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 ofa(the largest integer less than or equal toa).
- ⊕orxor
- Bitwise exclusive OR.
- Binomial coefficient or combination(α∈R,α≥0;k∈N∪{0}).Forbinomial coefficient is defined asα≥kIf, thenα<k
- Φ(x)
- Cumulative Gaussian distribution functiondefined over-∞<x< +∞..Φ(-∞) = 0,Φ(+∞) = 1
- Γ(α)
- The complete gamma functionwhere.α> 0
- B(p,q)
- The complete beta functionwhereandp>0.q>0
- LCG(a,c,m)
- Linear Congruential Generator, wherex= (n+1ax+nc) modmais called the multiplier,cis called the increment, andmis called the modulus of the generator.
- MCG(a,m)
- Multiplicative Congruential Generatoris a special case of Linear Congruential Generator, where the incrementx= (n+1ax) modnmcis taken to be 0.
- GFSR(p,q)
- Generalized Feedback Shift Register Generatorxn= xn-p⊕xn-q.