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\maketitle
+ \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}
\[ \Pr(A|B) = \frac{\Pr(A \cap B)}{\Pr(B)} \quad \text{where } \Pr(B) \ne 0 \]
\[ \Pr(A) = \Pr(A|B) \cdot \Pr(B) + \Pr(A|B^{\prime}) \cdot \Pr(B^{\prime}) \tag{law of total probability} \]
+
+ \[ \Pr(A \cap B) = \Pr(A|B) \times \Pr(B) \tag{multiplication theorem} \]
For independent events:
\begin{itemize}
- \item \(\Pr(A \cap B) = \Pr(A) \cdot \Pr(B)\)
+ \item \(\Pr(A \cap B) = \Pr(A) \times \Pr(B)\)
\item \(\Pr(A|B) = \Pr(A)\)
\item \(\Pr(B|A) = \Pr(B)\)
\end{itemize}
+ \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}.
+
+ \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.
+
+ \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{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}
+
+ \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*}
+
+
+
+
+
\end{document}