[methods] expand probability (conf. intervals, sample proportions)

diff --git a/methods/methods-collated.pdf b/methods/methods-collated.pdf
index ba671ee199b7dc2aa4800790c03ea9c9ee2df8ff..776aa119c23ac4e55c0d1a609e1dbf9f6805e2f3 100644 (file)
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diff --git a/methods/methods-collated.tex b/methods/methods-collated.tex
index b3b29d52f96feaab97b4bbcd2641a718ca93295b..a3a1e796dc6cd2ae5b1aa0b637147ffbc0a1a394 100644 (file)
--- a/methods/methods-collated.tex
+++ b/methods/methods-collated.tex
@@ -42,6 +42,7 @@
   decorations.text,
   decorations.pathreplacing,
   decorations.text,
   decorations.text,
   decorations.pathreplacing,
   decorations.text,

+  patterns,

   scopes
 }
 
   scopes
 }
 


@@ -89,6 +90,7 @@
 
 \definecolor{cas}{HTML}{e6f0fe}
 \definecolor{important}{HTML}{fc9871}
 
 \definecolor{cas}{HTML}{e6f0fe}
 \definecolor{important}{HTML}{fc9871}

+\definecolor{highlight}{HTML}{ffb84d}

 \definecolor{dark-gray}{gray}{0.2}
 \definecolor{peach}{HTML}{e6beb2}
 \definecolor{lblue}{HTML}{e5e9f0}
 \definecolor{dark-gray}{gray}{0.2}
 \definecolor{peach}{HTML}{e6beb2}
 \definecolor{lblue}{HTML}{e5e9f0}

diff --git a/methods/statistics-ref.pdf b/methods/statistics-ref.pdf
index 2cbc960ae2df8b897dda5ebda739d36bde7af713..b0484e061fad7b0d5c627cf4500d3cbf2163e8fe 100644 (file)
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diff --git a/methods/statistics-ref.tex b/methods/statistics-ref.tex
index 196455f4f6559b2ade4da64913724b3515508f2d..87319996ce78586398711d71132ff817f9babf6d 100644 (file)
--- a/methods/statistics-ref.tex
+++ b/methods/statistics-ref.tex
@@ -13,7 +13,7 @@
   \Pr(A) &= \Pr(A|B) \cdot \Pr(B) + \Pr(A|B^{\prime}) \cdot \Pr(B^{\prime})
 \end{align*}
 
   \Pr(A) &= \Pr(A|B) \cdot \Pr(B) + \Pr(A|B^{\prime}) \cdot \Pr(B^{\prime})
 \end{align*}
 

-Mutually exclusive \(\implies \Pr(A \cup B) = 0\) \\
+Mutually exclusive: \(\Pr(A \cap B) = 0\) \\

 
 Independent events:
 \begin{flalign*}
 
 Independent events:
 \begin{flalign*}


@@ -24,17 +24,37 @@ Independent events:
 
 \subsection*{Combinatorics}
 
 
 \subsection*{Combinatorics}
 

-\begin{itemize}
+\begin{itemize} \tightlist

   \item Arrangements \({n \choose k} = \frac{n!}{(n-k)}\)
   \item Arrangements \({n \choose k} = \frac{n!}{(n-k)}\)

-  \item \colorbox{important}{Combinations} \({n \choose k} = \frac{n!}{k!(n-k)!}\)
+  \item \colorbox{highlight}{Combinations} \({n \choose k} = \frac{n!}{k!(n-k)!}\)

   \item Note \({n \choose k} = {n \choose k-1}\)
 \end{itemize}
 
 \subsection*{Distributions}
 
   \item Note \({n \choose k} = {n \choose k-1}\)
 \end{itemize}
 
 \subsection*{Distributions}
 

-\subsubsection*{Mean \(\mu\)}
+\begin{tikzpicture}
+  \begin{axis}[axis lines=left,
+    ticks=none,
+    xmin=0,
+    ymax=0.5,
+    enlargelimits=upper,
+    ylabel={\(\Pr(X=x)\)},
+    xlabel={\(x\)},
+    every axis x label/.style={at={(current axis.right of origin)},anchor=north west},
+    every axis y label/.style={at={(axis description cs:-0.02,0.5)}, anchor=south west, rotate=90},
+    ]
+    \fill[pattern=north east lines, pattern color=orange] (0,0)  -- plot[domain=0:1.68, samples=50] function {abs(x)*exp(-x)} -- (1.68,0) -- cycle;
+    \fill[pattern=north west lines, pattern color=red] (1.68,0)  -- plot[domain=1.68:5, samples=50] function {abs(x)*exp(-x)} -- (5,0) -- cycle;
+    \draw[dashed, blue, very thick] (axis cs:1.68,0) -- (axis cs:1.68,0.31) node [above, anchor=south west, black] {Median};
+    \draw[dashed, blue, very thick] (axis cs:2,0) -- (axis cs:2,0.27) node [above, anchor=west, black] {Mean};
+    \draw[dashed, blue, very thick] (axis cs:1,0) -- (axis cs:1,0.365) node [above, black] {Mode};
+    \node at (1,0.18) {\textbf{50\%}};
+    \node at (3.1,0.08) {\textbf{50\%}};
+    \addplot[thick, black, no markers, samples=200, domain=0:5] {abs(x)*exp(-x)};
+  \end{axis}
+\end{tikzpicture}

 
 

-\textbf{Mean} \(\mu\) or \textbf{expected value} \(E(X)\)
+\subsubsection*{Mean \(\mu\)}

 
 \begin{align*}
   E(X) &= \frac{\Sigma \left[ x \cdot f(x) \right]}{\Sigma f} \tag{\(f =\) absolute frequency} \\
 
 \begin{align*}
   E(X) &= \frac{\Sigma \left[ x \cdot f(x) \right]}{\Sigma f} \tag{\(f =\) absolute frequency} \\


@@ -44,13 +64,31 @@ Independent events:
 
 \subsubsection*{Mode}
 
 
 \subsubsection*{Mode}
 

-Most popular value (has highest probability of all \(X\) values). Multiple modes can exist if \(>1 \> X\) value have equal-highest probability. Number must exist in distribution.
+Value of \(X\) which has the highest probability
+
+\begin{itemize} \tightlist
+  \item Most popular value in discrete distributions
+  \item Must exist in distribution
+  \item Represented by local max in pdf
+  \item Multiple modes exist when \(>1 \> X\) value have equal-highest probability
+\end{itemize}

 
 \subsubsection*{Median}
 
 
 \subsubsection*{Median}
 

-If \(m > 0.5\), then value of \(X\) that is reached is the median of \(X\). If \(m = 0.5 = 0.5\), then \(m\) is halfway between this value and the next. To find \(m\), add values of \(X\) from smallest to alrgest until the sum reaches 0.5.
+Value separating lower and upper half of distribution area

 
 

-\[ m = X \> \text{such that} \> \int_{-\infty}^{m} f(x) dx = 0.5 \]
+\textbf{Continuous:}
+\[ m = X \> \text{such that} \> \int_{-\infty}^{m} f(x) \> dx = 0.5 \]
+
+\textbf{Discrete:} (not in course)
+\begin{itemize} \tightlist
+  \item Does not have to exist in distribution
+  \item Add values of \(X\) smallest to largest until sum is \(\ge 0.5\)
+  \item If \(X_1 < 0.5 < X_2\), then median is the average of \(X_1\) and \(X_2\)
+  \begin{itemize}\tightlist
+    \item If \(m > 0.5\), then value of \(X\) that is reached is the median of \(X\)
+  \end{itemize}
+\end{itemize}

 
 \subsubsection*{Variance \(\sigma^2\)}
 
 
 \subsubsection*{Variance \(\sigma^2\)}
 


@@ -72,7 +110,7 @@ If \(m > 0.5\), then value of \(X\) that is reached is the median of \(X\). If \
 \subsection*{Binomial distributions}
 
 Conditions for a \textit{binomial distribution}:
 \subsection*{Binomial distributions}
 
 Conditions for a \textit{binomial distribution}:

-\begin{enumerate}
+\begin{enumerate} \tightlist

   \item Two possible outcomes: \textbf{success} or \textbf{failure}
   \item \(\Pr(\text{success})\) (=\(p\)) is constant across trials
   \item Finite number \(n\) of independent trials
   \item Two possible outcomes: \textbf{success} or \textbf{failure}
   \item \(\Pr(\text{success})\) (=\(p\)) is constant across trials
   \item Finite number \(n\) of independent trials


@@ -115,7 +153,17 @@ A continuous random variable \(X\) has a pdf \(f\) such that:
 
 \begin{cas}
   Define piecewise functions: \\
 
 \begin{cas}
   Define piecewise functions: \\

-  Math3 \(\rightarrow\) 
+  \-\hspace{1em}Math3 \(\rightarrow\)
+  \begin{tikzpicture}%
+    \draw rectangle (0.5,0.5); 
+    \node at (0.08,0.25) {\(\{\)};
+    \filldraw [black] (0.15, 0.4) rectangle(0.25, 0.3);
+    \draw (0.35, 0.4) rectangle(0.45, 0.3);
+    \node [font=\footnotesize] at (0.3,0.3) {\verb;,;};
+    \draw (0.15, 0.2) rectangle(0.25, 0.1);
+    \node [font=\footnotesize] at (0.3,0.1) {\verb;,;};
+    \draw (0.35, 0.2) rectangle(0.45, 0.1);
+  \end{tikzpicture}

   % TODO: finish this section
 \end{cas}
 
   % TODO: finish this section
 \end{cas}
 


@@ -197,6 +245,20 @@ For a new distribution with mean of \(n\) trials, \(\operatorname{E}(X^\prime) =
 
 \end{cas}
 
 
 \end{cas}
 

+\subsection*{Population sampling}
+
+\subsubsection*{Population proportion}
+
+\[ p = \dfrac{n \text{ with attribute in population}}{\text{population size}} \]
+
+Constant for a given population.
+
+\subsection*{Sample proportion}
+
+\[ \hat{p} = \dfrac{n \text{ with attribute in sample}}{\text{sample size}} \]
+
+Varies with each sample.
+

 \subsection*{Normal distributions}
 
 
 \subsection*{Normal distributions}
 
 


@@ -217,6 +279,11 @@ Normal distributions must have area (total prob.) of 1 \(\implies \int^\infty_{-
   \item \(C\)\% confidence interval \(\implies\) \(C\)\% of samples will contain population mean \(\mu\)
 \end{itemize}
 
   \item \(C\)\% confidence interval \(\implies\) \(C\)\% of samples will contain population mean \(\mu\)
 \end{itemize}
 

+\begin{cas}
+  Menu \(\rightarrow\) Stats \(\rightarrow\) Calc \(\rightarrow\) Interval \\
+  Set \textit{Type = One-Sample Z Int} \\ \-\hspace{1em} and select \textit{Variable}
+\end{cas}
+

 \subsubsection*{95\% confidence interval}
 
 For 95\% c.i. of population mean \(\mu\):
 \subsubsection*{95\% confidence interval}
 
 For 95\% c.i. of population mean \(\mu\):


@@ -230,10 +297,9 @@ where:
   \item \(n\) is the sample size from which \(\overline{x}\) was calculated
 \end{description}
 
   \item \(n\) is the sample size from which \(\overline{x}\) was calculated
 \end{description}
 

-\begin{cas}
-  Menu \(\rightarrow\) Stats \(\rightarrow\) Calc \(\rightarrow\) Interval \\
-  Set \textit{Type = One-Sample Z Int} \\ \-\hspace{1em} and select \textit{Variable}
-\end{cas}
+\subsubsection*{Confidence interval of \(p\) from \(\hat{p}\)}
+
+\[ x \in \left( \hat{p} \pm Z \sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}} \right) \]

 
 \subsection*{Margin of error}
 
 
 \subsection*{Margin of error}
 

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