Chapter 14 Mathematics

14.1 Notation

• Variables and Subscripts
  Variable \(Y\) with \(n\) sample values denoted \(y_1, y_2, ..., y_n\) in order of entry; The "1", "2", ... are called subscripts or indices. We use the letter i (or j) and the range "1 to n" to denote the n different \(y\) values and refer to the value of the ith \(y\) as "\(y_i\)".

• Summation
  The term \(\Sigma y\) (spoken: "sigma y" or "sum of y's") is used as a shorthand for the sum \(y_1 + y_2 + \dots + y_n\).

14.2 Powers, Logarithms and Anti–logarithms

• The term \(y^{1/2}\) is shorthand for the square root of \(y\) or \(\sqrt{y}\). Likewise, \(y^{1/n}\) denotes the n-th root of \(y\).

• \(\ln (y)\) denotes the "natural log of \(y\)" or "log of \(y\) to the base e" i.e. log\(_e(x)\), where e is 2.718.
  Note: y must be positive; ln (\(y\)) ranges from -\(\inf\) to +\(\inf\).
   
  ln (0.1) = -2.30; ln (1) = 0; ln (2) = 0.69; ln (10) =2.30

• \(\ln(A \times B) = \ln(A) + \ln(B); \ \ \ \ln(\frac{A}{B}) = \ln(A) - \ln(B)\)

• exp(\(y\)) is shorthand for \(e^y\) or "exponential of \(y\)" or the natural anti-log of \(y\). \(y\) ranges from -\(\infty\) to +\(\infty\). and exp(\(y\)) yields a positive value. eg. exp(-1) = 0.36; exp(0) =1; exp(.5) =1.64; exp(1) = 2.71...