2 Algebra Review

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
Quadratic equations
Graphing equations Once you get to multiple regression, some elements of vectors/linear algebra are helpful, for example the vector or dot product, also called the inner product:
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

2.2 Some Resources

# Algebra Review ## 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 * Quadratic equations * Graphing equations Once you get to multiple regression, some elements of vectors/linear algebra are helpful, for example the vector or dot product, also called the inner product: * 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 <em>linear predictor</em> ## Some Resources * [tutorial.math.lamar.edu/pdf/Algebra_Cheat_Sheet.pdf](http://tutorial.math.lamar.edu/pdf/Algebra_Cheat_Sheet.pdf) * [www.khanacademy.org/math/algebra](https://www.khanacademy.org/math/algebra) * [biostat.app.vumc.org/PrereqAlgebra](http://biostat.app.vumc.org/PrereqAlgebra) * [www.purplemath.com/modules/index.htm](http://www.purplemath.com/modules/index.htm)