10.715856 8.985266 <br>
6.2940806 10.376389 7.8221558<br>
8.0939865 9.2484944 13.215993<br>
6.6502306 8.7791304 8.812858 <br>
9.9907016 7.7903253 5.9802778<br>
12.750144 7.1200267 5.900484 <br>
11.611694 6.6661928 5.9750098<br>
5.2862072 6.0566186 11.102784<br>
12.220889 5.4940195 6.1479044<br>
12.575508 5.410083 10.976422<br>
12.158487 5.2062957 11.651957<br>
8.3135823 4.8094984 7.1769228<br>
10.343892 4.2714818 4.7199587<br>
<br>
m = <br>
8.0939865 9.2484944 13.215993<br>
12.158487 5.2062957 11.651957<br>
5.2862072 6.0566186 11.102784<br>
12.575508 5.410083 10.976422<br>
7.5354006 10.715856 8.985266 <br>
6.6502306 8.7791304 8.812858 <br>
<br>
<br>
--> p = size(m,1)/ns // requested probability<br>
p = <br>
0.3<br>
<br>
Samuel<br>
<br>
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