[methods] binomial theorem

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+++ b/methods/statistics.tex
@@ -1,5 +1,6 @@
 \documentclass[a4paper]{article}
 \usepackage[a4paper,margin=2cm]{geometry}
 \documentclass[a4paper]{article}
 \usepackage[a4paper,margin=2cm]{geometry}

+\usepackage{array}

 \usepackage{amsmath}
 \usepackage{amssymb}
 \usepackage{tcolorbox}
 \usepackage{amsmath}
 \usepackage{amssymb}
 \usepackage{tcolorbox}


@@ -51,17 +52,27 @@
 
   \begin{itemize}
     \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.
 
   \begin{itemize}
     \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 $\mu$} - measure of central tendency. Also known as \textit{balance point} or \textit{expected value} of a distribution. Centre of a symmetrical distribution.
-    \item \textbf{Mode} - most popular value (has highest probability of \(X\) values). Multiple modes can exist if \(>1 \> X\) value have equal-highest probability.
+    \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} 
+      \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.

-    \item \textbf{Variance $\sigma^2$} - measure of spread of data around the mean. Not the same magnitude as the original data. Represented by $\sigma^2=\operatorname{Var}(x) = \sum (x=\mu)^2 \times p(x) = \sum (x-\mu)^2 \times \Pr(X=x)$. Alternatively: $\sigma^2 = \operatorname{Var}(X) = \sum x^2 \times p(x) - \mu^2$
+      \[ m = X \> \text{such that} \> \int_{-\infty}^{m} f(x) dx = 0.5 \]
+    \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*}
+        \sigma^2=\operatorname{Var}(x) &= \sum_{i=1}^n p_i (x_i-\mu)^2 \\
+        &= \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: $\sigma =\operatorname{sd}(X)=\sqrt{\operatorname{Var}(X)}$
   \end{itemize}
 
   \subsubsection{Expectation theorems}
 
     \item \textbf{Standard deviation $\sigma$} - measure of spread in the original magnitude of the data. Found by taking square root of the variance: $\sigma =\operatorname{sd}(X)=\sqrt{\operatorname{Var}(X)}$
   \end{itemize}
 
   \subsubsection{Expectation theorems}
 

-  \[ \overline{x} = \frac{\Sigma(xf)}{\Sigma(f)} = \Sigma (x p(x)) \tag{expected value} \]
-

   \begin{align*}
     E(aX \pm b) &= aE(X) \pm b \\
     E(z) &= z \\
   \begin{align*}
     E(aX \pm b) &= aE(X) \pm b \\
     E(z) &= z \\


@@ -71,7 +82,38 @@
   \end{align*}
 
 
   \end{align*}
 
 

+  \section{Binomial Theorem}
+
+  \begin{align*}
+    (x+y)^n &= {n \choose 0} x^n y^0 + {n \choose 1} x^{n-1}y^1 + {n \choose 2} x^{n-2}y^2 + \dots + {n \choose n-1}x^1 y^{n-1} + {n \choose n} x^0 y^n \\
+    &= \sum_{k=0}^n {n \choose k} x^{n-k} y^k \\
+    &= \sum_{k=0}^n {n \choose k} x^k y^{n-k}
+  \end{align*}

 
 

+  \begin{enumerate}
+    \item powers of \(x\) decrease \(n \rightarrow 0\)
+    \item powers of \(y\) increase \(0 \rightarrow n\)
+    \item coefficients are given by \(n\)th row of Pascal's Triangle where \(n=0\) has one term
+    \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) 
+  \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}
+
+  \begin{tabular}{>{$}l<{$\hspace{12pt}}*{13}{c}}
+    n=\cr0&&&&&&&1&&&&&&\\
+    1&&&&&&1&&1&&&&&\\
+    2&&&&&1&&2&&1&&&&\\
+    3&&&&1&&3&&3&&1&&&\\
+    4&&&1&&4&&6&&4&&1&&\\
+    5&&1&&5&&10&&10&&5&&1&\\
+    6&1&&6&&15&&20&&15&&6&&1
+  \end{tabular}

 
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