Conditional probability:

\[\begin{equation*} P(B|A)=\frac{P\left( A\text{ and }B\right) }{P\left( A\right) }. \end{equation*}\] Bayes’ Theorem: \[\begin{equation*} P(B|A)=\frac{P\left(A|B\right) \times P(B) }{P\left( A\right) }. \end{equation*}\]

Binomial probability distribution:

\[\begin{equation*} P(X = k) = ^{n}C_{k} \times p^{k} \times \left( 1-p\right) ^{n-k}\qquad \left( \text{where } ^{n}C_{k} =\frac{n!}{k!\left(n-k\right) !}. \right) \end{equation*}\]

Poisson probability distribution:

\[\begin{equation*} P(X = k) =\frac{m^{k}\mathrm{e}^{-m}}{k!}. \end{equation*}\]

Exponential probability distribution:

\[\begin{equation*} P(X \leq k) = \begin{cases} 1-e^{- k/\mu}, & k \ge 0, \\ 0, & k < 0. \end{cases}\qquad \left( \text{where } \mu = \frac{1}{\lambda}\right) \end{equation*}\]

Sums of Squares Identities

\[\begin{eqnarray*} S_{XY} &=& \sum x_iy_i - \frac{\sum x_i\sum y_i}{n}\\ S_{XX} &=& \sum x_i^2 - \frac{(\sum x_i)^2}{n}\\ S_{YY} &=& \sum y_i^2 - \frac{(\sum y_i)^2}{n}\\ \end{eqnarray*}\]

Slope Estimate

\[\begin{eqnarray*} b_1 = \frac{S_{XY}}{S_{XX}} \end{eqnarray*}\]

Intercept Estimate

\[\begin{eqnarray*} b_0 = \bar{y} -b_1\bar{x} \end{eqnarray*}\]

Pearson’s correlation coefficient

\[\begin{eqnarray*} r = \frac{S_{XY}}{\sqrt{S_{XX} \times S_{YY}}} \end{eqnarray*}\]

Standard error of the Slope

\[\begin{eqnarray*} S.E.(b1) = \sqrt{\frac{s^2}{S_{XX}}} \end{eqnarray*}\]

where \(s^2 = \frac{SSE}{n-2}\) and SSE \(= S_{YY} - b_1S_{XY}\)