5 random variates

8/9/2019 5 Random Variates

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ICS 2307ICS 2307Modelling and SimulationModelling and Simulation

Random VariatesRandom Variates


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Random VariatesRandom Variates

There are several techniques for generating randomThere are several techniques for generating randomvariatesvariates

These include:These include:

Inverse MethodInverse Method

Convolution MethodConvolution Method

Acceptance-rejection methodAcceptance-rejection method

The inverse method is articularl! suited for anal!ticall!The inverse method is articularl! suited for anal!ticall!tracta"le ro"a"ilit! densit! functions#tracta"le ro"a"ilit! densit! functions#

The remaining methods deal $ith more com le% casesThe remaining methods deal $ith more com le% casessuch as the normalsuch as the normal


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The Inverse MethodThe Inverse Method

Method a lica"le to cases $here the C&' can "eMethod a lica"le to cases $here the C&' can "einversed anal!ticall!inversed anal!ticall!

Assume that we wish to generate stochasticAssume that we wish to generate stochasticvariates from a probability density function pdfvariates from a probability density function pdf

f(x)f(x)!et "(x) be the cumulative density function!et "(x) be the cumulative density function

"(x) is de#ned in the region $%&'"(x) is de#ned in the region $%&'

"irst we generate a random variable r which we"irst we generate a random variable r which weset to be e ual to "(x) i e "(x) *rset to be e ual to "(x) i e "(x) *r

+he uantity x is obtained by inverting " i e x*" +he uantity x is obtained by inverting " i e x*" ,,'' (r)(r)

+he inverse function is applied on both sides of +he inverse function is applied on both sides of


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ExampleExample

Assume that we wish to generate random variates withAssume that we wish to generate random variates withthe pdf f(x)=2x, 0the pdf f(x)=2x, 0 ≤≤ xx≤≤ 11

We first generate an expression for the CDFWe first generate an expression for the CDF

∫ = x

tdt x F 0

2)( xt x F 0

2 ][)( =2)( x x F =

!ow if we "now that F(x)=r, then!ow if we "now that F(x)=r, then

xx22=r=r

r x =


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r is o"tained "! generating a random varia"le "et$een 0r is o"tained "! generating a random varia"le "et$een 0and ( i)e) sam ling the C&'and ( i)e) sam ling the C&'

Remem"er that the C&' for an! df ta*es valuesRemem"er that the C&' for an! df ta*es valuesuniforml! distri"uted in the interval 0 to (uniforml! distri"uted in the interval 0 to (

This is illustrated in the figures "elo$This is illustrated in the figures "elo$

"ig ' .ampling A continuous


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"ig / .ampling a discretedistribution


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1)1) Sampling from a uniform distributionSampling from a uniform distribution


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,et the inverse of the C&' e% ression,et the inverse of the C&' e% ression

aba x

x F r −−== )(

aabr x +−= )(


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2)2) Sampling from an exponential distributionSampling from an exponential distribution

+e get the C&' e% ression+e get the C&' e% ression


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,et the inverse of the C&' e% ression,et the inverse of the C&' e% ression

t

et F λ −

−=1)(t et F r λ −−== 1)(

λ

λ

λ

λ

λ

)1ln(

)1ln(

)ln()1ln(

1

1

r t

t r

er

er

er

t

t

t

−−=

−=−=−

=−

−=

−

−

−

-ecause r is a com lement of (.r $e have-ecause r is a com lement of (.r $e have

λ

)log( r t −=


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3)3) Sampling from an Erlang distributionSampling from an Erlang distribution

The m./rlang random varia"le is the statistical sumThe m./rlang random varia"le is the statistical sumconvolutions1 of m inde endent and identicall!convolutions1 of m inde endent and identicall!

distri"uted e% onential random varia"lesdistri"uted e% onential random varia"les

If ! re resents the m./rlang random varia"le# tenIf ! re resents the m./rlang random varia"le# ten! !! ! (( !! 22 4 !4 ! mm

$here !$here ! i#i#i (#2#4#m are inde endent and identicall!i (#2#4#m are inde endent and identicall!

distri"uted e% onential random varia"les $hosedistri"uted e% onential random varia"les $hosero"a"ilit! densit! function is defined asro"a"ilit! densit! function is defined as


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i y

i e y f

λ λ −=)( y i>0, i=1,2,…,m

The iThe i thth e% onential sam le is given "!e% onential sam le is given "!

mir yii

..1),ln(1

=−=

λ

The m./rlang sam le is com uted asThe m./rlang sam le is com uted as

{ }

)...ln(1

)ln(...)ln()ln(1

21

21

m

m

r r r y

r r r y

××−=

+++−=

λ

λ


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4)4) The Normal DistributionThe Normal Distribution 5 random varia"le % $ith df 5 random varia"le % $ith df

is said to have a normal distri"ution $ith meanis said to have a normal distri"ution $ith mean µµ andand

standard deviationstandard deviation σσ The e% ectation and variance areThe e% ectation and variance are µµ andand σσ22 res ectivel!res ectivel!

IfIf µµ 0 and0 and σσ (# then the varia"le has a standard normal(# then the varia"le has a standard normaldistri"utiondistri"ution


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5 random varia"le % follo$s a normal distri"ution $ith 5 random varia"le % follo$s a normal distri"ution $ithmeanmean µµ and standard deviationand standard deviation σσ then the randomthen the randomvaria"le 6varia"le 6

σ µ −= x z

follo$s a standard normal distri"utionfollo$s a standard normal distri"ution

In order to generate variates from the normal distri"utionIn order to generate variates from the normal distri"ution$ith arameters$ith arameters µµ andand σσ# $e use the central limit theorem# $e use the central limit theorem


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The central limit theorem states thatThe central limit theorem states thatif xif x '' & x& x// & 0x& 0x nn are n independent random variatesare n independent random variateswith parameters meanwith parameters mean µµ and stdand std σσ then& and wethen& and wehave y de#ned ashave y de#ned as

∑=n

i x y 1y approaches a normal distribution as n becomesy approaches a normal distribution as n becomeslargelarge

1(y)*n1(y)*n µµ2ar(y)*n2ar(y)*n σσ//


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The rocedure for generating random varia"les from theThe rocedure for generating random varia"les from thenormal distri"ution $ill this require generation of n valuesnormal distri"ution $ill this require generation of n valuesof the uniform distri"utionof the uniform distri"ution

The random varia"les associated $ith these values haveThe random varia"les associated $ith these values have

mean ( 2 and variance ( (2mean ( 2 and variance ( (2

Therefore $hen the varia"le got from a summation of n ofTherefore $hen the varia"le got from a summation of n ofthese varia"les $ill have a normal distri"ution $iththese varia"les $ill have a normal distri"ution $ith

µµ n 2 andn 2 and σσ22

n (2n (2i)e ! has the normal distri"ution given "elo$i)e ! has the normal distri"ution given "elo$

)12

,2

()( nn

N y f =


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! can "e ma ed into the standard normal distri"ution to! can "e ma ed into the standard normal distri"ution toroduce a ne$ varia"le 6 usingroduce a ne$ varia"le 6 using

σ µ −= x z

Thus 6 is given "!Thus 6 is given "!

12/

2/

n

n y z

−=

8o$ if $e consider the normal distri"ution $e $ant to8o$ if $e consider the normal distri"ution $e $ant toroduce $ith arametersroduce $ith arameters µµ andand σσ# $e have# $e have

unn y

x

u xnn y

+−

=

−=−

12/))2/((

12

2/

σ

σ


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If $e ta*e n (2# this sim lifies toIf $e ta*e n (2# this sim lifies to

u y x +−= σ )6(In summar!# this means thatIn summar!# this means that

(1(1 +e generate ! "! generating and summing (2 random+e generate ! "! generating and summing (2 randomvaria"les r varia"les r (( 4r 4r (2(2 1# ! r 1# ! r (( 4 r 4 r (2(2

2121 +e su"tract 9 from !#+e su"tract 9 from !#

3131 Multi l! the difference in t$o $ith the standard deviationMulti l! the difference in t$o $ith the standard deviationfor the distri"ution $e $ant to sam lefor the distri"ution $e $ant to sam le

11 5dd the mean 5dd the mean µµ to the roduct of 3 to get our randomto the roduct of 3 to get our randomvariate %variate %


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This method is slo$ "ecause of the need to generate (2This method is slo$ "ecause of the need to generate (2random valuesrandom values

5 more efficient rocedure calls for using the 5 more efficient rocedure calls for using thetransformationtransformation

),2sin())ln(2(),2cos())ln(2(

212

211

r r z and r r z

π π

−=−=

This method is -o%.Mueller rocedure Taha#20031This method is -o%.Mueller rocedure Taha#20031

T$o values in the $ith the standard normal distri"utionT$o values in the $ith the standard normal distri"utionare generated using t$o random varia"lesare generated using t$o random varia"les

These can "e ma ed into the distri"ution of interest $ithThese can "e ma ed into the distri"ution of interest $ithmean 8mean 8 µµ##σσ11


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5)5)

Poisson DistributionPoisson Distribution

If the distri"ution of the time "et$een the occurrence ofIf the distri"ution of the time "et$een the occurrence ofsuccessive events is e% onential# then the distri"ution ofsuccessive events is e% onential# then the distri"ution ofthe num"er of events er unit time is ;oissonthe num"er of events er unit time is ;oisson

This relationshi s is used to sam le the ;oissonThis relationshi s is used to sam le the ;oissondistri"utiondistri"ution

If the ;oisson distri"ution has a mean value ofIf the ;oisson distri"ution has a mean value of λλ eventseventser unit time#er unit time#

The time "et$een events is e% onential $ith mean (The time "et$een events is e% onential $ith mean ( λλ time unitstime units

This means that a ;oisson sam le#n# $ill occur during tThis means that a ;oisson sam le#n# $ill occur during t

time units iff time units iff


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The time "et$een events is e% onential $ith mean (The time "et$een events is e% onential $ith mean (

λλ

time unitstime units

This means that a ;oisson sam le#n# $ill occur during tThis means that a ;oisson sam le#n# $ill occur during ttime units iff time units iff

3eriod until event n occurs3eriod until event n occurs ≤≤t4period until eventt4period until eventn5' occursn5' occurs

this is translated tothis is translated to

t t 11 +t +t 22 +…+t +…+t nn ≤ ≤ t < t t < t 11 +t +t 22 +…+t +…+t n+1 ,n+1 , n>0n>000 ≤ ≤ t


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The follo$ing discussion sho$s ho$ this is im lementedThe follo$ing discussion sho$s ho$ this is im lemented


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In summar!# this means thatIn summar!# this means that

λλ is the num"er of events er unit time#is the num"er of events er unit time#

5nd $e $ant a sam le of ho$ man! events $ill occur in a 5nd $e $ant a sam le of ho$ man! events $ill occur in atime ttime t

'irst $e o"tain the value of e'irst $e o"tain the value of e .. λλtt

ase 1ase 1

If $e generate the first random value r If $e generate the first random value r (( and it is less thanand it is less thanee .. λλtt# then $e have a sam le of 0# I)e) our ;oisson random# then $e have a sam le of 0# I)e) our ;oisson randomvaria"le n ta*es the value 0)varia"le n ta*es the value 0)

8ote that if e8ote that if e .. λλtt is greater than (# $e $ill never get ais greater than (# $e $ill never get a

sam le of 0 co6 the relevant inequalit! $ill never "esam le of 0 co6 the relevant inequalit! $ill never "esatisfiedsatisfied


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ase 2 ase 2

i)i) +e generate a series of random varia"les r +e generate a series of random varia"les r ii until theuntil theroduct of the the random varia"les $e have generated isroduct of the the random varia"les $e have generated is

less than eless than e .. λλtt# these varia"le $ill "e r # these varia"le $ill "e r (( 4r 4r m (m (

ii)ii) +e then chec* if the roduct of r +e then chec* if the roduct of r (( 4r 4r mm is greater or equalis greater or equalto eto e .. λλtt# if so# then out ;oisson random varia"le n ta*es the# if so# then out ;oisson random varia"le n ta*es thevalue m)value m)

iii)iii) IfIf ii ii

is not satisfied# $e have to discard the m values andis not satisfied# $e have to discard the m values andstart generating ne$ r start generating ne$ r ii valuesvalues


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5cce tance Re>ection Method 5cce tance Re>ection Method

!)!) Sampling the binomial distributionSampling the binomial distribution

is the ro"a"ilit! of success and q is the ro"a"ilit! of is the ro"a"ilit! of success and q is the ro"a"ilit! offailurefailure

The C&' for one trial is as sho$n "elo$The C&' for one trial is as sho$n "elo$

"ig 6 C7" of a binomialdistribution


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If $e assume 0 corres onds to a success and ( to aIf $e assume 0 corres onds to a success and ( to afailure from the diagram1failure from the diagram1

To generate a random sam le# * assuming that $e haveTo generate a random sam le# * assuming that $e haven trialsn trials

i)i) +e set * to 0+e set * to 0

ii)ii) If $e generate a random num"er r If $e generate a random num"er r ii "et$een 0 and (1 and"et$een 0 and (1 andit is less or equal to # set * * (# else * doesn?t changeit is less or equal to # set * * (# else * doesn?t change

iii)iii) Re eat from ii until !ou have generated n random valuesRe eat from ii until !ou have generated n random valuesThe value of * $ill hold the value of the "inomial variateThe value of * $ill hold the value of the "inomial variate

This is an e%am le of acce tance re>ection methodThis is an e%am le of acce tance re>ection method


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The acce tance.re>ection method is designed forcom le% dfs that are hard to deal $ith anal!ticall!

The general idea is to re lace this com le% df f %1 $ith amore anal!ticall! managea"le ro%! df h %1

Sam les from h %1 can then "e used to sam le theoriginal df

Pro"edurePro"edure

&efine the ma>ori6ing g %1 such that it dominates f %1 in&efine the ma>ori6ing g %1 such that it dominates f %1 inits entire rangeits entire range

g %1g %1≥≥f %1# .f %1# .∞∞@%@.@%@.∞∞


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ExampleExample

#se the a$$eptan$e%re&e$tion method to generate at one#se the a$$eptan$e%re&e$tion method to generate at oneva'ue from the fo''owing distri utionva'ue from the fo''owing distri ution

f(x)= x(1%x), 0f(x)= x(1%x), 0≤≤xx ≤≤11

Assume the fo''owing random va'ues are generatedAssume the fo''owing random va'ues are generated

0*+0*+

-..-..

/-/-

/+/+

.1

.1


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Sam ling 'rom an /m irical &istri"utionSam ling 'rom an /m irical &istri"ution

Buite often# an em irical ro"a"ilit! distri"ution ma! notBuite often# an em irical ro"a"ilit! distri"ution ma! not"e a ro%imated $ell "! one of the $ell *no$n"e a ro%imated $ell "! one of the $ell *no$ntheoretical distri"utionstheoretical distri"utions

In such a case# $e have to generate variates from theIn such a case# $e have to generate variates from the

em irical distri"utionem irical distri"utionSampling from a discrete probability distributionSam pling from a discrete probability distribution

et % "e a discrete random varia"le ta*ing the valueset % "e a discrete random varia"le ta*ing the values%%(( #%#%22 #4%#4%nn # $ith ro"a"ilities # $ith ro"a"ilities (( # # 22 #4##4# nn

et ; %et ; % ≤≤%%ii1 ;1 ; ii# i 0# (#4#n Cumulative ;ro"a"ilit!1# i 0# (#4#n Cumulative ;ro"a"ilit!1

et r "e a random varia"le generated from the uniformet r "e a random varia"le generated from the uniformdistri"ution D0#(Edistri"ution D0#(E


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Then % %Then % %ii $hen ;$hen ;

i.(i.(@r @r ≤≤;;

ii

"ig 8 .ampling a discreteempirical distribution "rom the

#gure& x*x /


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Sampling from a continuous probability distributionSam pling from a continuous probability distribution

Sometimes $e need to grou the data into classesSometimes $e need to grou the data into classeses eciall! $hen the sam les ta*e continuous values1es eciall! $hen the sam les ta*e continuous values1

/ach sam le $ill have the lo$er class "oundar! and the/ach sam le $ill have the lo$er class "oundar! and the

u er class "oundar!u er class "oundar!+e $ill use the u er class "oundar! of each class to+e $ill use the u er class "oundar! of each class to

lot the C&'#lot the C&'#

et us assume that the u er class "oundar! values areet us assume that the u er class "oundar! values are%%(( # %# %22 # 4%# 4% nn $here %$here % i.(i.( is the lo$er class "oundar! for theis the lo$er class "oundar! for theclass $ith the u er "oundar! %class $ith the u er "oundar! % ii

ets also assume that the %ets also assume that the % ≤≤%%ii1 ;1 ; ii


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If $e generate a random value r in the interval D0#(EIf $e generate a random value r in the interval D0#(E

Then the associated value % is o"tained using theThen the associated value % is o"tained using thefollo$ing formulafollo$ing formula

1

111 )(

−

−−−

−

−−+=

ii

iiii

P P

P r x x x x

+here r is greater than ;+here r is greater than ; i.(i.( "ut less or equal to ;"ut less or equal to ; ii

5 random variates

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