Algebra Review

2 Algebra Review

2.1 Overview

Algebra and probability are underlying frameworks for basic 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