raneys: start pf of lemma 9, add g_k outline img.

I found it challenging to get the image close to a quality level I'm
ok with. I wish I knew how to set it up entirely as a vector-based
image. I also wish I knew how to exclude the figure caption in the pdf
output; in that case I'd have excluded the caption entirely in both
output formats.

raneys_lemmas/images/g_k_outlines.key/Metadata/BuildVersionHistory.plist

@@ -0,0 +1,8 @@
+<?xml version="1.0" encoding="UTF-8"?>
+<!DOCTYPE plist PUBLIC "-//Apple//DTD PLIST 1.0//EN" "http://www.apple.com/DTDs/PropertyList-1.0.dtd">
+<plist version="1.0">
+<array>
+	<string>Template: White (2014-02-28 09:41)</string>
+	<string>M6.2.2-1878-1</string>
+</array>
+</plist>

raneys_lemmas/images/g_k_outlines.key/Metadata/DocumentIdentifier

@@ -0,0 +1 @@
+760BE4B3-99A5-4BB6-81E8-65B3CC76955A

raneys_lemmas/raneys_lemmas.html

@@ -251,8 +251,15 @@ <h1 id="random-sequences"><span class="header-section-number">3</span> Random se
                      = \sum_{k=0}^{n-1}\text{Pr}(\rho_k(x)\text{ is sum-positive}).\]</span> Since the elements of <span class="math">\(x\)</span> are chosen independently, and with an identical distribution, we also have <span class="math">\[\text{Pr}(\rho_i(x)\text{ is sum-positive}) =
         \text{Pr}(\rho_j(x)\text{ is sum-positive}) \quad \forall i,j.\]</span> In other words, <span class="math">\[E\,\sigma(x) = n \, \text{Pr}(x\text{ is sum-positive})
                      = n \, \frac{(2n-1)!!}{(2n)!!},\]</span> using lemma 9 for the final equation. <span class="math">\(\Box\)</span></p>
+      <p>Now we're ready to dive into the proof of lemma 9.</p>
+      <p><strong>Proof of lemma 9</strong>  Our goal is to study the event <span class="math">\(e_n = (s_k &gt; 0 \,\forall\, k:1\le k \le n)\)</span>. To this end, we'll study the distribution of the random variable <span class="math">\((s_k | e_{k-1})\)</span>; that is, the value of <span class="math">\(s_k\)</span> given that <span class="math">\(s_j &gt; 0\)</span> for <span class="math">\(j &lt; k\)</span>. Let <span class="math">\(g_k\)</span> be the probability density function of <span class="math">\((s_k | e_{k-1})\)</span>, characterized by <span class="math">\(\text{Pr}(s_k \in A \,|\, e_{k-1}) = \int_A g_k(x)dx\)</span>. The first few <span class="math">\(g_k\)</span> functions are illustrated below.</p>
+      <div class="figure">
+      <img src="images/g_k_outlines.png" alt="Outlines of the probability density functions g_k." />
+      <p class="caption"><em>Outlines of the probability density functions <span class="math">\(g_k\)</span>.</em></p>
+      </div>
       <hr />
       <p>(TODO add code that checks this; consider the case <span class="math">\(x_i\in[-1, 1]\)</span> but don't spend too much time on it if it's tricky)</p>
+      <p>(TODO show that lemmas 1 and 2 are corollaries of property 5)</p>
       <div class="references">
       <h1 id="references" class="unnumbered">References</h1>
       <p>Knuth, Donald E., Oren Patashnik, and Ronald L. Graham. 1998. <em>Concrete Mathematics: A Foundation for Computer Science</em>. addison-wesley.</p>

raneys_lemmas/raneys_lemmas.md

@@ -598,6 +598,20 @@ $$E\,\sigma(x) = n \, \text{Pr}(x\text{ is sum-positive})
 using lemma 9 for the final equation.
 $\Box$
 
+Now we're ready to dive into the proof of lemma 9.
+
+**Proof of lemma 9**\ 
+Our goal is to study the event
+$e_n = (s_k > 0 \,\forall\, k:1\le k \le n)$.
+To this end, we'll study the distribution of the random variable
+$(s_k | e_{k-1})$; that is, the value of $s_k$ given that
+$s_j > 0$ for $j < k$.
+Let $g_k$ be the probability density function of $(s_k | e_{k-1})$,
+characterized by $\text{Pr}(s_k \in A \,|\, e_{k-1}) = \int_A g_k(x)dx$.
+The first few $g_k$ functions are illustrated below.
+
+![*Outlines of the probability density functions $g_k$.*](images/g_k_outlines.png)
+
 ---
 
 (TODO add code that checks this; consider the case $x_i\in[-1, 1]$ but