Contemporary aapcctt of Monte Carlo ... 139 Olowofcla st 01 
I 
Contemporary Aspects of Monte Carlo Methods and Sirnulatioil in 
Physics 
J A. OLOWOFELA, J .  A .ADEGOICE, 0. P.A D E W W  and 1. C. KAMIYOLE 
Deparhnenr of Physics, University of lbndan. Ibarlnn, Nigeria. 
Deparhnsnt of Physlrs. Oja-State College o /Ed~~cnt ion ,O~~~ .A '~ger~n  
Abstract 
Thls artlcle introduces Monte Car lomethds which are duerent from c o n r e n t i o ~nl  ~mieticanl lethods andshow bow 
some of the methods can be applled in Physlcs,ro simrclate or solve physical proble~its. t h t w r ~ hco rrilvrrer ~vrgrrrr>rr' 
(wrltten In FORTRAN In thls ca.r@, by ruing a few cxnml~lrrt oltchinff ,n doi.11 arid c/ns.~l[:olI 'lr~!!:ricrsh r ~ t ~ . rrr~s ~ g ,  
r~lrf ihess.A  functional approach to  probab//ity and statisticsclis  describeJJor the p,1rynre o / tbn  work r*.st~ndo f 
. comp/eto treatment. The fmpqflancp oJg&'saqrrencu of random n~rntbersw lth I n g e  p a r i o d ~is  demor~xrrnredn i~d 
the methods, In same fnstaniss are compared ~vitha conventional type and dflnrenccs poinred imr. 
. , 
Hislory and Introduction Monte Carlo methods necessitate a fast and effcct~ue 
The expression "Monte Carlo" is actually way to generate random numbers uni fornil y 
very general and called s o  after a popular city of distributed on the interval [ The outcotnes of'these 
Monte Carlo in Europe, famous for its cosmopolitan random samplings must be accumttlated in an 
character and gambling casinos, as a result of the appropriate manner to produce Ule des~ccdr esult, hut 
stochastic nature of gambling and of the methods. the essential characteristics of Monte Cnrlo is the use 
Thr:y are based on the use of random numbers anrl of random snnlplin~t ceh~~ic~l(incnst i ~rrrlrnlwc ltl\ct 
probnbility stntistics to invcstignlc~p rob1Ctn.c. 011c cnti nlgcl\~nt o ~~~nrlipul*tlilcc  oulcoihcs) I t \  :il~iven t  a 
find Monte Carlo methods used in everything from solution of the physical problem. 
economics to nuclear physics. Monte Cwlo methods It is natural to think that Monte Carlo 
have been. used for centuries, but only in the past methods are used to simulate rnndom processes, ns 
several decades has the technique gained the status of these can be described by pdf s. However, inrmy 
a full-fledged numerical method capable of Monte Carlo applications have no apparent stochastic 
addressing the most complex applica'tions. The content, such as the evaluation of definite integrnl In 
applications vary fiom field to field, and thtre are spite' of this, one call pose the desiretl ~oluti(x1in  
dozr:ns' of subsets of Monte Carlo, even within terms of pdFs, nnti \vllilc tlus trnnslirtr~~ationr ny 
phytiics. To call soincthing a "Monte Carlo" seem artificial, this step nllOws thc s~stcnit o Ile 
experiment, all one needs to do is use random treated ns a stochnslic process for the purpose of 
numbers to examine some problems. . simulation and hence Monte Carlo methods can be 
The use of Monte Garlo methods to model applied to simulate t l~es ystem One can, thereffire. 
physical problems allows us to examine more take a broad view of the definition o f  Monte Carlo 
complex systcms than we otherlvise can. Solving nictl~odsb y inclliditil? ri11,ric 1111 tnrtl~trtls[ I I : \ ~  ~l~\'crl\.c 
cqiiations which dcscribc the interactions for statistical sii~iulation of semc u~iticrlvttig slatc.tn, 
huldrcds or thousands of atom is im possible but, whether or not the system reprcscnts n ~c:tl pliystc;tl 
with Monte Carlo methods, a large system canobe process. 
. sampled in o number of random configurations, and , The 111njor components of n l\.ltrlitc Cnt lo 
that dnta cnn be used to describe the system as n mclhod, comprising the foundatiot~o f   no st Molitc 
whole. Statistical simulation methods may be Cq lo  applications, we: '(i) Probability distribulton 
contrasted to conventional numerical discretisation functions (pdf's), (ii) Random number generator, (iii) 
methods , which typically are applied to ordinary or Sampling rule - rule for sampling from the specified 
partial differential equations that describe some pdf s must be given, (iv) Scoring- the out con!cs nnist 
underlying physical or msthcmatical system. In mnny be nccutnulnted into overall tnllies or scores for llto 
applications of Monte Carlo, the physical process is quantities o f , i nterest (v) Error ostitt~ntion - mi 
simulntcd directly, with the only requirement that tlle estimate of ,the statisbicnl error (variance) as a 
physical (or mathematical) system be described by function of the number of trials and other quantit~es 
probability density functions (pdt's) and no need to must be determined 'nnd (vi) Vnrinnce aeductlon 
even write down the differential equations thnt techniqr~cs- this can lend to reduction in the 
describe the behnviour of the system. Many computational time for Monte Carlo simulation .a 
applications are then pcrfonned (multiple .."trials") well. 
and thedesired resultistaken-as an average over the What is intended to be shown in this article will be 
number of observations. In many practical illustration of the applicnti?n of some .Monte Carlu 
applicntians, one can predict the statisticnl error Methods and their major coniponents in the follow~ng 
(variance) in this average result, and hence on areas (1) Simple .Monte G ~ l oev aluation of ml 
estimate of the nunlber of  Monte Carlo trials thnt ore integral (2) Multidimensionnl Montc Cnslo 
necded to achieve a given efior. integration (3) Monte Cmio !Srror Annlvstr. (J) .l'l~c 
- 
Zumo Jaurrt. of Pure and Applied Science1 6(1) 200% 
UNIVERSITY OF IBADAN LIBRARY
Approach to Equifibrium - Monte.Car10 Method; and noted that the ~alunt ionh appens to be equivalent to 
.(S)Radioactive Decay. the value of pi. 
The first method. This method, can be v~ewed 
Ran dotn number generation and.~orfteM onte Carlo graphically such that the area of a pnrticular fi~nctiocl. 
Methods. which is to be determitred, is represenlcd \vitIii~ln  
Since Monte Carlo Methods are procedures that make larger, regular area or shape whose area is kno\m and 
use of random numbers, generation of random an algorithm developed so that ratio of number of 
numbers of immense importance to the methods. , rnndouily picked coocdit~ntcs ~vllich Tnll \v i l t \cr l  
A sequence of random numbera is a set of numbers represented area of the filnctiorl to tolnl n~i~nhcorf 
that have nothing to do with the other numbers in the coordinates generated (which should nllvnys fnll In 
sequence. The idea of a single rmdom number is thpse for the larger area) gives the nren of thc 
incorrect. .In a uniform distribution of random hc t ion  when multiplied by area of the largcr shape. 
numbers in the range [ To be precise, the algorithm The evaluation of the function of the first quadrnnt 
gencmtc integers between 0 and, say M, and divide (AT3CD) (Pigr~ru1 ) of n ctrclc of \~n!tr :ldit~s\ V I ~ I I I I I  n 
by M to return a real value within the range. An square, using tlus ii~ethodi n the coinl>trtrrp rogrnnr 
example called "middle square", describes generating algorihn. reveals thnt incrcnse .in tri:tl uunrher$ 
a sequence of ten digit integers by sMing with. one, increases @I$ ac curacy of lhe method. The table 1 of 
squaring it and then taking the middle ten digits from results for a particlilar sequence of'random numbers 
the nnswer as the next nwber  in the sequence. The of over 8 million corlnts ns period is givrcl Irrlo~v. 
sequcnce is, however, not random, as each n~unberi s V~luco f pi using Ilil-Miss Metllatl 
mmpletely determined from the previous. Actunl Value of pri=3. 
The power residue and linear congruent method; are, I415926535897932384626433832795 
respectively. de scriw by equations: The logarithm of the errors in the value. when plotted 
I = a1 + c mOdM (1) against the total trials made, gives a graph (figure 2) 
I = (ah + c)modM (2) showing that the greater the total trial number, the 
The starting value (seed) of I is I where a, c an$ M better the accuracy, that is,.for large trial total, sny N, 
are constants specifically chosen such that a and c error approaches zero as inverse of square root of N. 
and greater than or equal to 0 and M is greater than I This is revealed from the values "DEP". \vllich is the 
a and c. A poor choice of thb constants can lead to ratio of the error in the \~oldco f pi to the 11 it11 told N, 
vcry poor scquences e.g. ones with short periods. The cnlc~ilatedf or two inslnnces, as seen in the result. It 
chhice c=O, in the case of linear congruential method, establishes the relotionship betwcen error nnd ?rial 
obviously leads to a eomewhat faster algorithm, and total. 
can also result in long sequehces. This is called The second and final method for the purpose of this 
"Multiplicative congruential". M should be as large article is the "Sample Mean" method (F~gure2 )- In 
es possible since the period can never be longer than definite integration, with liniits a and b manipulation 
M. One's choice of M should be one near the largest is made such that il probability distribution function' 
integer that can be represented by the computer. A f(x) is defined to be Il(b-a) for a c x < nnd zero for 
snmple nlgorithm for the power residue method is any other vnlues and inlrodu~cecli nto tl~c~ r\t~pri(Il>l\ . 
given below: dividing the originnl fi~nction) nit11 1I1r c\ripr~~nl 
STEP I :specify the three constants to be used a,c and integrnl vicwcd ns the csprctntin~l valtrr of tlrc 
m and the seed. STEP 2:substitute them into the functiori in it $0 that n final cxprcssiorl of lllc for 
equation to obtain a value, say x. @-a)M*4 -x (3) 
STEP 3:if this is greater than m: set the value back to is sbtained, where b=l and a=O for tlus particular 
that of Ulc seed. example. USING SAMPLE MEAN METHOD 
STEP 4:whatever the case, divide the value x by m to (Table 2). 
get next number in sequence. It can be seen that the values fluctuate for smail Gal 
STEP 5: substitute into the seed's position the new x totals but steadily approach thg true value as trials 
nnd return to step 2. increase. With trials of 1,000,000, accurady to 2 
For any program making use of random number decimal places can be snid to be ncl~ievedn nd higher 
sequences it must be enswed that thq amount 'of trials accufacies obtained wilh increase in trial total (N). By 
used does not ~ c e e tdhe  periods of the sequences for dividing variance by total trial number ("DEP") and 
reliable results to be obtained, other factors in order. the .fact.t hat there is correfpondence between error 
Two methods are demonstrated for the purpose of and variance, one can observe, as shown by the la$ 
illustration by integrating four lines of data in the in the result, t h ~ tt he 
the function: 4* Vf- x within limits 0 and 1 with approximate hctional' dependence of error on N is 
respect to x. The function is from the equation of the such that error approache's zero as inverse of square 
first quadrant of a circle of unit radius. The integer 4 root N, for large N. When a trial ,total is squared. 
in the expression ensures the area of the circle is what square root of its error is obtained. "ERROR 2" show 
is obtained when the integration is cniried out since ap proximate deviation from true pi ohlnincd by 
fhcre nro four quadrants in any circla. It ahould be 
Zumr Journ'of Pure and Applied Selenns ql) 2M14 
UNIVERSITY OF IBADAN LIBRARY
Contemporary arpectr of Monto Carlo ... . 141 
dividing standard deviation by square root of tiial obtained in ~ a b j e4  arc consistent with the error 
total. obtained in. previ ous evaluations where nctual pi 
When the lines of algbrithm or actual code lines used value minus calculated ones gave the efror involved. 
in the two methods ate compared the sample mean Also Table 4 shows that any set o~iiicnsurei~ienctsa n 
method is seen to take a shorter time to exe cute and be approprinte by sn\.ing conijlutcr t111ic .llie 
this, in addition to the fact that it gives better resat interpretation of the standard devintioii for, say I0 
for the same number of -trials, ,makes it a better measureinents. is Ulat F has 68.3GYn chancc of hc111g 
choice. Typically, the hit-miss method takes more within 0.006836 of the trui tilean, where 1: is ttic 
than twice the computational time required by the estimafe. 
sample mean method for a considerable trial total. It is observed, from Table 4, that sets of mensrirernent 
Sitnpson's rule, .a conventional numerical method, ahove 20 give larger stnndnrd tlcvintion, ~ I I Inl l  Iinvc 
requires lots partitions, or iterntions, to be able to more or less same "2ND MEASUIZE OF E R R O R  
obtain a very accurate value of pi. A particular result "2ND MEASURE" gives values of errors determined 
is shown below , for each in orlother way. It is by di\-itling stncldn~;l 
VALUE OF PT USMO SIMPSON'S RULE: devintion for n singlc iiicnsurc~nc~int  11 svI ll~c 
PARTITIONS MADE = 11 9 sqlidre root of lrinl.lc~lcil( 10,000 in Illis cnsc). I r o 11 
SI'Ml'SONS OITJES PI- 3.13861 1 should be nolcti that "S'I' 1)13VIA'I'ION" of I':~t!lc I is 
However, it gives a good result for a small partition calculated fiom the mean of the number or scts uscd. 
total of 1 19, comparing with values for the trial of ,each set having 10 measuremects. 
1000 obtained in the previous methods. Judging from With the second procedtlre siniilnr steps nrc hkcn l~ut 
the code needed to have used Simpson% ~ l eit, c an trial total (10,000) .is divided into equal pnrtitiorn. 
be considered to be inefticient with computer time The standard dewation of the'suh-divisions is given 
compared with the pravious Monte Carlo methods. by square root of the expregsion below: 
The clloice of using one method instead of the other (7). 
then depends on the objectives of the programmer. It As expecled, Eom the' third table. stnlidnrti dc\.inlioil 
would however be a bad choice to use conventional ,of the functions of a single division does riot give 
methods for multidimensional integration as the-error accurate error involved. . Table 4 reveals that the 
introduced .standard deviation of each set of sub-divisions is also 
would increase and considerable computer time will ' incol1si"sent with previous errors, hut the expressioris 
be wasted. gives consistency.a nd is the error involved rccordcii 
3 Error Analysis as "ERROR. a 
This involves a way of obtaining the error present It ig observed that 10 subsets might he Ule llrcfernhle 
from any evaluation so that the degree of accuracy of choice of sub-divisions. ns 20 Iaas ligller probnhle 
a parlicular method used can be verified. This will be error and picking niost above 20 only tnke coinpiltcr 
dcrnonstrated by integrating the function mentioned time without improving on Ule error. 
earlier for evaluating pi, taking trial to bg 10,000 To drive lio~neth e pcriiil or using n gopd sccliicltcc or 
trials, and obtaining the standard deviation of the random numbers, a bad sequence of period ol' period 
measurements. The second is by dividing the 10,000 two is generated and the results analpsed tnking the 
trials into sub-sets and the standard de.viation also following constants: 5, 0 and 32 as a. c and In. 'fie 
obtained of the measurements. The second is by power residue method is used for this pllrposc First, 
dividing the 10,000 trials into sub-sets and the to cnlculnte pi for varying trial ~iuln\)crs$.1 1 tr cd>\*io~is 
standard deviation also obtained. In the first case, the that the.values ohtnined for pi are oTf the ninrk, and 
question "How.can one tell if only 10 measurements even got worse for large total trials, as denioristrated 
will give 'the desired accuracy?".still remains, and, by sample Mean method and the integrnl used 
appropriately,'can one tell whether 10 sub-divisions previously. 
are appropriate in the second case? These suggested The two procedures mentioned earlier nre also 
trying various options and andysing the results. It affected as values turned deterministic. For Uie first 
should be stated that, as observed fiom previous procedure (10,000 trials each), it can be seen that the 
work, standard deviation of B single measurement is values of pi repeat tliemsbl'ves with the same period. 
incorrect as error in value of pi obtained at any time, two, the sequence used and Table 6 couldn't bc 
hence the need for the above procedures. generated (unlike before) as variance of the 
It is worth remembering that the variance, square of measurements for each group is negative which 
standard deviation, of m measurements is given by: wouldn't give real standard deviations. 
(4) Similar problems (error in value of pi and repetition 
where are observed when the second procedure is used and 
( 5 )  its Table 10 shows "calculated error" for only two 
('4 sets of subdivisions, since the variance of the 
Results to demonstrate the first procedure reveal that remaining yield negative values. 
standnrd deviation of each set'of measurements does 
not give the darrect error (Table 3). but values Application to Some l'l~uicolE .rcrt~~ples 
Zunir .lourn, or Pure and Apptlad Selsncclr 6(1) 2004 
UNIVERSITY OF BADAN LIBRARY
Contemporary arpcctr of Monte Carlo ... 142 C)louiilch rl ol 
The Apprwch to Equilibrium It is observed thnt the systenl approacllcd equ~lihrlu~li 
An ideal gas of identical. distinguishable (Figtke 3) rnrd for more initial, total pnrticles it took 
particles in a box for which inter action might be longer time to approach equilibrium (Figures 4 and 
ignored and the box is isolated is considered. The 5), as expected. Since the tendency in nature is 
initial state of the system is . m h f hat dlpglticles are towards disorder (equilibradon, in this case) 
in the leA half of the box and distribution occurs The next graph (Fig~ue6 ) tias gcnerntcd Iiy n 
when a small slide. covering a hole in th? middle, the program twitten to mlnpare Ulc appcwcli tq 
point of division of the box in two, is removed. One eq~iilibriumf or five diflerent trials nnci thcir averngcd 
assumption wade is that one padcle passes though value using constant to91 particles or 10. Again their 
the hole in a unit time. This section answers questions initial position is tile len Iinlf of thc hou n r d  otltcr 
likc, "nt time t, what is the average nurnher of nssutnptioiis nlc ns in IIlc cntlicr csn~rllrlr 1 lr~lcrl l~r  
particles IcR in the left half of the box?". "Whnt is Ule . ~netllodo f nchievi~~g'rcs~i~s lstisl ltilrl~t o  t l r  ~ ~ I I I I C I  
probable equilibration time?': This illustrations The relative magnitude'of the fluctuntions can he 
demonstrates for total'particles of 10, 20 and 40, three determined for each unit time after equilibril~rni s 
different systems. Tht'equilibrium state is achieved reached, where " " represents -n  time avernge. the 
by a system when, afler some time, its macroscopic siandard devintiort a is tlie squnre rocrt of (<?I2  - < 
state does not change in time except for small raodom >2) and n is the svcrngc nun~hcro f patitlcs i n  tltc 
fluctuations. left half of the box. 
Tho algorithm of the program is such thnt it enswcs 'I'he grnpli of thc e\.crngctl t ~ ~ l(1~:it~cu lr 6 )  is scc11 lo 
only the first time when the particles are equally lie mid-wny ns espectctl. 
spread out is r ccded  as equilibrntion time since only 
small fluctuetions follow later. A sample algorilhm is Rodinactive Decw 
given below: Radioactive decay is truly n rnntlotn process 
STEP 1:  Specify the constant initial amount in the left and it provided the first evidence that the laws Ulat 
half of the box. govern the sub-atomic world are statistical. With the 
STEP 2: Allow appreciable time for equilibration, choice of a system of initial, parent particle numhcr . 
say, ten times mount. of 100, decay constant 0.01 nnd dumtion of 300 
STEP 3: Initialize variable to hold equilibration time seconds, radioactive decay was simtdated using tile 
STEP 4: Find the ratio of &les in the lelbhalf to . 'sequence of rand0111 lilrlnbcrs gcrlcrntcd hy tlic 
totnl and compare with random number computer. Two si~n~tla'tions 
STEP 5: If ratio is less than or equal, then a particle . were pcrfog~ed. ?he hrst contpnrcs 111c-r nndom 
moves to the right, else, it moves left. numbers with the constant prohahility of decny. ) of n 
STEP 6: Repeat step8 4 and 5, each time ch%cking pyticle (where \) is ihe decay constant nnd L is in I 
when, first, half of the total remains in the left ubtil second in this case) with the assumption that each 
time nllowed elapse particle has constant probability of dccny p a  unit 
STEP 7: Another system may be worked for, say, 'time, and this probnbility must be tnucli lesser U~nri I .  
doubling the former, and step 2 la r e m e d  to: While the second uses poisson distribution. \ \ i t l ~  
The results below are for a particular sequence of random numbers conlpnred with vnryiitg probnbilit ics 
tundom numbers. of decay, NA At, where each pnrticlc l ~ cso~ ulnnt 
l'ARTICLf7,TOTAL = 10 prc1lrnl1ilily rind N is tllc I U I I ~ I I I  ptt111cIt. tt1111t1tcl1~ * ~ l  
PARTICLE LEFT lN LEFT PART 4 nncr litlrc t 'I'l~r, r nisson tlist~i1 ~11ltr l t l  111 t r r r c l l l ~r 
M P R .  T W S ) T O REACH EQUlLIBKIUM = 9 involvpd dividing. 191 3,000 i n  lllis cilsc. rllc i l l ~ r r ~ t ~ ~ r t  
PARTICLE-TOTAL P 20 into smnller periods ensuring tllc prnbnllllily for 1s a 
PARTICLE LEFT IN LEFT PART = 10 lot lesser than 1, as a rule. The probability of 
APPR. TIME(S) TO REACH EQUILIBRIUM = 32 obskving, say. n decays is P = P - P)"' rn for m 
PARTICLETOTAL = 40 divisions. Where p is the probability of a siugle decny 
P m n c L E L E m w  EFT PART = 18 per division &d C represents Cornhination 
APPR. TIME!@) TO W H EQU ILIBRIUM = 50 
- 
Zumr Journ. of hire  ind Applied Sclenecr 6(1) 2004 
UNIVERSITY OF IBADAN LIBRARY
Table 2 
Zumr Journ of Pure mnd Applled &Ienesr 6(1) 2004 , 
UNIVERSITY OF IBADAN LIBRARY
Contemporary a~psctr~ f Monte Car10 ... . 144 ( ~ ~ I I I \ V01~ 01I ~ C ~ ~  
.Error Ax~slyeliel or 10 Measurerhnnt~ 
csl.  i0.000 'fii~bea ch 
Z u m n  Journ. at Purr md Applled Bclcncan 6(1) 2Oh4 
UNIVERSITY OF IBADAN LIBRARY
Contemporary aapccts of Monte Carlo ... 145 (>lnrtofcla rr nt 
I 
- - - - - -- 
n b l k  f i  
Error Annlysis for 70,0011 Triztls 
diviricd into 11) Siil~TJivisiorls 
Z u m a J ourn. of Pure rnd Applied Sciences 6(1) 2004 
UNIVERSITY OF IBADAN LIBRARY
'khla 7 
I:~iigS n mpla !vlifl?nn hlct,had with Sctlilanc.e of F'ctiorl 2. 
'J!able 8 
Error .Jia~l~sfiosr  10 r~~eavurer~~uel rl~U ,lOsO O  ' k i ~ l n  
eadi t l s i n ~S equeilce uf Pcriocl 2 
Zumr Journ. olPure and Applied Sdonces 6(1) 2004 
UNIVERSITY OF IBADAN LIBRARY
- -  
Table 9 
Error A11~1,yvhir~  10,000 Trials Dividcti into 
10 .iI(?uurt~metlCsf or Sequcnar? of Pariot] 2. 
Zuma Journ. of Purr and Applled Sclencsr q1) 2004 
UNIVERSITY OF IBADAN LIBRARY
loi 
Graphs of both were plettsd and a third graph showed A little trick was lased to draw thc decay graph 
the paltcm obtained fiom the equation of radioactive (Figure 7) of the poisson distrihdii~np roccdtue on 
decay N(t) = NoeJ\t. the same scale as the others, an nwrnged .particle for 
every ten steps was taken and recorded as number of 
Z u m a  Journ. oTPur8 and Applled &ieneea6(1) 2004 
e 
UNIVERSITY OF IBADAN LIBRARY
Contemporary aspectr of Monte Carlo .. .  149 ~ ~ l ~ ~ ~ctv ..o ~ c l a  
parent particles left per unit time. It worked because achieved. Special plapose nlgorithms (fc~uuid iri 
of the large value 3,000 used in dividing. n~unericnll ibrnries) are used iri this silrrnhcw Gctlernl 
Tho result (Figure 7) shows the mean and variance purpose method nccottlirig to nri nrllitrnrv ~ I~s t t i \~ i t t t (~ t~  
for the exponential distribution of which the decay filnctio~ic nti also I?c I I < ~ F Ii rislc:t(I. 1111 c ~ : t ~ ~ r1l1jctlr~lg 
equation is an example. the variance being the square "rejection technique", i.e. flit-Miss ~ncth~wl\v. llicli 
of the mean which is itself the inverse of the decay was described earlier I.Io%vever tliis metllcxi IS not 
constant. The same sequence was used to generate the suitable for distribution with one or more large peaks 
simulated decays but .the one through "unaided" Another method which is suitable for relntivclv 
application deviated quite differently. Hence Uie'need simple tlistrihr~tiori litrictior~s is llic "inw.crsio~ 
to generate random numbers according to particular technique". The inversion techniques has Ute 
distribution in some cases. The Poisson distribution, following steps: (1) Normnlisntion of the distribution 
however, faired much better with its graph and report function to get a pdf, (2) integration of  the pdf 
on remaining prent particles; as confirmed by analytically fkotn the rninitnrrln s to nn nlllihnry s. 
calculation tluough the decay formula. representing tile prnb nhility or choosing n \pnlitel ess 
than x and (3) Equnting,this to a uniform nrlrnber and 
Conclusion solving for x. The resulting x will be distributed 
Working with Monte Carlo methods can' be according to the pdf. This method is considered 
easy, representative of the de sired process or efficient. 
opcrotion nnd fun but requires great care in some 
instances. This is because not the same sequence'of 
random numbers can be npplied in all physical References 
processes and the same degree of accuracy expecteb l ln~veyG OIIIII n11d Jnn 'Vt!lyclti~tk ( IQSS) .  :\(I I I I I I ~ ~ I I I ~ ! ~$II*~ I  
ill nll cnqcs. Tlia way most, if not nll, c o t ~ i p d c ~ s  Cnlnptller 
gencralc random numbers is such thnt at thc cntl of n Simttlnlinm: Applle~lin~11a1 ~ ' l l~*icnlS'  SS~CII( I'II I I~  1 ?) ..\~I~IsoII- 
ccrtnin program which uses random numbers, when n Wesley ll~hlirliingC nlttpntry 
ncw program which uses random numbers.as well, is Sen, S.K. (1967): Simulation of (Ire Raylciglt-Pcarr~rnr andom walk, J. md. Inst. S~i~49.19-27 
started it makes use of the same sequence used by the Malvin H Kaloa and Paula A \VhiUock (1986): h.fontc Carlo 
earlier program picking each number in the same Methoda. Vol. I: Basics, John \Vilcy Sons. 
order.. Rcwen Y.Ruhingtein (19Rl).Sin~t1lntion ant1 Ille h1011te ( '~ r to  
Spccinl sequences can be developed (sequences Mcthod John \Villcy So~,*,'( 'om~t~~lntioI'l~ly~riacl. John R. Tnylor (1982).,\11 Itttrod~~ctiont o 61~crrA ~tntysis. 
according to specific distributions) for different University Science Rooks. OxFord I!nivcrsity Press. 
processes or operations so that better results are 
UNIVERSITY OF IBADAN LIBRARY