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<h1>The moment of truncated random variables</h1>
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<span> Posted on Sat 08 December 2018 in <a href="https://newptcai.github.io/category/math.html" style="font-style: italic">math</a>
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<p>Given a random variable <span class="math">\(Y\)</span>, sometimes we want to compute the expectation of <span class="math">\(Y[Y\le a]\)</span>, where
<span class="math">\([P]=1\)</span> if the <span class="math">\(P\)</span> and true and <span class="math">\([P]=0\)</span> if <span class="math">\(P\)</span> is false. (The notation is called <a href="https://en.wikipedia.org/wiki/Iverson_bracket">Iverson
Bracket</a>.) </p>
<p>In Patrice, it can be convenient to write <span class="math">\(\mathbb E(Y[Y \le a])\)</span> in terms of the left and right tails of
<span class="math">\(Y\)</span>. With just a bit calculus, it is not difficult to see that if <span class="math">\(Y \ge 0\)</span> and <span class="math">\(p \ge 1\)</span>,
</p>
<div class="math">\begin{align}
\mathbb E(Y^p[Y \le a])
&
=
a^p \mathbb P(Y \le a)-\int_0^ap y^{p-1} \mathbb P(Y \le y) \, dy
\\\\
&
=
\int_0^ap y^{p-1} P(Y>a) \, dy-a^p \mathbb P(Y>a)
\end{align}</div>
<p>Similarly
</p>
<div class="math">\begin{align}
\mathbb E(e^{-Y}[Y \le a])
&
=
\int_0^ae^{-y} \mathbb P(Y \le a) \mathbb \, d y+e^{-a} \mathbb P(Y \le a)
\\\\
&
=
1-\int_0^a e^{-y} P(Y > a) \, dy-e^{-a} P(Y > a)
\end{align}</div>
<p>I recently put these two qualities in a Mathematica package <code>CalcMoment.m</code>. You can find it <a href="https://github.com/newptcai/Zeno">here</a>.
You can load it and try it a Mathematica notebook like this</p>
<div class="highlight"><pre><span></span><code><span class="n">SetDirectory</span><span class="p">[</span><span class="n">NotebookDirectory</span><span class="p">[]];</span><span class="w"></span>
<span class="n">Get</span><span class="p">[</span><span class="s">"CalcMoment`"</span><span class="p">]</span><span class="w"></span>
<span class="n">gdist</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">GammaDistribution</span><span class="p">[</span><span class="mi">3</span><span class="p">,</span><span class="w"> </span><span class="mi">7</span><span class="p">]</span><span class="w"></span>
<span class="n">gtest</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">TruncatedMoment</span><span class="p">[</span><span class="n">gdist</span><span class="p">,</span><span class="w"> </span><span class="n">p</span><span class="p">,</span><span class="w"> </span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">TruncatedMoment</span><span class="p">[</span><span class="n">gdist</span><span class="p">,</span><span class="w"> </span><span class="n">p</span><span class="p">,</span><span class="w"> </span><span class="n">a</span><span class="p">,</span><span class="w"> </span><span class="n">MomentForm</span><span class="w"> </span><span class="o">-></span><span class="w"> </span><span class="s">"Left"</span><span class="p">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">TruncatedMoment</span><span class="p">[</span><span class="n">gdist</span><span class="p">,</span><span class="w"> </span><span class="n">p</span><span class="p">,</span><span class="w"> </span><span class="n">a</span><span class="p">,</span><span class="w"> </span><span class="n">MomentForm</span><span class="w"> </span><span class="o">-></span><span class="w"> </span><span class="s">"Right"</span><span class="p">]</span><span class="w"></span>
<span class="n">Assuming</span><span class="p">[</span><span class="mi">0</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">&&</span><span class="w"> </span><span class="n">p</span><span class="w"> </span><span class="o">>=</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">gtest</span><span class="w"> </span><span class="o">//</span><span class="w"> </span><span class="n">Activate</span><span class="w"> </span><span class="o">//</span><span class="w"> </span><span class="n">FullSimplify</span><span class="p">]</span><span class="w"></span>
<span class="n">edist</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">ExponentialDistribution</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="w"></span>
<span class="n">etest</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">TruncatedExpMoment</span><span class="p">[</span><span class="n">edist</span><span class="p">,</span><span class="w"> </span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">TruncatedExpMoment</span><span class="p">[</span><span class="n">edist</span><span class="p">,</span><span class="w"> </span><span class="n">a</span><span class="p">,</span><span class="w"> </span><span class="n">MomentForm</span><span class="w"> </span><span class="o">-></span><span class="w"> </span><span class="s">"Left"</span><span class="p">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">TruncatedExpMoment</span><span class="p">[</span><span class="n">edist</span><span class="p">,</span><span class="w"> </span><span class="n">a</span><span class="p">,</span><span class="w"> </span><span class="n">MomentForm</span><span class="w"> </span><span class="o">-></span><span class="w"> </span><span class="s">"Right"</span><span class="p">]</span><span class="w"></span>
<span class="n">Assuming</span><span class="p">[</span><span class="mi">0</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">a</span><span class="p">,</span><span class="w"> </span><span class="n">etest</span><span class="w"> </span><span class="o">//</span><span class="w"> </span><span class="n">Activate</span><span class="w"> </span><span class="o">//</span><span class="w"> </span><span class="n">FullSimplify</span><span class="p">]</span><span class="w"></span>
</code></pre></div>
<p>The output is of course <code>True</code>.</p>
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