Contents

# 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
`xn  = xn-p⊕xn-q.`

#### Product and Performance Information

1

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Notice revision #20110804