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z!DxFwkhL}g=O#8Atvi7i{d{6EB2vQZ&|?yvY#lPTz% zMBUq1Xq-VY_^O}f0|Oa5(cX*T%6-Se|K-kfE7*rma|qR-!{iL1Vu^IB?nbkRS9#6% zdVUVX`IBrCkuuG9J5?y_ae76khwzPL_XnhFv;M}=MF-4Z7HVP@yv9WQzvj&D9{8qi X9^P)2Rv5fI0^FP!bac`xG8q300uC!b diff --git a/methods/statistics.tex b/methods/statistics.tex index 590e547..04a1903 100644 --- a/methods/statistics.tex +++ b/methods/statistics.tex @@ -20,18 +20,40 @@ \date{} \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. \textit{Balance point} or \textit{expected value} of a distribution. Centre of a symmetrical distribution. + \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} + \end{document} -- 2.47.1