Random Number Generators Mathematical Notation

The following notation is used throughout the text:

N: The set of natural numbers
N = {1, 2, 3 ...}
.
Z: The set of integers
Z = {... -3, -2, -1, 0, 1, 2, 3 ...}
.
R: The set of real numbers.
: The floor of a (the largest integer less than or equal to a).
⊕
or xor: Bitwise exclusive OR.
: Binomial coefficient or combination
(
α
∈
R,
α
≥
0; k
∈
N
∪
{0}).

For
α
≥
k
binomial coefficient is defined as

If
α
< k
, then
Φ (x): Cumulative Gaussian distribution function

defined over
-
∞
< x < +
∞
.
Φ
(-
∞
) = 0,
Φ
(+
∞
) = 1
.
Γ ( α ): The complete gamma function

where
α
> 0
.
B(p, q): The complete beta function

where
p>0
and
q>0
.
LCG(a,c, m): Linear Congruential Generator
x_n+1 = (ax_n + c) mod m
, where a is called the multiplier, c is called the increment, and m is called the modulus of the generator.
MCG(a,m): Multiplicative Congruential Generator
x_n+1 = (ax_n) mod m
is a special case of Linear Congruential Generator, where the increment c is taken to be 0.
GFSR(p, q): Generalized Feedback Shift Register Generator
x_n = x_n-p⊕x_n-q.