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.
Equation
The floor of
a
(the largest integer less than or equal to
a
).
or
xor
Bitwise exclusive OR.
Equation
Binomial coefficient or combination
(
α
R
,
α
0;
k
N
{0}).
Equation
For
α
k
binomial coefficient is defined as
Equation
If
α
<
k
, then
Equation
Φ
(
x
)
Cumulative Gaussian distribution function
Equation
defined over
-
<
x
< +
.
Φ
(-
) = 0,
Φ
(+
) = 1
.
Γ
(
α
)
The complete gamma function
Equation
where
α
> 0
.
B
(
p
,
q
)
The complete beta function
Equation
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
.

Product and Performance Information

1

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Notice revision #20110804