densidad probabilidad

7/29/2019 densidad probabilidad

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330 sTMULATToN MoDELTNG AND ANALysrsTABLE 6.3Continuous distributions {,BLE 6.3 (continued)4mential expo( jUniform rJfu,b)Possible applications Used as a "first" model for a quantity that is felt to be randoml;rvarying between a and b but about which little else is known. TheU(0,1) distribution is essential in generating random values from allother distributions (see Chaps. 7 and 8)(tfG):l b- a if a=x


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SELECTING INPUTPROBABILITY DISTRIBUTIONS 331

T{3LE 6.3 (continued)L+onential expo( )F@=lL-' ifx>0otherwise'srributionrr3metef;-:ge,-,:i:lance,.{:uleULE'rments

Scale parameter B >0[o'-)BB'0p = *t")T ff,J "l


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332 sIMULATToN MoDELING AND ANALysrsTABLE 6.3 (continued)Gamma gamma(c,B)

#-][L":.

./"LJ- t/T/Density (see Fig. 6.3)Distribution

ParametersRangeMeanVarianceModeMLE

Comments

( B-axd-te-,tp1'):l-- r(")- if x>olo otherwisewhere f(a) s the gamma function, defined by l(z): li t'-'c^for any real number z > 0. Some properties of the gammaf(z + 1) : zl(z) for any z>0, l(k + 1): kr for anyinteger /c, I( + ) = l . I. 3. 5. . .(2k - \ /2k for anyinteger /r, l(ll2)=1-,If a is not an integer, there is no closed form. If a is a porhninteger, then*':{; -'-',7"W if x>oll otherwiseShape parameter a ) 0, scale parameter p ) 0[0,-)aBoB'9@-l)ifa>1,0ifarn,h 4 +.rr() aB: *@)which could be solved numerically. [V():f'()/f()called the digamma function; f' denotes the derivative ddiAlternatively, approximations to and B canbe obtained by lct7:[lnX(n)-l,i=,lnX,lnl-', using Table 6.19 (see App. Al

123m[xE 6.3ppm,at a,1) density functions.

E[Lf 6.3 (continued)*r[obtain as a function of T, and letting B = X1" n. [See Cbri dWette (1969) for the derivation of this procedure and of Table1. The expo(B) and gamma(l,p) distributions are the same2. For a positive integer m, the gamma(z,B) distribution is crruthe m-Erlang( B ) distribution3. The chi-square distribution with & df is the same as &gamma(k I 2, 2) distribution4. If Xt, X2, . , X- are independent random variables wirh -f, -gamma(a,,B), then X, + X2+ -..+ X^-gamma(a, t a- -- -,*+ "^' F)5. lf Xl and X, are independent random variables with f -gamma(a,, p), then Xrl (X, + Xr) - beta(a,ar)6. X-gamma(a,B) if and only if Y:llX has a Pearson tr6 Wdistribution with shape and scale parameters a and I I B, deodP"t(a,t t p)7.

Weibull(c.,

(a ifa1

Fm;ole applicationsIhmutl (see Fig. 6.4)IlmrbutionpuTmeterst:xru_eMm"ffiamce

fMrrJe

Time to cgamma der/('): {;p"(r: {-Shape para[0,-)4 r(r ra \aa{zr(t-,dt alu("-'lo' a


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SELECTING INPUT PROBABILIfi DISTRIBUTIONS 333f(x)

,9,-l

o=1

FIGURE 6.3pnma(o,l) densitY functions.

lf"{BLE 6.3 (continued)

Io= 2

ffieibull Weibull(c,B)P:ssible applications

)cnsity (see Fig. 6.a)lsrributionPr-ametersfl"rngehean-eiace

lll'Lad

Time to complete some task (density takes on shapes similar togamma densities), time to failure of a piece of equipmentn:["9-"x"-te ('tP)a if x > oJ\'-' t0 otherwise{1-"-"'P'" ifx>0lxl = 1L 0 otherwlseShape parameter a )0, scale parameter B >0[0,-)P rl1)a \a/({,(?)- :['(:)]']lu("-t\''" ira>l1'\ d /Lo ifacl


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334 sTMULATToN MoDELTNG AND ANALysrsTABLE 6.3 (continued)Weibull Weibull(c,B) il4MLE

Comments

The following two equations must be satisfied:2,*i rn", _ !"ard

i*1The first can be solved for numerically by Newton's method. mdthe second equation then gives B directly. The general recursive sc$for the Newton iterations is, A+7ldk-cktBku,l_-k+r ** lldo'+lBLHk- co')lBo'

Bo=), x,uu, cr:i xiuklnx,t=l /=1ar=i x,k(rn x,)2[See Thoman, Bain, and O"ri" trn6nl for these forrnulas, as well r*for confidence intervals on the true a and B.] As a starting point fmthe iterations, the estimate, l\lrrnxl'-(* x,\' /^ll-''o- \ 7f-LFt \,-, 't / I i ,-l J

[due to Menon (1963) and suggested in Thoman, Bain, and A(1969)] may be used. With this choice of 0, it was repoed nThoman, Bain, and Antle (1969) that an average of only _5Newton iterations were needed to achieve four-place accurac!-.1. The expo(B) and Weibull(1,p) distributions are the same2. X-Weibull(a,B) if and only if X" -expo(B") (see prob. J3. The (natural) logarithm of a Weibull random variable ha Idistribution known as the extreme-value or Gumbel distribrw[see Law and Vincent (1990), Lawless (1982), and Prob. 8.lt{fl4. The Weibull(2, p) distribution is also called a Rayleigh distrifution with parameter p, denoted Rayleigh(B). If Y and Z qindependent normal random variables with mean S d yi*lmB2 (see the normal distribution), then X: (Y2 + Z2)': -Rayleigh(21l!)5. As a+@, the Weibull distribution becomes degenerate at $,Thus, Weibull densities for large a have a sharp peak at rhmode6.

+ B(+) '") rn{

n

.l.d -

ll 4 -,11

0.5 1.0M6[}E .4rtji,ebcril(a,1) density functions.

:,ILE 6.3 (continued)qfrt[allmsble applications

lensir-v (see Fig. 6.5)lmrbutionPns.metersnmgefufuff,[1mrcelfffl-Elmments

N(r,o

**':{i:i;:i

Errosquanrrtue ,f(x):No clolLocatic(--,-)It 2olL: *r1. If f\

latedortr


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SELECTINGINPUTPROBABILITYDISTRIBUTIONS 335f(x)

1.0.5IIIGI,]RE 6.4'n ibull(a,1) density functions.

T{BLE 6.3 (continued)qmsd N(t,o')hssible applications

Ie:nsitv (see Fig. 6.5)lrmrbutionun-metersillmgEWsunriw:relMi3ermLEl;nnments

Errors of various types, e.g., in the impact point of a bomb;quantities that are the sum of a large number of other quantities (byvirtue of central limit theorems)f(x\: -L e-G-Ptzt2o2 for all real numbers !2o'No closed formLocation parameter r (-w,e), scale parameter o ) 0(--,-)p

2p, -- *@) . 6:l' -nt s'tu)]' '1. If two jointly distributed normal random variables are uncone-

lated, they are also independent. Fo distributions other thannormal, this implication is not true in general


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336 STMULATToN MoDELTNG AND ANALysrsTABLE 6.3 (continued)Normal N(p,o') UbmLE 53 (continued)lfiSrel

il'nnmlrre applications

Ilrns:] tse Fig. 6.6)lmr:i:,donMtr:elersfr*mgtlt{ta'i(t&:r.f]{elll0irp

,nd[.-E

l-mcrents

F-lCrrnE 6.6.}'t 0,o'z) density functions

2. Suppose that the joint distribution of X,, Xr, ..,X- is mul_tivaiate normal and let p.,_: E(X,) and C,,:'Cov(X,,i,). Thenfor any real numbes a, br, b2, . . . , b^,'th. .uniorn variablea -t brX, + b2X2+ . . . + b^X^ has a nrmal distribution with1-cdo

\SIre

:ee

mean = a + ti, b,p., and varianceo'= ) ) b,b,c,,=l t=lNote that we need not assume independence of the X,,s. If theX,'s are independent, then

o, :i b,z yar(x,)3. The N(0,1) distibution ,.'lr,"n called the standard or uninormal distribution4. lf Xt,Xz, . ,X* are-independent standard nomal randomvljalte thgy Xr2 + X22 + . . . t Xorhas a chi_square distributionwith tdf, which is also the gu ^1ktZ,Z.) distibution5. If X-N(p,at), then et has the logno:rmal distribution withparameters p, and o, denoted LN(r.,or). If X- N(0,1), if y has a chi_square qistribution with k df, and ifX and Y are independent, then X/\/-yTE has a distribution wirhk df (sometimes called Student's t distribution)7. If the normal distribution is used to represent a nonnegativequantity (e.g., time), then its density should be tuncated at x : 0(see Sec. 6.8)8. As o+0, the normal distribution becomes degenerate at &

1.0.5FIGURE 6.5N(0,1) density function.

f(r)


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SELECTINGINPUTPROBABILITYDISTRIBUTIONS 337T.{BLE .3 (continued)Lognormal LN(r,o2)Fssible applications

)ensity (see Fig. 6.6))istribution?aametersRange\fean'r riance\fodeULE--Jmments

Time to perform some task [density takes on shapes similar togamma(o,B) and Weibull(a,B) densities for a-> L, but can have aiarge "spike;' close to .x:0 that is often useful]; quantities that arethe proiuct of a large number of other quantities (by virtue ofcentral limit theorems)f r .---1lnx_r)' ifx>o-exD-= ,ft')={ *xfr}'"v 2/l.o otherwiseNo closed formShape parameter o > 0, scale parameter p (-o,o)[0,-)" +n2/2e',rr*,'7"-' - l)e, -o' " f " lttzjrnx, l),'"x,-)'lp=7'i =L=- , l

1. X-LN(.r.,o' if and only if lnX-N(p,u'?)' Thus' if one hasdata Xt, Xr, . . . , X^ that are thought to be lognormal' thelogarithms tf th" dutu points, ln Xr,lnXz, " ',lnX,, can betrJated as normally distributed data for purposes of hypoth-esizing a distribution, parameter estimation' and goodness-of-fittesting2. As a-0, the lognormal distribution becomes degenerate at ep'Thus, lognormal densities for small o have a sharp peak at themodeS. Itq /(t):0, regardless of the parameter values

0.5 l.uFIGURE 6.6-\(0,o'z) densitY functions


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338 STMULATToN MoDELTNc AND ANALysrsTABLE 6.3 (continued) TABLE 6.3 (continued)Beta beta(arrc.r) Beta bfa(alPossible applications Used as a rough model in the absence of data (see Sec. 6.9!distribution of a random proportion, such as the proportion ddefective items in a shipment; time to complete a task, e.g., in ePERT network

for any real numbers zr ) 0 and z, > 0. Some properties of the betefunction:B(zr,zr)= B(zr,zr), B(z,,zr)=W

No closed form, in general. If either dl on d.2 is a positive integer, ebinomial expansion can be used to obtain F(.r), which will be polynomial in .r, and the powers of .r will be, in general, positircreal numbers ranging from 0 through a, I a, - '!-Shape parameters a, ) 0 and a, ) 0[0,1]dlar1 a,

dtQzG,+"rt@,+",+1)

4. X-5. X-\.I d1.*6. Theisa7Density (see Fig. 6.7)

Distribution

ParametersRangeMeanVariance

Mode

Comments

I8. Thethe r

I a.-7I;-:t- if ar)l,arlII d.rt a2- ]Oanar if a,11,ar11.l0 if (o,t)| 1 if (a, > 1, ar


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SELECTINGINPUTPROBABILITYDISTRIBUTIONS 339TABLE 6.3 (continued)Beta bela(a'o.r)

6.7.

4. X-beta(a,ar) if and only if 1- X-befa(a',ar)5. X-beta(a,,cr) if and only if Y: Xl(l- X) has a Pearson typeVI distribution with shape parameters c' cr and scale parameter1, denoted PT6(ar,ar,1)The beta(1,2) density is a left triangle, and the beta(2,1) densityis a right triangle(e ifa,(l l- ifa,l 'r lo ifa,>lThe density is symmetric about : I if and only if ar : 4.. Also,the mean and the mode are equal if and only if ar: a,

0.4 (.),L RE 6.7, :..a,) density functions.

,:son type V PTS(a,)rie applications

::v (see Fig. 6.8)

Time to perform some task (density takes on shapes similar tolognormal, but can have a larger "spike" ciose to x :0)( x to'r\e P \l" if y)0frx:1 B-"1(a)l.o olherwise

al=5, q2=1.5

-0.8, cr=8

ot-0.8,cr=0.2


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FlE-l1Sh[0.pa:8.,(af8t'ItoIfctheresmaF=1. .2. 1

!3. 1t

3.6

3.0

W sIMULATIoNMoDELINGANDANALYSISTABLE 6.3 (continued)Pearson type V YlS(a,B) t63 rcowinued)tgr 11

4pticationsmt@ry *e Fig. 6.9)fllmimniitmrian

mMIEmEm"ErlilfrmlftmceffitEIffi-E

,iflnments

Distribution ,@=[r-."(+) irx>ol.o otherwisePaametersRangeMeanVarianceModeMLE

Comments

FIGURE 6.8PTs(a,1) density functions.

where Fo(x) is the distribuvariable _rtion function of a gamma(a,l/B) ratrfuShape parameter o ) 0, scale parameter B > 0# for a)lpt(a -1)'(a -2)pa+7If one has data X,, X,, . .,*1'".*i.1; i# 1 r' ";;;"jlirl,,l" : iil:tr"i:i-mthe PT5(a, B ) are '=r^ lf tu :l-lmum-likelihood estimators tuI. x-P"15(a,B) ir an'J"" P - 'tPG (see comment t below):;^x;x:#i:,"f i,:;11,ffi ;'#;j:rui,:?i:th2. Note that the mean andthe shape p;;;;;r** va'ance exist only for certain value d

for a>2

lriryular

0.5 1.0 rls 2.0 )0lO otherwisewhere Fr(x) is the distribution function of a beta(a'ar) randomvariableShape parameters a, >0 and a, >0, scale parameter B )0[o'*)I for a,>1d- |E'"J".,,I,",- )) for a.)2(a,- l)"(a,- 2)Ig@,--t) ifa,>11 dzr tl.o otherwiseIf one has data X, X2,. . . , X,fhatare thought to be PT6(4,'or,1),then fit a beta(a,,or) distibution to X'l(1+ -{) for i=1,2'. .. 'n'resulting in the maximumlikelihood estimators , and r' Then themaximum-likelihood estimators for the PT6(a',4r,1) (note thatF : 1) distribution are also , and , (see comment 1 below)1.. X - PJ6(a'ar,1) if and only it Y = X I G + X) - beta(a',o')2. lf Xt and X, are independent random vaiables with X, -gamma(a,,B), then Y= X,lXr-PT6(a,dr,F) (see Prob. 6'3)3. Note that the mean and variance exist only for certain values ofthe shape parameter 42

Triangular triang(arbrc)Possible apPlications

Density (see Fig. 6.10)

Distribution

PaametersRange

Used as a rough model in the absence of data (see Sec' 6'9)[ 2(x..a) if.a-x


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342 sTMULATToN MoDELTNc AND ANALysrs

(.) oFIGURE.I.gPT6(a,,a.,1) density functions.TABLE 6.3 (continued)

Lwless (1982) for other aion function (if it exists i"Jescription of the param(mdicates the interval where-{lso listed are the mean (er which the density funolnkelihood estimator(s) of tomments include relations:utions. Graphs are givennotation following the nemJistribution, which includestd as."Note that we have incTpe VI distributions, becaprovide a better fit to dataFr_e. 6.19) than standard dis5.2.3 Discrete DistributiThe descriptions of the sixattern as for the continuol

T-TBLE 6.4lXcrete distributionst rtroulli BrtrTriangular triang(ar,c)MeanVarianceModeMLEComment

aIb*cJa'+b2+ct-ab-ac-bc------- l

cOur use of the triangular distribution, as described in Sec. 6.9, is aa rough model when there are no data. Thus, MLEs are ndrelevantThe limiting cases as c+ and c+a are called the right triangulgand left trinngular distributions, respectively, and are discussed inProb. 8.7. For =0 and :1, both the left and right trianguladistributions are special cases of the beta distribution

Pssible applications

M"ass (see Fig. 6.11)trustribution}Br'meter*-rngeMean-rbiance!,{odeh{LEJmments

Ra


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]TING INPUT PROBABILITY DISTRIBUTIONS 343Lawless (1982) fo. o,rn:l apprications]. Then the density function and disrribu_ion function (if it exists ln ri.pr""'.i.*d {orm) ur" iir,""'exr is a shortescription of the parameters, including their'possibi"- 1u".. The rangendicates the intervaf where ," ""."1"i"i random'variaute can-tate on values.iso listed are the mean (expe.r"-""i""j, u3ri31_"? and mode, i.e., the valuet which the density runctin rr n'*irr"d. MLE ,"t ., -'rne maximum_ikelihood estimator(s) .of rl: p";;;;i;,.tr"ut"Jlui; ;""". 6.5. Generalomments incrude relationshipi of the oisiriution """, ,r"iu to other distri_utions' Graphs are.givn "r'r" j"^iiil;il;;r"#"#Tr,r,oution. Theotation foilowins rhe name of each distribtion ""i"ul.*iation for that:f[:,ytt*' whic incrud"r trr" pu.u-Jr.. rn" symbol - is read ,,is distribut_Note that we have included the less fam'iar pearson type v and pearsonype vI distributions, because *" rruu" found that these dlstributions oftenrovide a better fit to crta r",. *rro." rr-iragru-, are skewed to the right (seeig' 6'19) than standard distributio; -";, gamma, weibuil, and rognormal.6.2,3 Discrete DistributionsThe descriptions of the six discrete distributios in Table 6.4 follow the sameattern as for the continuous distributionsln Table 6.3.T.1,BLE 6.4Discrete distributionsBernoulli Bernoulli(p)Possible applications

\lass (see Fig. 6.11)DistributionParameteRange\{eanVariance

Mode\fLEComments

Random occurrence with two r.."Fother discrete random urll-lltttot" outcomes; used to g"n"tut"negative binomial lnates' e'g'' binomial' geometiic, andl1-p ifx=opG)=lp ifx=1|.O orherwisefo ifx


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Y4 SIMULATIoN MoDELING AND ANALYSIsTABLE 6.4 (continued)Bernoulli Bernoulti(p) TABLE 6.4 (continued)failsandx=lif ir-experiment, ort"n "ul"TT*s' then x-BetnoufiJ- s-*ilway of relating ,ru"ruro 1l"'n,lulli trial' provides u "onu"oi.olnoulli distribution I other discrete distributions to the Be_t f.i"i'",i,,?Tffi.'ffi*"T;ili I l,; -,. .x, ut: independent

ll'1"ll'l:Hi,'.""#'L lf::T::i+, 2; o, +ff::T'ffi"#;, ,:::ir:::" .{;:n:*'#:ff H:i|.. of ,u""..,"rr

Discrete uniform

3. Suppose we begin ,.rirg iJ"pla",""r *li:il"", of a Bernour_;"fi1,-# #:?::,,;,i., or.',,....,' ...""n 1.,", rhen thegeometric distribution o/.,o-J" ob"-ing the first success has ai" nutn.;'oi'lli;;.Jli.o"rameter p. For a posirive integer s_negative binomial otr,.,n?.ll-o:t observing the sth success has a+. rni ernouil;i ;#liron with parameters s and pi.t.iuuio"*i*]ln'rTi"uout'on is a special case of rhe binomial.nd the same value for p)

Mean\arianceMode\{LECommenl

I (i-t+ IP (x)

Binomial

x FIGURE .11Bernoulli(p) mass function (p > 0.5 here).Possible applications

Mass (see Fig. 6.13)

Distribution

PaametersRangeMeanVarianceMode\fLE

Discrete uniform DU(i,j)Possible applications Random occurrencets equally likety; used as a.-fi.st,. ;;"ru;'"luanrity rhar ,*fi'#T among rhe integers i through, bri;;;;,il;h litre else isMass (see Fig. 6.12)

Distribution

ParametersRange

(tn@:l/=r lTT if x {i, i + t.... , }[o otherwise(0 ifx


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345'i1i,ti1r

illrll TABLE 6'4 (continued)Discrete uniform DU(r,i)MeanVarianceModeMLEComment

1/(i-t+ I

i+j2(i-i+t)'z-1t2

Does not uniquelY existi: min X^. j= ,q,u*-&ffr.'=oto'f l and Bernoulli(!) distributions are the same

FIGURE 6.12DU(t,i) mass functionBinomialPossible aPPlications

Mass (see Fig' 6'13)

Distribution

ParametersRangeMeanVarianceModeMLE

Number or suuucrrvr'^' ' T;;;"r of "defective" items in a batchty p of success on each trialof size ; number ol rtems in a batch.(e-g'' u *11-ot people) oft"""*'itr"t .t-ber of items demanded from an inventory[(')o'1r-rl' ifx{o'1""'f}p(x)=l'x' orherwise*h*" (i) is the binomial cofficient' defined bv('):5if


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346 sIMULATToN MoDELTNG AND ANALysrsTABLE 6.4 (continued)Binomial bin(f,p) TABLE 6.4 (continued)

following approach could be taken. Let M : max X,, and fork=0,1,...,M,Iet /o be the number of X,'s>iltt"n it can beshown that and p are the values for and p that maximize ,lpfunctionM

s(t,p): ) rn(-k+r)+nttn(t-p)+ nilnm j- o.'o .Pr r,is easy to see that for a fixed value of t, say /0, the value f p tt,*,maximizes g(ro,p) is X1ntto, so i and p ari the values of adX(n) I t that lead to the largest value ot gV. *@) t tl for t e {M, M -I,. . . , M' j, where M' is given by [see DeRiggi (19g3)]''=l *vtu-tt ILt-t (,'-v("yx-(dlNote also that g[t,*@)/t] is a unimodal function of 1. If Yt,Yr, .., Y, are independent Bernoulli(p) random vari-ables, then Yt + Y2 + ... + yt-bin(r,p)2 If X1, Xr, . . , X- are independent random variables and .{ _^ \n(t,,p), then X, + X2+...+ X^^-bin(r, + tz+...+ t^,p)3. The bin(r,p) distribution is symmetric if and only if p-'t^'4. X -bin(t,p) if and onty if I - X -bin(t,t - p)5. The bin(l,p) and Bernoulli(p) distributions are the same

Geometric geom(

Comments

FIGURE 6.13bin(r,p) mass functions.

4 5 6 7 8 9t0

Possible applications

Mass (see Fig. .14)DistributionParameterRangeMeanVarianceModeMLEComments

0t[IGI.]RE 6.14eom(p) mass functions.

Numbtdent Enumbeilem; rdemanp(r):r("):pe(0{0, 1.1-pp1,-p

2p0'x1. If Ivar2. Ifthepar3. Thtialtio4. Thbu

p (x) p (x)

PG)', ()


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SELECTINGINPUTPROBABILITYDISTRIBUTIONS 347TABLE 6.4 (continued)Geometric geom(p)Possible applications Number of failures before the first success in a sequence of indepen-dent Bernoulli trials with probability p of success on each trial;number of items inspected before encountering the first defectiveitem; number of items in a batch of random size; number of items

demanded from an inventory^,., -[n(t - tl' ifxe {0. 1.. . .}Y\^' to otherwiseFtrt=[ I - (l -p)r'r'I if >ot 0 otherwisep e (0,1){0,1,...}1,- pp

Mass (see Fig. 6.14)DistributionParameterRangeMeanVariance

ModeMLEComments

l- p2p

0"1:-' X(n)+ll. lf Yt, yr, . . . is a sequence of independent Bernoulli(p) randomvariables and X: min{i: { = 1} - 1, then X- geom(p).2. lf Xt, Xr, . . , X, are independent geom(p) random variables,then X, + X2 + .. . + X, has a negative binomial distribution withparameters s and p3. The geometric distribution is the discrete analog of the exponen-tial distribution, in the sense that it is the only discrete distribu-tion with the memoryless property (see Prob. 4.27)4. The geom(p) distribution is a special case of the negativebinomial distribution (with s = L and the same value for p)

0.60.50.40.3n)0.1

34FIGURE 6.14_leom( p) mass functions.

p (x) p (x)


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348 sTMULATToN MoDELTNG AND ANALysrsTABLE 6.4 (continued)Negative binomial negbin(s,p)Possible applications Number of failures before the sth- success ln u ."qu*."1iliIf,!I-dent Bernoulli trials with probability p ot ,,r"""r. n each trial;number of good items inspected befoie encountering the sth defec-tlve rtem; number of items in a batch of random slze; number ofitems demanded fom an inventoryMass (see Fig. 6.15)

DistributionParametersRangeMean

VaianceMode

MLE

Comments

p@) =[('* i- 1) n'


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p (x)SELECTING INPUT PROBABILITY DISTRIBUTIONS 349

p (x)0.300.250.20indepen-each trial;defec-number of

urtknown,X(n). Letnumber offor s and p

- p)ft(so,p) is+ 1l),the values ofvalue ofa unimodalwhen tovariables,

randomand X,-

same

"o | 2 3 4 5FIGURE 6.15negbin(s,p) mass functions'

0.150. l0

TABLE 6.4 (contnued)Poisson Poisson(,\)Possible applications

Mass (see Fig. 6.16)

Distribution

ParameterRangeMeanVarianceModeMLEComments

Numbe of events that occur in an interval of time when the eventsare occuring at a constant rate (see Sec' 6'10); number of items in abatch of ,undo- size; number of items demanded from an inventory( -^'

e(,r) = t xr ifxe{0,1,...}otherwiseifr


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P@\ p (x')350 STMULATToN MoDELTNG AND ANALysrs

012 34 5 6FIGURE 6.1Poisson(,\) mass functions.

6.2.4 Empirical DistributionsIn some situations we might want to use the observed data themselves tospecify directly (in some sense) a distribution, called an empirical distribuon.from which random values are generated during the simulation, rather thanfitting a theoretical distribution to the data. For example, it could happen thatwe simply cannot find a theoretical distribution that fits the data uiqu"t"ly(s.ee.Secs. 6.4 through 6.6). This section explores ways of specifying "-piriistributions.For continuous random variables, the type of empirical distribution thacan be defined depends on whether we have the actual values of the individualoriginal observations x1, x2,. . . , xn rather than only the number of x,'s tharfall into each of several specified intervals. (The latter case is called groupeddata or data in the form of a histogram.) lt the original data are availble, wecan define a continuous, piecewise-linear distribution function F by first sortingthe x,'s into increasing o-rder. L"t \,1denote the ith smallest oi the 4'r, Jthat X1r = X(r)

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Figure 6.17 gives:aprdll' over those rangernssred. Also, for each inrge n) the proportion cw,mld like a continurus,oiruussion of other q'arinucring this particular:rm it during a simularisee Sc. 8.3.12). Also. tm the .{'s (see Prob. 6.If. on the other har:a"hen since we do not krr -{.'s are grouped into I:trt the /th interval corCften the ar's will be eq.d, reasonable piece*iseryrecified by first letting: l....,k.Then,inte4

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