Random-Number Generation, Random-Variate Generation

Unit 3

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contents• Random number generation• Properties of random numbers• Generation of pseudo-random numbers• Techniques for generating random numbers• Tests for Random Numbers

• Random-Variate Generation:• Inverse transform technique• Acceptance-Rejection technique• Special properties

Random-Number Generation

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Properties of random numbers• the main properties of random numbers are• Uniformity• Independence• Maximum density• Maximum period

• Maximum density means that the gaps between random numbers should not be large, can be achieved by having maximum period.• Maximum period refers the length of the sequence of random

numbers which are going to repeat after a certain random numbers.

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• Each random number Ri must be an independent sample drawn from a continuous uniform distribution between zero and 1• The pdf of the given by• f(x)=• The expected value of each is given by• E( R) = =

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• The variance is given by • V( R) = = • The following figure shows the pdf for random numbers

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Generation of pseudo Random Numbers• Pseudo means false , here it implies generating random numbers by

known method to remove the potential for true randomness.• If the method is known then set of random numbers can be repeated.• Which means that numbers are not random• The main goal of random generation technique is to produce a

sequence of numbers between 0 and 1 that simulates or imitates the ideal properties of uniform distribution and independence• Random numbers are generated by digital computer as part of

simulation, there are numerous ways to generate these values

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• The following are few important considerations• The method should be fast, simulation process requires millions of random

numbers hence it has to be fast• The method has to be portable to different computer• The method should have sufficiently long cycle, means there should be long

gap between the random numbers once generated getting repeated.• The random numbers should be repeatable• The generated random numbers should closely approximate the ideal

statistical properties of uniformity and independence

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Errors or departures of pseudo random numbers• The generated random numbers might not be uniformly distributed.• Generated numbers might be discrete value instead of continuous value.• The mean of generated random numbers might be too high or too low• The variance of generated numbers might be too high or too low• There might be dependence• Authentication between numbers• Numbers successively higher or lower than adjacent numbers• Several numbers above the mean followed by several numbers below the mean.

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Techniques for generating random numbers• Linear congruential method• Combined linear congruential generators• Random number streams

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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

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• If c then form is called mixed congruential method• When c=0, the form is called multiplicative congruential method• The selection of the values for a, c, m and X0 affects the statistical

properties and the cycle length.• Random numbers Ri between 0 and 1 can be generated by setting• Ri = , i=1,2,…

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examples

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Combined linear congruential generators• Combine two or more multiplicative congruential generators in such a

way that the combined generator has good statistical properties and longer period.• The following result from L’ Ecuyer suggest how this can be done:

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Combined linear congruential generators

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Combined linear congruential generators• The maximum possible period is given by

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examples

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Tests for random numbers• Number of tests are performed to check the uniformity and independence of

random numbers• Two types of tests are• Frequency test : compares the distribution of the set of numbers generated to a

uniform distribution. Few are:• Kolmogorov-Smirnov Test• Chi-square Test

• Autocorrelation test: tests the correlation between the two numbers and compares the sample correlation to the desired correlation, zero• Runs test• Gap test• Pokers test

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Kolmogorov- Smirnov Test –for uniformity (Procedure)1. Formulate the hypothesis

H0:Ri ~U[0,1]H1:Ri ~U[0,1]

2. Rank the data from smallest to largestR(1)≤R(2) ≤R(3)…

3. Calculate the values of D+ and D-

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4. Find D=max(D+,D-)5. Find the critical value Dα from the K-S table6. If D> Dα then

reject the hypothesis H0

else If D < Dα then accept the hypothesis H0

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K-S TABLE

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examples

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Chi-square Test –for uniformity (Procedure)1. Formulate the hypothesis

H0:Ri ~U[0,1]H1:Ri ~U[0,1]

2. Divide the data into different class intervals of equal intervals

3. Find out how many random numbers lie in each interval and hence find Oi

(observed frequency) & expected frequency Ei using the formulaEi = where N is the total no of observation n is the no of class intervalL=n-1 is known as degree of freedom

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4. Calculate

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Chi-Square table

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examples

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Tests for autocorrelation (Test for independence of random numbers)• The test for autocorrelation are concerned with the dependence

between numbers in a sequence.• The autocorrelation between every m numbers starting with ith number

i.e. Ri,Ri+m,Ri+2m, … , Ri+(m+1)m is ρ im

• The value M is the largest integer such that i+(m+1)≤N where N is the total number of values in the sequence• A non-zero autocorrelation implies a lack of independence

H0: ρ im = 0H1: ρ im ≠ 0

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• For large value of M, the distribution of the estimator of ρ im is denoted as ρ im

• the test statistics is as follows

• Which is distributed normally with a mean of 0 and variance of

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• And standard deviation of

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• After computing Z0, do not reject the null hypothesis of independence if

• ≤ Z0 ≤ where is the level of significance and• is obtained from a following table A-3

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Runs test• Definition: The runs test is defined as sequence of similar preceded

and followed by different events• Eg. Suppose tossing a coin 10 times results in the following sequence• H H T T T H T H T T• Here are 6 runs, first one of length 2, 2nd length of 3, 3rd ,4th ,5th of

length 1 and 6th of length 2• Two points to be considered while performing runs test• No of runs• Length of each run

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Runs up and runs down• A run is said to be up if its followed by a bigger number and down if

the number is followed by a smaller number• Since last number is not followed by any number, the maximum

number of runs is n-1 where n is the number of observation• Procedure of the runs up and runs down is as follows• Step 1 :

H0: Ri is independentH1: Ri is not independent

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• Step 2: find runs up and runs down by assigning the + sign to every random number that is followed by bigger number and – sign to a number that is followed by a smaller number• Step 3: find the total number of runs (a)• Step 4: calculate where =(2N-1)/3 and =sqrt((16N-29)/90)

where N is total number of observation• Step 5: find the critical value from the normal table• Step 6: reject H0 if |Z|≥ otherwise accept H0

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examples

Random-Variate Generation

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Inverse-transform technique• It can be used to sample from the exponential , uniform, Weibull and

triangular distributions and from empirical distributions.• Underlying principle for sampling from a wide variety of discrete distributions.• Most straightforward but not always efficient technique. Few are• Exponential distribution• Uniform distribution• Weibull distribution• Triangular distribution• Empirical discrete distribution• Discrete uniform distribution• Geometric distribution

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Exponential distribution• Probability density function (pdf) is given by

• The cumulative distribution function (cdf) is given by• F(x)= =

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Procedure for inverse transform technique

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examples

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Uniform distribution• Consider a random variable X i.e. uniformly distributed on the interval

[a,b]. • Step 1 : the cdf is given by

F(x) = • Step 2: Set F(x)= = R• Step 3: on solving we get , X=a+(b-a)R which is the equation for

random variate generation using uniform distribution

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Weibull distribution• The pdf is given

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Triangular distribution

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examples

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A discreteuniform distribution

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Geometricdistribution

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Acceptance – rejection technique• Devising a method for generating random numbers ‘X’ uniformly

distributed between ¼ and 1 follows three steps1. Generate a random number R2. a) If R ≥ ¼ accept X=R the goto step 32. b) if R < ¼ reject R and return to step 13. If another uniform random variate on [1/4,1] is needed, repeat the procedure beginning at step 1 , if not stop.

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Poisson distribution• Step 1 : Set n=0, P=1• Step 2 : generate a random number Rn+1 , replace P by P.Rn+1

• Step 3: If P < then accept N=n, otherwise reject the current n, increase n by one and return to step 2• With N=n poison of average number is given by

E(N+1)=α+1

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examples

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Non stationary Poisson Process

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• For the arrival function in the table generate the 1st two arrival times t(mins) mean time b/n Arrival rate arrival (mins) A(t)

0 15 1/15 60 12 1/12 120 17 1/17 180 5 1/5 240 8 1/8 300 10 1/10 Given the random no’s are: 0.2130, 0.8830, 0.5530, 0.0240, 0.0001, 0.1443

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Gamma Distribution

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Special properties• They are variate generation techniques that are based on features of

particular family of probability distributions , rather than general purpose techniques like inverse transform or acceptance-rejection technique.• Direct transformation for the normal and lognormal distributions• Convolution method• Erlang distribution

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Direct transformation for the normal and lognormal distributions

• The standard normal cdf is given by ɸ (x) =

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Convolution method• The probability distribution of a sum of two or more independent

random variable is called convolution of distribution of the original variable• The convolution method refers to adding together two or more

random variables to obtain a new random variable with a desired distribution

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Erlang distribution

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End of unit 3Thank you