[methods] expected values

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   \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. \textit{Balance point} or \textit{expected value} of a distribution. Centre of a symmetrical distribution.
+    \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{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: $\sigma^2 = \operatorname{Var}(X) = \sum x^2 \times p(x) - \mu^2$
     \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}
 
     \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: $\sigma^2 = \operatorname{Var}(X) = \sum x^2 \times p(x) - \mu^2$
     \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 \\
+    E(X+Y) &= E(X) + E(Y) \\
+    E(X)^n &= \Sigma x^n \cdot p(x) \\
+    &\ne [E(X)]^2
+  \end{align*}
+
+
+
+
+

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