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<h1>Another version of Chernoff's bound and a Mathematica package for it</h1>
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<span> Posted on Sun 02 December 2018 in <a href="https://newptcai.github.io/category/math.html" style="font-style: italic">math</a>
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<p><a href="">Chernoff's bound</a> is a group of quite well-known concentration-inequalities. I though I know it
but last week my collage <a href="http://katalog.uu.se/empinfo/?id=N18-1440">Gabriel Berzunza Ojeda</a> taught
me another version of it. Let <span class="math">\(X\)</span> be a binomial <span class="math">\((n,p)\)</span> random variable. Then
</p>
<div class="math">$$
\mathbb P \left(X \ge (1+\delta) n p + \epsilon \right) \le (\delta +1)^{-\epsilon } \left(e^{\delta } (\delta +1)^{-\delta -1}\right)^{n p}
$$</div>
<p>
and
</p>
<div class="math">$$
\mathbb P \left(X \le (1-\delta) n p - \epsilon \right) \le (1-\delta )^{\epsilon } \left(e^{-\delta } (1-\delta )^{\delta -1}\right)^{n p}
$$</div>
<p>A simpler version is
</p>
<div class="math">$$
\mathbb P \left(X \ge (1+\delta) n p + \epsilon \right) \le (\delta +1)^{-\epsilon } e^{\frac{1}{2} (\delta -1) \delta ^2 n p}
$$</div>
<p>
and
</p>
<div class="math">$$
\mathbb P \left(X \le (1-\delta) n p - \epsilon \right) \le (1-\delta )^{\epsilon } e^{\frac{1}{2} \delta ^3 n p-\frac{1}{2} \delta ^2 n p}
$$</div>
<p>An even simpler version is
</p>
<div class="math">$$
\mathbb P \left(X \ge (1+\delta) n p + \epsilon \right) \le (\delta +1)^{-\epsilon } e^{-\frac{1}{3} a^2 n p}
$$</div>
<p>
and
</p>
<div class="math">$$
\mathbb P \left(X \le (1-\delta) n p - \epsilon \right) \le (1-\delta )^{\epsilon } e^{-\frac{1}{3} a^2 n p}
$$</div>
<p>These inequalities can be handy if you are deal with a <span class="math">\(\epsilon\)</span> which is <span class="math">\(o(n p)\)</span> and <span class="math">\(\delta\)</span> is a constant.
The proof simply follows a small modification of <a href="https://en.wikipedia.org/wiki/Chernoff_bound#Multiplicative_form">this proof on
Wikipedia</a>.</p>
<p>Anyway, as a practice for programming in Mathematica, I created a Mathematica package to collect the
forms of Chernoff's bound that now I know. You can find the package here. I put it on GitHub and
you can find it <a href="https://github.com/newptcai/Zeno/tree/master/ProbChopper">here</a>.</p>
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