random number and it's generating techniques

1. Random number and It's
generating techniques
By
Md. Fazle Rabbi
16CSE057

2. 4.2
Random numbers
What is random numbers ?
• A random number is a number generated by a process,
• Whose outcome is unpredictable.
• And which cannot be sub sequentially reproduced.
Random number plays important role in simulation task

3. 4.3
Properties of random numbers
The main properties of random numbers are
• Uniformity
They are equally possible every where
• Independence
The current value of a random variable has no relation
with the previous value

4. 4.4
Generation of Pseudo-Random Numbers
• “Pseudo”, because generating numbers using a known method
removes the potential for true randomness.
• Goal: To produce a sequence of numbers in [0,1] that
simulates, or imitates, the ideal properties of random numbers
(RN).
• Important considerations in RN routines:
• Fast
• Portable to different computers
• Have sufficiently long cycle
• Replicable
• Closely approximate the ideal statistical properties of
uniformity and independence.

5. 4.5
Techniques for Generating Random Numbers
• Linear Congruential Method (LCM).
• Combined Linear Congruential Generators (CLCG).
• Random-Number Streams.

6. 4.6
Linear congruential method
Proposed by Lehmer, produces a sequences of integer numbers X1 ,X2 ,
… between zero and m-1 by following the recursive relationship:
X i+1= (aXi+c) mod m, i=0,1,2,3…
• The initial value i.e. x0 is called seed
• a is called multiplier
• c is called the increment
• m is called the modulus
• If c ≠ 𝟎 then form is called mixed congruential method
• When c=0, the form is called multiplicative congruential method
,...
2
,
1
, 
 i
m
X
R i
i

7. 4.7
Example [LCM]
• Use X0 = 27, a = 17, c = 43, and m = 100.
• The Xi and Ri values are:
X1 = (17*27+43) mod 100 = 502 mod 100 = 2, R1 = 0.02;
X2 = (17*2+43) mod 100 = 77, R2 = 0.77;
X3 = (17*77+43) mod 100 = 52, R3 = 0.52;

8. 4.8
Combined linear congruential generators
• Reason: Longer period generator is needed because of the increasing
complexity of stimulated systems.
• Approach: Combine two or more multiplicative congruential generators.
• Let Xi,1, Xi,2, …, Xi,k, be the ith output from k different multiplicative
congruential generators.
• The jth generator:
• Has prime modulus mj and multiplier aj and period is mj-1
• Produces integers Xi,j is approx ~ Uniform on integers in [1, m-1]
• Wi,j = Xi,j -1 is approx ~ Uniform on integers in [1, m-2]

9. 4.9
Combined linear congruential generators
• Suggested form:
• The maximum possible period is:
1
mod
)
1
( 1
1
,
1










 


m
X
X
k
j
j
i
j
i










0
,
1
0
,
Hence,
1
1
1
i
i
i
i
X
m
m
X
m
X
R

1
2
1
2
)
1
)...(
1
)(
1
(




 k
k
m
m
m
P
The coefficient:
Performs the
subtraction Xi,1-1

10. 4.10
Combined linear congruential generators
• Example: For 32-bit computers, L’Ecuyer [1988] suggests combining k = 2
generators with m1 = 2,147,483,563, a1 = 40,014, m2 = 2,147,483,399 and a2
= 20,692. The algorithm becomes:
Step 1: Select seeds
• X1,0 in the range [1, 2,147,483,562] for the 1st generator
• X2,0 in the range [1, 2,147,483,398] for the 2nd generator.
Step 2: For each individual generator,
X1,j+1 = 40,014 X1,j mod 2,147,483,563
X2,j+1 = 40,692 X1,j mod 2,147,483,399.
Step 3: Xj+1 = (X1,j+1 - X2,j+1 ) mod 2,147,483,562.
Step 4: Return
Step 5: Set j = j+1, go back to step 2.
• Combined generator has period: (m1 – 1)(m2 – 1)/2 ~ 2 x 1018














0
,
563
2,147,483,
562
2,147,483,
0
,
563
2,147,483,
1
1
1
1
j
j
j
j
X
X
X
R

11. 4.11
Random-Numbers Streams
• The seed for a linear congruential random-number generator:
• Is the integer value X0 that initializes the random-number sequence.
• Any value in the sequence can be used to “seed” the generator.
• A random-number stream:
• Refers to a starting seed taken from the sequence X0, X1, …, XP.
• If the streams are b values apart, then stream i could defined by starting seed:
• Older generators: b = 105; Newer generators: b = 1037.
• A single random-number generator with k streams can act like k distinct
virtual random-number generators
• To compare two or more alternative systems.
• Advantageous to dedicate portions of the pseudo-random number sequence to
the same purpose in each of the simulated systems.

12. 4.12
Thank you