[methods] clean up statistics notes

diff --git a/methods/statistics.pdf b/methods/statistics.pdf
index bccb7d23b36684c039b60729dde560a93c1c6a95..0929ba09c6bc5f4402728543f81cac3f1a88f066 100644 (file)
Binary files a/methods/statistics.pdf and b/methods/statistics.pdf differ

diff --git a/methods/statistics.tex b/methods/statistics.tex
index 4c398f00c4d32db31f36dba0a9432d3317a354d6..56cd566e151553f4d58f293dae4b6a6fc016c311 100644 (file)
--- a/methods/statistics.tex
+++ b/methods/statistics.tex
@@ -11,9 +11,10 @@
 \usepackage{listings}
 \usepackage{xcolor} % used only to show the phantomed stuff
 \definecolor{cas}{HTML}{e6f0fe}
 \usepackage{listings}
 \usepackage{xcolor} % used only to show the phantomed stuff
 \definecolor{cas}{HTML}{e6f0fe}

+\usepackage{mathtools}

 
 \pagestyle{fancy}
 
 \pagestyle{fancy}

-\fancyhead[LO,LE]{Unit 3 Methods Statistics}
+\fancyhead[LO,LE]{Unit 3 Methods --- Statistics}

 \fancyhead[CO,CE]{Andrew Lorimer}
 
 \setlength\parindent{0pt}
 \fancyhead[CO,CE]{Andrew Lorimer}
 
 \setlength\parindent{0pt}


@@ -23,34 +24,33 @@
   \title{Statistics}
   \author{}
   \date{}
   \title{Statistics}
   \author{}
   \date{}

-  \maketitle
+  %\maketitle

 
   \section{Probability}
 
   \section{Probability}

-
-  \[ \Pr(A \cup B) = \Pr(A) + \Pr(B) - \Pr(A \cap B) \]
-  \[ \Pr(A \cup B) = 0 \tag{mutually exclusive} \]
-
-  \section{Conditional probability}
-
-  \[ \Pr(A|B) = \frac{\Pr(A \cap B)}{\Pr(B)} \quad \text{where } \Pr(B) \ne 0 \]

   
   

-  \[ \Pr(A) = \Pr(A|B) \cdot \Pr(B) + \Pr(A|B^{\prime}) \cdot \Pr(B^{\prime}) \tag{law of total probability} \]
-  
-  \[ \Pr(A \cap B) = \Pr(A|B) \times \Pr(B) \tag{multiplication theorem} \]
+  \subsection*{Probability theorems}

 
 

-  For independent events:
+  \begin{align*}
+    \textbf{Union:} &&\Pr(A \cup B) &= \Pr(A) + \Pr(B) - \Pr(A \cap B) \\
+    \textbf{Multiplication theorem:} &&\Pr(A \cap B) &= \Pr(A|B) \times \Pr(B) \\
+    \textbf{Conditional:} &&\Pr(A|B) &= \frac{\Pr(A \cap B)}{\Pr(B)} \\
+    \textbf{Law of total probability:} &&\Pr(A) &= \Pr(A|B) \cdot \Pr(B) + \Pr(A|B^{\prime}) \cdot \Pr(B^{\prime}) \\
+  \end{align*}
+
+  Mutually exclusive \(\implies \Pr(A \cup B) = 0\) \\

   
   

-  \begin{itemize}
-    \item \(\Pr(A \cap B) = \Pr(A) \times \Pr(B)\)
-    \item \(\Pr(A|B) = \Pr(A)\)
-    \item \(\Pr(B|A) = \Pr(B)\)
-  \end{itemize}
+  Independent events:
+  \begin{flalign*}
+    \quad \Pr(A \cap B) &= \Pr(A) \times \Pr(B)& \\
+    \Pr(A|B) &= \Pr(A) \\
+    \Pr(B|A) &= \Pr(B)
+  \end{flalign*}

 
 

-  \subsection{Discrete random distributions}
+  \subsection*{Discrete random distributions}

 
   Any experiment or activity involving chance will have a probability associated with each result or \textit{outcome}. If the outcomes have a reference to \textbf{discrete numeric values} (outcomes that can be counted), and the result is unknown, then the activity is a \textit{discrete random probability distribution}.
 
 
   Any experiment or activity involving chance will have a probability associated with each result or \textit{outcome}. If the outcomes have a reference to \textbf{discrete numeric values} (outcomes that can be counted), and the result is unknown, then the activity is a \textit{discrete random probability distribution}.
 

-  \subsubsection{Discrete probability distributions}
+  \subsubsection*{Discrete probability distributions}

   
   If an activity has outcomes whose probability values are all positive and less than one ($\implies 0 \le p(x) \le 1$), and for which the sum of all outcome probabilities is unity ($\implies \sum p(x) = 1$), then it is called a \textit{probability distribution} or \textit{probability mass} function.
 
   
   If an activity has outcomes whose probability values are all positive and less than one ($\implies 0 \le p(x) \le 1$), and for which the sum of all outcome probabilities is unity ($\implies \sum p(x) = 1$), then it is called a \textit{probability distribution} or \textit{probability mass} function.
 


@@ -58,13 +58,12 @@
     \item \textbf{Probability distribution graph} - a series of points on a cartesian axis representing results of outcomes. $\Pr(X=x)$ is on $y$-axis, $x$ is on $x$ axis.
     \item \textbf{Mean $\mu$} or \textbf{expected value} \(E(X)\) - measure of central tendency. Also known as \textit{balance point}. Centre of a symmetrical distribution.
       \begin{align*}
     \item \textbf{Probability distribution graph} - a series of points on a cartesian axis representing results of outcomes. $\Pr(X=x)$ is on $y$-axis, $x$ is on $x$ axis.
     \item \textbf{Mean $\mu$} or \textbf{expected value} \(E(X)\) - measure of central tendency. Also known as \textit{balance point}. Centre of a symmetrical distribution.
       \begin{align*}

-        \overline{x} = \mu = E(X) &= \frac{\Sigma(xf)}{\Sigma(f)} \\
-        &= \sum_{i=1}^n (x_i \cdot P(X=x_i)) \\
-        &= \int_{-\infty}^{\infty} x\cdot f(x) \> dx \quad \text{(for pdf } f \text{)}
-        &= \sum_{-\infty}^{\infty} 
+        \overline{x} = \mu = E(X) &= \frac{\Sigma \left[ x \cdot f(x) \right]}{\Sigma f} \tag{where \(f =\) absolute frequency} \\
+        &= \sum_{i=1}^n \left[ x_i \cdot \Pr(X=x_i) \right] \tag{for \(n\) values of \(x\)}\\
+        &= \int_{-\infty}^{\infty} (x\cdot f(x)) \> dx \tag{for pdf \(f\)}

       \end{align*}
     \item \textbf{Mode} - most popular value (has highest probability of \(X\) values). Multiple modes can exist if \(>1 \> X\) value have equal-highest probability. Number must exist in distribution.
       \end{align*}
     \item \textbf{Mode} - most popular value (has highest probability of \(X\) values). Multiple modes can exist if \(>1 \> X\) value have equal-highest probability. Number must exist in distribution.

-    \item \textbf{Median \(m\)} - the value of \(x\) such that \(\Pr(X \le m) = \Pr(X \ge m) = 0.5\). 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.
+    \item \textbf{Median \(m\)} - the value of \(x\) such that \(\Pr(X \le m) = \Pr(X \ge m) = 0.5\). 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.

       \[ m = X \> \text{such that} \> \int_{-\infty}^{m} f(x) dx = 0.5 \]
     \item \textbf{Variance $\sigma^2$} - measure of spread of data around the mean. Not the same magnitude as the original data. For distribution \(x_1 \mapsto p_1, x_2 \mapsto p_2, \dots, x_n \mapsto p_n\):
       \begin{align*}
       \[ m = X \> \text{such that} \> \int_{-\infty}^{m} f(x) dx = 0.5 \]
     \item \textbf{Variance $\sigma^2$} - measure of spread of data around the mean. Not the same magnitude as the original data. For distribution \(x_1 \mapsto p_1, x_2 \mapsto p_2, \dots, x_n \mapsto p_n\):
       \begin{align*}


@@ -72,17 +71,23 @@
         &= \sum (x-\mu)^2 \times \Pr(X=x) \\
         &= \sum x^2 \times p(x) - \mu^2
       \end{align*}
         &= \sum (x-\mu)^2 \times \Pr(X=x) \\
         &= \sum x^2 \times p(x) - \mu^2
       \end{align*}

-    \item \textbf{Standard deviation $\sigma$} - measure of spread in the original magnitude of the data. Found by taking square root of the variance: $\sigma =\operatorname{sd}(X)=\sqrt{\operatorname{Var}(X)}$
+    \item \textbf{Standard deviation $\sigma$} - measure of spread in the original magnitude of the data. Found by taking square root of the variance:
+      \begin{align*}
+        \sigma &= \operatorname{sd}(X) \\
+        &= \sqrt{\operatorname{Var}(X)}
+      \end{align*}

   \end{itemize}
 
   \end{itemize}
 

-  \subsubsection{Expectation theorems}
+  \subsubsection*{Expectation theorems}
+
+  For some non-linear function \(g\), the expected value \(E(g(X))\) is not equal to \(g(E(X))\).

 
   \begin{align*}
 
   \begin{align*}

-    E(aX \pm b) &= aE(X) \pm b \\
-    E(z) &= z \\
-    E(X+Y) &= E(X) + E(Y) \\
-    E(X)^n &= \Sigma x^n \cdot p(x) \\
-    &\ne [E(X)]^2
+    E(X^n) &= \Sigma x^n \cdot p(x) \tag{non-linear function} \\
+    &\ne [E(X)]^n \\
+    E(aX \pm b) &= aE(X) \pm b \tag{linear function} \\
+    E(b) &= b \tag{for constant \(b \in \mathbb{R}\)}\\
+    E(X+Y) &= E(X) + E(Y) \tag{for two random variables}

   \end{align*}
 
 
   \end{align*}
 
 


@@ -94,6 +99,7 @@
     &= \sum_{k=0}^n {n \choose k} x^k y^{n-k}
   \end{align*}
 
     &= \sum_{k=0}^n {n \choose k} x^k y^{n-k}
   \end{align*}
 

+  \subsubsection*{Patterns}

   \begin{enumerate}
     \item powers of \(x\) decrease \(n \rightarrow 0\)
     \item powers of \(y\) increase \(0 \rightarrow n\)
   \begin{enumerate}
     \item powers of \(x\) decrease \(n \rightarrow 0\)
     \item powers of \(y\) increase \(0 \rightarrow n\)


@@ -101,14 +107,19 @@
     \item Number of terms in \((x+a)^n\) expanded \& simplified is \(n+1\)
   \end{enumerate}
 
     \item Number of terms in \((x+a)^n\) expanded \& simplified is \(n+1\)
   \end{enumerate}
 

-  Combinations: \(^n\text{C}_r = {N\choose k}\) (binomial coefficient) 
+  \subsubsection*{Combinatorics}
+
+  \[ \text{Binomial coefficient:} \quad ^n\text{C}_r = {N\choose k} \]
+

   \begin{itemize}
     \item Arrangements \({n \choose k} = \frac{n!}{(n-r)}\)
     \item Combinations \({n \choose k} = \frac{n!}{r!(n-r)!}\)
     \item Note \({n \choose k} = {n \choose k-1}\)
   \end{itemize}
 
   \begin{itemize}
     \item Arrangements \({n \choose k} = \frac{n!}{(n-r)}\)
     \item Combinations \({n \choose k} = \frac{n!}{r!(n-r)!}\)
     \item Note \({n \choose k} = {n \choose k-1}\)
   \end{itemize}
 

-  \subsubsection{Pascal's Triangle}
+  \colorbox{cas}{On CAS:} (soft keyboard) \keystroke{\(\downarrow\)} \(\rightarrow\) \keystroke{Advanced} \(\rightarrow\) \verb;nCr(n,cr);
+
+  \subsubsection*{Pascal's Triangle}

 
   \begin{tabular}{>{$}l<{$\hspace{12pt}}*{13}{c}}
     n=\cr0&&&&&&&1&&&&&&\\
 
   \begin{tabular}{>{$}l<{$\hspace{12pt}}*{13}{c}}
     n=\cr0&&&&&&&1&&&&&&\\


@@ -120,50 +131,47 @@
     6&1&&6&&15&&20&&15&&6&&1
   \end{tabular}
 
     6&1&&6&&15&&20&&15&&6&&1
   \end{tabular}
 

-  \colorbox{cas}{On CAS:} (soft keys) \keystroke{\(\downarrow\)} \(\rightarrow\) \keystroke{Advanced} \(\rightarrow\) \verb;nCr(n,cr);
-

   \section{Binomial distributions}
 
   (aka Bernoulli distributions)
 
   \begin{align*}
   \section{Binomial distributions}
 
   (aka Bernoulli distributions)
 
   \begin{align*}

-    \Pr(X=x) &= {n \choose x} p^x (1-p)^{n-x} \\
+    \text{Defined by} \quad X &\sim \operatorname{Bi}(n,p) \\
+    \implies \Pr(X=x) &= {n \choose x} p^x (1-p)^{n-x} \\

     &= {n \choose x} p^x q^{n-x}
   \end{align*}
 
     &= {n \choose x} p^x q^{n-x}
   \end{align*}
 

+  where:
+  \begin{description}
+    \item \(n\) is the number of trials
+    \item There are two possible outcomes: \(S\) or \(F\)
+    \item \(\Pr(\text{success}) = p\)
+    \item \(\Pr(\text{failure}) = 1-p = q\)
+  \end{description}
+   
+  \subsection*{Conditions for a binomial variable/distribution}

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

-  If these conditions are met, then it is a Binomial Random Variable. This variable is said to have a \textit{binomial probability distribution}.
-
-  \begin{itemize}
-    \item \(n\) is the number of trials
-    \item There are two possible outcomes: \(S\) or \(F\)
-    \item \(\Pr(\text{success}) = p\)
-    \item \(\Pr(\text{failure}) = 1-p = q\)
-    \item Shorthand notation: \(X \sim \operatorname{Bi}(n,p)\)
-  \end{itemize}
-
-  \colorbox{cas}{On CAS:} Main \(\rightarrow\) Interactive \(\rightarrow\) Distribution \(\rightarrow\) \verb;binomialPDf; \\
-  Input \verb;x; (no. of successes), \verb;numtrial; (no. of trials), \verb;pos; (probbability of success)
-
-  \subsection{Applications of binomial distributions}
-
-  \[ \Pr(X \ge a) = 1 - \Pr(X < a) \]
-
-  \subsection{Expected value of a binomial distribution}
+  \subsection*{\colorbox{cas}{Solve on CAS}}
+  
+  Main \(\rightarrow\) Interactive \(\rightarrow\) Distribution \(\rightarrow\) \verb;binomialPDf;

 
 

-  \[ E(X \sim \operatorname{Bi}(n,p))=np \]
+  \hspace{2em} Input \verb;x; (no. of successes), \verb;numtrial; (no. of trials), \verb;pos; (probbability of success)

 
 

-  \subsection{Variance}
+  \subsection*{Properties of \(X \sim \operatorname{Bi}(n,p)\)}

 
 

-  \[ \sigma^2(X) = np(1-p) \]
+  \begin{align*}
+    \textbf{Mean} \hspace{-4cm} &&\mu(X) &= np \\
+    \textbf{Variance} \hspace{-4cm} &&\sigma^2(X) &= np(1-p) \\
+    \textbf{s.d.} \hspace{-4cm} &&\sigma(X) &= \sqrt{np(1-p)}
+  \end{align*}

 
 

-  \subsection{Standard deviation}
+  \subsection*{Applications of binomial distributions}

 
 

-  \[ \sigma(X) = \sqrt{np(1-p)} \]
+  \[ \Pr(X \ge a) = 1 - \Pr(X < a) \]

 
 \end{document}
 
 \end{document}