fixed small errors

exercises/exercises.Rmd

@@ -12,12 +12,14 @@ knitr::opts_chunk$set(echo = TRUE, warning = FALSE, message = FALSE, dev.args =
 
 # Probability
 ## 1. Axioms
+
 $$
 \begin{split}
-P(\mathcal{S}) = 1, \,  P(\emptyset) = 0 \\
-P(\bigcup^{\infty}_{i = 1}\mathcal{A}_i) = \sum^{\infty}_{i=1}P(\mathcal{A}_i)
+&P(\mathcal{S}) = 1, \,  P(\emptyset) = 0 \\
+&P(\bigcup^{\infty}_{i = 1}\mathcal{A}_i) = \sum^{\infty}_{i=1}P(\mathcal{A}_i) \qquad A_1, A_2, \ldots \text{are disjoint}
 \end{split}
 $$
+
 From these two axioms, three important properties can be deduced.
 
 $$

slides/model-graphs/binomial-model-bayes.tex

@@ -19,10 +19,9 @@
   \draw[->] (theta)--(y);
   \draw[->] (n)--(y);
 
-  \begin{scope}[xshift = 4.5cm, node distance = 1cm]
+  \begin{scope}[xshift = 4.5cm, yshift = -2cm, node distance = 1cm]
       \node[] at(0, 0) (a) {$\alpha = \beta = 1$};
-      \node[below of = a] (n) {$n = 20$};
-      \node[below of = n] (theta) {$\theta \sim \mbox{Beta}(\alpha, \beta)$};
+      \node[below of = a] (theta) {$\theta \sim \mbox{Beta}(\alpha, \beta)$};
       \node[below of = theta] (y) {$y \sim \mbox{Binomial}(\theta, n)$};
    \end{scope}
 

slides/model-graphs/binomial-model.tex

@@ -8,17 +8,15 @@
 \begin{document}
 
 \begin{tikzpicture}[node distance = 2cm, double distance = 2pt, minimum size=.8cm, thick]
-  \node[circle, draw=black, double] (theta) {$\hat \theta$};
+  \node[circle, draw=black] (theta) {$\theta$};
   \node[rectangle, below of=theta, draw=black, fill=lightgray] (y) {$y$};
   \node[rectangle, draw=black, fill=lightgray, left of=y, node distance = 1.5cm] (n) {$n$};
 
   \draw[->] (theta)--(y);
   \draw[->] (n)--(y);
 
-  \begin{scope}[xshift = 2.5cm, node distance = 1cm]
-      \node[] at(0, 0) (n) {$n = 20$};
-      \node[below of = n] (theta) {$\hat \theta = \frac{y}{n}$};
-      \node[below of = theta] (y) {$y \sim \mbox{Binomial}(\hat \theta, n)$};
+  \begin{scope}[xshift = 2.5cm, yshift = -2cm, node distance = 1cm]
+      \node[] at(0, 0) (y) {$y \sim \mbox{Binomial}(\theta, n)$};
    \end{scope}
 
 \end{tikzpicture}

slides/slides.Rmd

@@ -252,8 +252,8 @@ $$
 
 $$
 \begin{align*}
-P(\mathcal{S}) = 1, \,  P(\emptyset) = 0 \\
-P(\bigcup^{\infty}_{i = 1}\mathcal{A}_i) = \sum^{\infty}_{i=1}P(\mathcal{A}_i)
+&P(\mathcal{S}) = 1, \,  P(\emptyset) = 0 \\
+&P(\bigcup^{\infty}_{i = 1}\mathcal{A}_i) = \sum^{\infty}_{i=1}P(\mathcal{A}_i) \qquad A_1, A_2, \ldots \text{are disjoint}
 \end{align*}
 $$
 <center>
@@ -272,7 +272,6 @@ P(\mathcal{A}) &\leq P(\mathcal{B}) \, \, \, \, \, \, \, \text{  if   } \mathcal
 $$
 
 ## How likely is today somebody's birthday?
-- The sample space is $\mathcal{S} = \{1, 2, \ldots, 365\}$
 - There are $k = 35$ people in the room
 - We assume that no people share birthdays, i.e., sampling without replacement ($k \leq 365$)
 - Then