Fix more typos

Author: Zentrik

Numbers and Sets/03.rmd

@@ -501,10 +501,10 @@ Hence we must have $b_K = a_K + 1$^[if $b_K > a_K + 1$, then $0.b_1 b_2 \dots b_
 
 2. A decimal expansion is *periodic* if, after a finite number of terms, say $\ell$ digits, it repeats in blocks, of length $k$. 
 I.e. $\exists \; \ell, k$ s.t. $d_{n + k} = d_n \; \forall \; n > \ell$. \
-A periodic decimal is rational, e.g. $x = 0.7832147147147 \dots$, $10^4 x - 7832 = 0.147147 \dots$
+A periodic decimal is rational, e.g.
 \begin{align*}
     x &= 0.7832147147147 \dots \\
-    10^4 x - 7832 &= 0.147147 \dots
+    10^4 x - 7832 &= 0.147147 \dots \\
     &= 147 \sum_{j=1}^{\infty} \frac{1}{10^{3j}} \\
     &= 147 \cdot \frac{1}{10^3} \cdot \frac{1}{1 - 10^3}
 \end{align*} so $x \in \mathbb{Q}$. \

Numbers and Sets/_book/search_index.json

@@ -533,10 +533,10 @@ <h2><span class="header-section-number">3.4</span> Decimal expansion</h2>
 Hence we must have <span class="math inline">\(b_K = a_K + 1\)</span><a href="#fn13" class="footnote-ref" id="fnref13"><sup>13</sup></a> and <span class="math inline">\(a_j = 9, b_j = 0 \; \forall \; j &gt; K\)</span>.</p></li>
 <li><p>A decimal expansion is <em>periodic</em> if, after a finite number of terms, say <span class="math inline">\(\ell\)</span> digits, it repeats in blocks, of length <span class="math inline">\(k\)</span>.
 I.e. <span class="math inline">\(\exists \; \ell, k\)</span> s.t. <span class="math inline">\(d_{n + k} = d_n \; \forall \; n &gt; \ell\)</span>.<br />
-A periodic decimal is rational, e.g. <span class="math inline">\(x = 0.7832147147147 \dots\)</span>, <span class="math inline">\(10^4 x - 7832 = 0.147147 \dots\)</span>
+A periodic decimal is rational, e.g.
 <span class="math display">\[\begin{align*}
  x &amp;= 0.7832147147147 \dots \\
- 10^4 x - 7832 &amp;= 0.147147 \dots
+ 10^4 x - 7832 &amp;= 0.147147 \dots \\
  &amp;= 147 \sum_{j=1}^{\infty} \frac{1}{10^{3j}} \\
  &amp;= 147 \cdot \frac{1}{10^3} \cdot \frac{1}{1 - 10^3}
 \end{align*}\]</span> so <span class="math inline">\(x \in \mathbb{Q}\)</span>.<br />

Numbers and Sets/_book/the-reals.html

@@ -23,7 +23,7 @@ \section{Continuous Probability}
 \end{definition} 
 
 \begin{example}[A 6-sided dice]
-ars
+    insertpicture
 \end{example} 
 
 \subsection{Properties of CDF}
@@ -1028,7 +1028,7 @@ \subsection{Inequalities for $\mathbb{E}[f(X)]$}
 
 \begin{proof}
     Convexity $\implies \forall \; x < y < z$
-    ars
+    insertpicture
     \begin{align*}
         \frac{f(y) - f(x)}{y - x} &\leq \frac{f(z) - f(y)}{z - y} \\
         \text{Let } M^- &= \sup_{x < y}  \frac{f(y) - f(x)}{y - x} \\
@@ -1135,8 +1135,7 @@ \subsection{Rejection Sampling}
 
 \begin{question}
     How do we sample uniformly on $A$?
-    ars
-    arst
+    insertpicture
 \end{question} 
 
 Goal: \begin{align*}
@@ -1175,9 +1174,11 @@ \subsection{Rejection Sampling}
     Then $U^{(1)} \sim X$.
 \end{claim} 
 
+insertpicture
+
 \begin{proof}
     \begin{align*}
-        \mathbb{P}(U^{1} \leq u) &= \mathbb{P}(U \in \text{ blue region}) \\
+        \mathbb{P}(U^{1} \leq u) &= \mathbb{P}(U \in \text{blue region}) \\
         &= \operatorname{area}(\{(x, y): x \leq u, y \leq f_X(x)\}) \\
         &= \int_{0}^{u} f_X(x) \,dx \\
         &= F_X(u).