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4 changes: 2 additions & 2 deletions 2018/03/07/ubuntu16.04.3中配置openface环境遇到的问题/index.html
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4 changes: 2 additions & 2 deletions 2018/08/03/常用的排序算法/index.html
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4 changes: 2 additions & 2 deletions 2019/08/14/Viewing-matrices/index.html
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4 changes: 2 additions & 2 deletions 2019/09/11/Point-Set-Registration-Coherent-Point-Drift/index.html
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4 changes: 2 additions & 2 deletions 2019/09/20/Euler-Lagrange方程/index.html
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4 changes: 2 additions & 2 deletions 2019/10/09/RT-with-Peter-Shirley/index.html
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4 changes: 2 additions & 2 deletions 2019/12/19/Functor-Applicative-Monad以及Monoid/index.html
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4 changes: 2 additions & 2 deletions 2020/10/03/6-828-xv6-Virtual-Memory/index.html
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4 changes: 2 additions & 2 deletions 2020/11/14/解释Qt的signal-slots的小例子/index.html
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23 changes: 17 additions & 6 deletions 2021/03/30/Inequalities/index.html
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<meta property="og:description" content="\(f(x)\),\(g(x)\) integrable on \([a,b]\)">
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<meta property="article:published_time" content="2021-03-30T02:29:35.000Z">
<meta property="article:modified_time" content="2024-05-09T09:45:18.873Z">
<meta property="article:modified_time" content="2024-07-14T09:56:47.010Z">
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Expand Down Expand Up @@ -343,9 +343,20 @@ <h3 id="schwarz">Schwarz</h3>
<h3 id="minkowski">Minkowski</h3>
<p><span
class="math display">\[\{\int_a^b[f(x)+g(x)]^2dx\}^\frac{1}{2}\leq\{\int_a^bf^2(x)dx\}^\frac{1}{2}+\{\int_a^bg^2(x)dx\}^\frac{1}{2}\]</span></p>
<h3 id="markovs-inequality">Markov's inequality</h3>
<p><span class="math inline">\(X\)</span> is nonnegative random variable
and <span class="math inline">\(a&gt;0\)</span> <span
class="math display">\[P(X\ge a)\leq\frac{E(X)}{a}\]</span></p>
<h3 id="chebychev">Chebychev</h3>
<p><span
class="math display">\[P(x&gt;t)\leq\frac{\sigma^2}{\sigma^2+(t-\mu)^2}\]</span></p>
<p><span class="math inline">\(k&gt;0\)</span> <span
class="math display">\[P(|X-\mu|\ge k\sigma) \leq
\frac{1}{k^2}\]</span></p>
<h4 id="cantellis-inequality">Cantelli's inequality</h4>
<p>to improve the sharpness of the bound, better at one-sided tail
bounds <span
class="math display">\[P(X-\mu\ge\lambda)\leq\frac{\sigma^2}{\sigma^2+\lambda^2},\lambda&gt;0\]</span>
<span class="math display">\[P(X\ge
t)\leq\frac{\sigma^2}{\sigma^2+(t-\mu)^2}\]</span></p>
<h3 id="the-classic-approximation">The Classic Approximation</h3>
<p><span class="math display">\[\int_\Omega
f(x)g(x)dx\approx\frac{\int_\Omega f(x)dx}{\int_\Omega
Expand Down Expand Up @@ -487,7 +498,7 @@ <h4 id="split-sum-integral---sum">(split sum) integral -&gt; sum</h4>

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Expand Down Expand Up @@ -542,7 +553,7 @@ <h4 id="split-sum-integral---sum">(split sum) integral -&gt; sum</h4>



<div class="post-toc-content"><ol class="nav"><li class="nav-item nav-level-3"><a class="nav-link" href="#schwarz"><span class="nav-number">1.</span> <span class="nav-text">Schwarz</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#minkowski"><span class="nav-number">2.</span> <span class="nav-text">Minkowski</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#chebychev"><span class="nav-number">3.</span> <span class="nav-text">Chebychev</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#the-classic-approximation"><span class="nav-number">4.</span> <span class="nav-text">The Classic Approximation</span></a><ol class="nav-child"><li class="nav-item nav-level-4"><a class="nav-link" href="#light-precompute-ibl"><span class="nav-number">4.1.</span> <span class="nav-text">light precompute (IBL)</span></a></li><li class="nav-item nav-level-4"><a class="nav-link" href="#shadow"><span class="nav-number">4.2.</span> <span class="nav-text">shadow</span></a></li><li class="nav-item nav-level-4"><a class="nav-link" href="#split-sum-integral---sum"><span class="nav-number">4.3.</span> <span class="nav-text">(split sum) integral -&gt; sum</span></a></li></ol></li></ol></div>
<div class="post-toc-content"><ol class="nav"><li class="nav-item nav-level-3"><a class="nav-link" href="#schwarz"><span class="nav-number">1.</span> <span class="nav-text">Schwarz</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#minkowski"><span class="nav-number">2.</span> <span class="nav-text">Minkowski</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#markovs-inequality"><span class="nav-number">3.</span> <span class="nav-text">Markov&#39;s inequality</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#chebychev"><span class="nav-number">4.</span> <span class="nav-text">Chebychev</span></a><ol class="nav-child"><li class="nav-item nav-level-4"><a class="nav-link" href="#cantellis-inequality"><span class="nav-number">4.1.</span> <span class="nav-text">Cantelli&#39;s inequality</span></a></li></ol></li><li class="nav-item nav-level-3"><a class="nav-link" href="#the-classic-approximation"><span class="nav-number">5.</span> <span class="nav-text">The Classic Approximation</span></a><ol class="nav-child"><li class="nav-item nav-level-4"><a class="nav-link" href="#light-precompute-ibl"><span class="nav-number">5.1.</span> <span class="nav-text">light precompute (IBL)</span></a></li><li class="nav-item nav-level-4"><a class="nav-link" href="#shadow"><span class="nav-number">5.2.</span> <span class="nav-text">shadow</span></a></li><li class="nav-item nav-level-4"><a class="nav-link" href="#split-sum-integral---sum"><span class="nav-number">5.3.</span> <span class="nav-text">(split sum) integral -&gt; sum</span></a></li></ol></li></ol></div>


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Expand All @@ -562,7 +573,7 @@ <h4 id="split-sum-integral---sum">(split sum) integral -&gt; sum</h4>

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