## 2.1 Overview

Algebra, arthmetic, and probability are underlying frameworks for statistics. The following elements of algebra are particularly important:

• Understanding symbols as variables, and what they can stand for

• Factoring out common terms: $$axw + bx = x(aw + b)$$

• Factoring out negation of a series of added terms: $$-a - b = - (a + b)$$

• Simplification of fractions

• Addition, subtraction, multiplication, and division of fractions

• Exponentiation with both fractional and whole number exponents

• Re-writing exponentials of sums: $$b^{u + v} = b^{u}\times b^{v}$$

• Logarithms

• log to the base $$b$$ of $$x$$ = $$\log_{b}x$$ is the number $$y$$ such that $$b^{y} = x$$
• $$\log_{b}b = 1$$
• $$\log_{b}b^{x} = x \log_{b}b = x$$
• $$\log_{b}a^{x} = x \log_{b}a$$
• $$\log_{b}a^{-x} = -x \log_{b}a$$
• $$\log_{b}(xy) = \log_{b}x + \log_{b}y$$
• $$\log_{b}\frac{x}{y} = \log_{b}x - \log_{b}y$$
• When $$b = e = 2.71828\ldots$$, the base of the natural log, $$\log_{e}(x)$$ is often written as $$\ln{x}$$ or just $$\log(x)$$
• $$\log e = \ln e = 1$$
• Anti-logarithms: anti-log to the base $$b$$ of $$x$$ is $$b^{x}$$

• The natural anti-logarithm is $$e^{x}$$, often often written as $$\exp(x)$$
• Anti-log is the inverse function of log; it ‘undoes’ a log
• Understanding functions in general, including $$\min(x, a)$$ and $$\max(x, a)$$

• Understanding indicator variables such as $$[x=3]$$ which can be thought of as true if $$x=3$$, false otherwise, or 1 if $$x=3$$, 0 otherwise

• $$[x=3]\times y$$ is $$y$$ if $$x=3$$, 0 otherwise
• $$[x=3]\times[y=2] = [x=3 \,\textrm{and}\, y=2]$$
• $$[x=3] + 3\times [y=2] = 4$$ if $$x=3$$ and $$y=2$$, $$3$$ if $$y=2$$ and $$x\neq 3$$
• $$x\times \max(x, 0) = x^{2}[x>0]$$
• $$\max(x, 0)$$ or $$w \times [x>0]$$ are algebraic ways of saying to ignore something if a condition is not met
• Let $$x$$ stand for a vector of quantities $$x_{1}, x_{2}, \ldots, x_{p}$$ (e.g., the values of $$p$$ variables for an animal such as age, blood pressure, etc.)
• Let $$\beta$$ stand for another vector of quantities $$\beta_{1}, \beta_{2}, \ldots, \beta_{p}$$ (e.g., weights / regression coefficients / slopes)
• Then $$x\beta$$ is shorthand for $$\beta_{1}x_{1}+\beta_{2}x_{2} + \ldots + \beta_{p}x_{p}$$
• $$x\beta$$ might represent a predicted value in multiple regression, and is known then as the linear predictor