# 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, ... , xN } have mean µx and variance sx2
Let Y = { y1, y2, ... , yN } have mean µy and variance sy2
Let Z = X union Y (Has 2N elements)

Then µz = ½(µx + µy)
Then sz2 = ½(sx2 + sy2) + ¼(µx - µy)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 = tn - tn-1
• Term: tn = a + (n-1)d
• Sum: Sn = ½n(2a + (n-1)d)
• Mean: B = (A + C)/2

### Geometric Series

• Ratio: r = tn / tn-1
• Term: tn = a rn-1
• Sum: Sn = a (1-rn) / (1-r)
• Sum to infinity: Soo = a / (1-r)
• Mean: B = ± sqrt(AC)