Useful Math Formulae

I have written this page because I got sick of David Powell asking me questions like "Torst, what's the formula for the sum of a geometric series?" over and over again. :-) Please note I am using the HTML extensions <SUP> and <SUB> for superscript and subscript, as well as some ISO8859-1 Latin-1 characters.

Geometric distribution

• Density: P(Y = n) = p (1 - p)^{n - 1} for n = 1, 2, ...
• Mean: E(Y) = 1/p
• Variance: Var(Y) = (1-p) / p²

Statistics

• µ = mean
  s = standard deviation
  s² = variance

• µ = E[X]
  µ = (1/N) sum(x)

• s² = E[X - E[X]]² = E[X - µ]² = E[X²] - E[X]²
  s² = (1/N)sum(x²) - {(1/N) sum(x)}²
  s² = (1/N)sum(x²) - µ²
  s = sqrt(s²)

• Let X = { x₁, x₂, ... , x_N } have mean µ_x and variance s_x²
  Let Y = { y₁, y₂, ... , y_N } have mean µ_y and variance s_y²
  Let Z = X union Y (Has 2N elements)
  
  Then µ_z = ½(µ_x + µ_y)
  Then s_z² = ½(s_x² + s_y²) + ¼(µ_x - µ_y)²
  

Non-Linear Algebra

• max(u,v) = ½(u+v) + ½|u-v|
• min(u,v) = ½(u+v) - ½|u-v|

Arithmetic Series

• Difference: d = t_n - t_n-1
• Term: t_n = a + (n-1)d
• Sum: S_n = ½n(2a + (n-1)d)
• Mean: B = (A + C)/2

Geometric Series

• Ratio: r = t_n / t_n-1
• Term: t_n = a r^n-1
• Sum: S_n = a (1-rⁿ) / (1-r)
• Sum to infinity: S_oo = a / (1-r)
• Mean: B = ± sqrt(AC)

Links

• Wolfram's Mathworld