Using symbolic capabilities of Maple Statistics package for enhancing the concepts of probability theory and statistics

Transcription

1 Using symbolic capabilitis of Mapl Statistics packag for nhancing th concpts of probability thory and statistics Mihály Klincsik, Phd Dpartmnt of Mathmatics, Univrsity of Pécs Pollack Mihály Faculty of Enginring, Hungary Abstract Th symbolic manipulations now ar availabl not only for th xprssions or functions howvr also with th random variabls by using Statistics packag of Mapl Computr Algbra Systm (CAS. Ths facilitis ar uniqu among th CAS programs. Such an implmntation of th random variabls is a similarly hard task as to raliz symbolic manipulations with algbraic xprssions. W us ths capabilitis of th Mapl CAS in ordr to bttr undrstand th mathmatical concpts of th randomnss by th studnts. Ths ar usful tools for studnts to Manipulat symbolically with diffrnt random variabls and asking from th Mapl CAS to answr for spcific faturs of thm Exprimnt with th random variabls to obtain som nw rsults So w can dmonstrat thortical rsults by using th symbolic facilitis of th computr program without proving mathmatically thm. What is th appropriat ratio btwn th rigorous mathmatical vrifications and th computr dmonstrations? Th answrs ar varying from cours to cours and from topic to topic. In this lctur w xamin th usability of th random variabls of th Mapl Statistics packag throughout xampls. Kywords: Mapl Computr Algbra Systm, larning procss, random variabls. Introduction. Whn th random phnomna ar invstigating and modling by using computr programs thn th numrical calculations ar powrful tools howvr it is a mor grat challng to utiliz th symbolic capabilitis of th systms. Th concpt of random variabls can b larnd by xampls. Whn a thortical distribution is usd without spcific xampls thn w hav to mak som mathmatical abstractions. In this procss, w highlight th common faturs of th xampls, such as formula of th probability mass function in discrt cas or dnsity function in continuous cas, and th paramtr(s, and so on. If w want to us probability distributions by using th Statistics packag in Mapl CAS thn w may choos from th built-in distributions. Lt us considr for xampl th wll-known binomial distribution with th paramtrs (n,p. W want know how can gt information about th typ of this distribution, th conditions for th paramtrs, th formula of th probability function and th man valu and so on. Now applying th following Mapl cods and th procdurs answr for ths qustions.

2 > B:=Statistics[Distribution](Binomial(n,p; B := modul( xp ortconditions, ParntNam, Paramtrs, CDF, CharactristicFunction, Kurtosis, Man, MGF, ProbabilityFunction, Skwnss, Support, Varianc, VariationCofficint, CDFNumric, QuantilNumric, RandomSampl, RandomSamplStup, RandomVariat ; option Distribution, Discrt ; nd modul So B is a modul in Mapl with th procdurs listd abov and th option discrt, i.. it is a discrt distribution. If w want to know th conditions for th paramtrs n and p thn us th Mapl cod > B:-Conditions; [ 0n, 0p, p ] W gt that n is positiv and th probability p is btwn 0 and. Th valu of th probability function at th plac k is giving by th calling squnc > B:-ProbabilityFunction(k; > `Expctd valu`=b:-man; Expctd valu n p W can gt also th xpctd valu of distribution binomial(n, p by th formula. Now w dmonstrat th possibilitis of th symbolic manipulations with random distributions and variabls giving two xampls which wr workd out at th univrsity cours in computr laboratory or homwork nvironmnt. 2. Symbolic manipulations with th paramtrs of th random variabl In th first xrcis w ar looking for on of th paramtr of th binomial distribution on th basis of a givn sid condition.. Exrcis. Cars arriv at a busy intrsction whr th traffic is controlld by traffic lights. It was found on avrag that ach fourth car turn lft to th sid road, with th maximum capacity 5 cars up to th nxt traffic light. What is th maximum numbr of th cars waiting at th traffic lights such that th numbr of th turning cars dos not xcd th capacity of th sid road with at last probability 0.9? (Assum that th cars turn indpndntly from ach othr. Solution. Lt us dnot by X th numbr of cars which ar turning lft at th traffic light from all of th n cars. Sinc w assumd that th cars ar turning lft by indpndntly from ach othr with th sam probability thrfor X is a random variabl with binomial distribution with th paramtrs n which is unknown intgr and { 0 k0 k p k ( p ( n k othrwis which is th known probability. On th basis of th givn condition w may writ th is th random numbr of th turning cars. 2 inquality, whr Dfin th random variabl X by using th command RandomVariabl from Statistics packag > with(statistics:

3 > X:=RandomVariabl(Binomial(n,/4; X := _R W can obtain th probability by using th CDF(X,5 Mapl command bcaus of this procdur computs th cumulativ distribution function F(u=P(X<u of th random variabl X at th spcifid location u=5. Sinc th variabl X contains th paramtr n without spcific valu thrfor w can dfin th function > G:=unapply(CDF(X,5,n: > Equation:=G(n=0.9; Equation := n n 4 ( n ( n ( n ( n This quation is a rathr complicatd non-linar quation so that w can t solv it by symbolically. Thrfor w hav to us numric and/or graphic mthods for finding th maximal valu of n. Calculating diffrnt valus of G(n in a squnc from th starting valu 5 up to final valu 20. > sq(['n'=n,valf(g(n],n=5..20; [ n5,. ], [ n6, ], [ n7, ], [ n8, ], [ n9, ], [ n0, ], [ n, ], [ n2, ], [ n, ], [ n4, ], [ n5, ], [ n6, ], [ n7, ], [ n8, ], [ n9, ], [ n20, ] ( n5 0.9 W can s that th probability quals whn only 5 cars ar waiting at th traffic lights; bcaus of in this cas it is sur that th numbr of turning cars cannot xcd th 5 cars capacity of th sid road sinc in th worst cas all of th 5 cars turn to th lft. Whn th numbr of th waiting cars is incrasing thn th probability of th vnt will bcom smallr and smallr. Similarly th probability of th complmntary vnt X>5 will b gratr and gratr. Whn th numbr of waiting cars n= thn th probability probability howvr in th cas n=4 th. Thrfor th maximum valu of n = such that th numbr of th turning cars dos not xcd th capacity of th sid road with at last probability 0.9. Plotting th stp function G(n and th probability lvl 0.9 and th location n= of intrsction of th two curvs w can visualiz th prvious calculations by th following figur. > lin:=plot([[,0],[,g(]],linstyl=,color=[blu]: > lin2:=plot([[5,g(],[,g(]],linstyl=,color=[blu]: > curv:=plot([g(floor(n,0.9],n=5..20: > plots[display]([lin,lin2,curv],titl=`th probability P(X<=5 and th lvl 0.9`;

4 Fig.. Dcrasing th probability whn th paramtr n is incrasing From this figur w can solv th abov givn non-linar quation by using th Mapl fsolv numric solving procdur bcaus of w know a prliminary intrval for th valu of n. > fsolv(equation,n=..4; Functions of random variabls W can us transformations on continuous random variabls and th Mapl can calculat symbolically th spcific faturs of th transformd variabl. In th nxt xrcis w want to calculat th probability distribution of th minimal valu of two indpndnt random variabls with xponntial distributions. 2. Exrcis. Considr a systm with indpndntly working componnts A and B arrangd in sris as shown in figur 2. Fig. 2. Systm with two sris componnts Th oprating livs of A and B ar dnotd by T and T 2, rspctivly, and thy follow indpndnt xponntial distributions with paramtrs λ and λ 2 T ~ Exponntial (λ, T 2 ~ Exponntial(λ Dtrmin th probability dnsity function of th whol systm lif T=min(T, T 2! 2.2. Show that th probability rlation 4

5 P(T>t=P(T >t P(T 2 >t is fulfilld btwn oprating lif of th componnts. Solution Lt us dfin th random variabls T and T 2 as xponntial distributions with two diffrnt paramtrs. > with(statistics: > T:=RandomVariabl(Exponntial(lambda; > T2:=RandomVariabl(Exponntial(lambda2; T := _R Considr th probability dnsity functions of th two givn variabls. > f,f2:=pdf(t,x,pdf(t2,x; Now w can s that th paramtr is th rciprocal valu of th usually givn paramtr. Dfin th minimal valu of T and T 2 with th min function of Mapl. > T:=min(T,T2; Now w want to know th dnsity functions of th variabl T. > PDF(T,x;f:=simplify(%; T2 := _R0 0 x0 f, f2 := x, othrwis T := min (_R, _R0 0 x0 Aftr th simplification command w gt th final outcom which is a thortical rsult: THEOREM: If T ~ Exponntial (λ and T 2 ~ Exponntial(λ 2 ar indpndnt variabls, thn th distribution of thir minimal valu T=min(T, T 2 is also an xponntial with paramtr x othrwis 0 x0 0 x0 undfind x0 0 x0 0 x0 undfind x0 x x x x othrwis othrwis 0x 0x f := 0 x0 undfind x0 x ( ( 0x 5

6 Prov th statmnt 2.2 also by using th Mapl function CDF (Cumulativ Distribution Function. First of all w assum that th tim t>0 is positiv. > assum(t>0; > (-CDF(T,t*(-CDF(T2,t=simplify((-CDF(T,t*(-CDF(T2,t; t~ t~ Now w ar rady to prov th quality in 2.2 without using computr for t>0 t~ ( /bcaus of indpndnc of T and T 2 / From this rlation howvr w can prov also th thorm without using computr. Author addrss: Mihály Klincsik, Phd Dpartmnt of Mathmatics, Univrsity of Pécs Pollack Mihály Faculty of Enginring, Boszorkány str. 2., H-7624 Pécs, Hungary -mail: klincsik@pmmk.pt.hu 6