3 Monte Carlo Methods

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1 3 Mote Crlo Methods Brodly spekig, Mote Crlo method is y techique tht employs rdomess s tool to clculte, estimte, or simply ivestigte qutity of iterest. Mote Crlo methods c e used i pplictios s diverse s estimtig defiite itegrls, performig sesitivity lysis o prmeters fit to oisy dt, d solvig differetil equtios. This sectio sketches the proility ckgroud ecessry to uderstd certi core Mote Crlo priciples, d sketches some cocrete pplictios i the settig of itegrtio. 3. Proility Bsics 3.. Rdom Vriles A rdom vrile is qutity X whose vlue is determied y the outcome of some rdom process. Rdom vriles c e discrete (meig tht the possile vlues they c ssume c e eumerted i fiite or coutly ifiite set) or cotiuous (meig tht the possile vlues they c ssume lie i itervl, or collectio of itervls.) The set of vlues rdom vrile c tke o is clled the smple spce. Exmple 3.. (Roll of 6-sided die) Let X e the vlue of sigle roll of fir six-sided die. The X is discrete rdom vrile whose smple spce is the set {, 2, 3, 4, 5, 6}, d the rdom process tht geertes X is the rollig of the die. Exmple 3.2. (Witig times) Let X e the mout of time perso who rrives t certi us stop must wit util her us rrives. The X is cotiuous rdom vriles whose smple spce is the set [0, ), d the rdom process tht geertes X is whtever comitio of trffic coditios, driver dispositio, d mechicl cpcity tht leds to the rrivl of the us. Note tht it could e rgued tht the rdom processes i these two cses re ot relly rdom t ll, ut the product of determiistic forces. The questio of whether or ot true rdomess exists is philosophicl poit tht shll ot cocer us here Distriutios Iformlly spekig, proility distriutio is specifictio of wht vlues rdom vrile c ssume, d how ofte is ssumes them. More formlly, let Ω deote the smple spce of rdom vrile X, d defie evet E s y suset of Ω. A proility distriutio is mppig P from the set of ll evets of Ω to the itervl [0, ] such tht the followig hold:. P (Ω) = (i.e. the proility tht the vlue of X lies somewhere i Ω is.) 2. If E d F re evets with E F =, thep (E F )=P (E)+P (F ). Proility distriutios re geerlly specified y desity fuctios. A desity fuctio is mp ρ :Ω R tht is used to defie the distriutio P. Desity fuctios tke differet forms depedig o whether the rdom vrile is discrete or cotiuous. For discrete rdom vrile, desity fuctio ρ is mp tht ssocites ech elemet x i Ωto proility p i. This mp yields P uder the covetio tht if E is suset of Ω, P (E) = p i. i:x i E Note tht P defied this wy utomticlly stisfies property (2) ove, d tht property () requires i= p i =. Exmple 3.3. For the fir 6-sided exmple give ove, the proility tht X ssumes y prticulr vlue is /6. Let E e the evet tht X is odd, i.e. E = {, 3, 5}. The the proility of E is /6 + /6 + /6 =/2. 7

2 For cotiuous rdom vrile, desity fuctio is mp ρ :Ω [0, ) such tht if E is suset of Ω, the the proility of E is give y P (E) = ρ(x)dx. Note P defied this wy utomticlly stisfies property (2) ove, d tht property () requires tht ρ(x)dx =. Ω Exmple 3.4. Let X e rdom vrile with smple spce [0, ) d desity fuctio ρ(x) =e x.ife is the evet tht X<2, the P (E) is give y 3..3 Expected Vlue P (E) = 2 0 E e x dx = e 2. Iformlly, the expected vlue (lso clled the me) of rdom vrile X is its verge, were you to ru the uderlyig rdom process ifiite umer of times. The expecttio of rdom vrile is geerlly deoted either y E(X) or y µ X. Formlly, the expectti of discrete rdom vrile is defied s µ X = E(X) = x i p i, (Expected Vlue of Discrete R.V.) i:x i Ω while the expecttio of cotiuous rdom vrile is µ X = E(X) = xρ(x)dx. (Expected Vlue of Cotiuous R.V.) 3..4 Vrice d Stdrd Devitio Ω Iformlly, the vrice of rdom vrile X is mesure of its spred, i.e. how fr X teds to e from its verge vlue. The vrice of X is geerlly deoted y σ 2 x, d is defied formlly s σ 2 X = E([X µ X ] 2 ). Note tht sice this defiitio is give strctly (i.e. i terms of the expecttio) it pplies to oth discrete d cotiuous rdom vriles. Note too tht the qutity [X µ X ] 2 is rdom vrile i its ow right: it will e discrete if X is discrete d cotiuous if X is cotiuous, ut i either cse will it e equl to X. The stdrd devitio of rdom vrile X is defied s the squre root of the vrice. The stdrd devitio is deoted σ X Specil Distriutios Here is short list of proility distriutios tht show up lot i pplictios. Note tht these distriutios re ll chrcterized y prmeters, i.e. certi costts tht eed to e specified. Chgig the vlues of these prmeters will chge the vlues of the desity fuctios, ut ot their sic shpes. Uiform: X tkes vlues i itervl [, ], d hs desity fuctio f X (x) = The me is µ X =( + )/2, d the vrice is σ 2 =( ) 2 /2. Expoetil: X tkes vlues i [0, ), d hs desity fuctio f X (x) =λe λx for some positive λ. The me is µ X =/λ, d the vrice is σ 2 X =/λ2. 8

3 Norml (Gussi): X tkes vlues i (, ), d hs desity fuctio f X (x) = 2πσ 2 e (x µ)2 /σ 2. The me is µ d the vrice is σ Idepedece Iformlly spekig, two rdom vriles X d Y will e idepedet if the outcome of oe hs o ifluece o the outcome of the other. Exmple 3.5. (Idepedece) Let X e the outcome of oe toss of die, Y the outcome of secod toss. Sice the result of the first throw hs o ifluece o the result of the secod, X d Y re idepedet. The ove exmple is prdigmtic: if the vlue of X is the result of some rdom process, d the the sme rdom process is ru gi to geerte Y,the X d Y will e idepededet d ideticlly distriuted (iid). This otio c e exteded (i ovious wy) to more th two rdom vriles. A very useful fct out idepedet rdom vriles X d Y is the followig: E(XY )=E(X)E(Y ). This simply sttes tht if the two rdom qutities X d Y do t ifluece oe other, the the verge of their product is the product of their verges. Exmple 3.6. Suppose you roll die twice d let Z e the product of the outcomes. Sice the rolls re idepedet, the expected vlue of Z is just the product of the mes of ech idividul roll, i.e. E(Z) = Note tht cosequece of the ove fct is tht if X d Y re idepeet d oth hve me 0, the E(XY ) = Fuctios of Rdom Vriles Suppose X is cotiuous rdom vrile with smple spce Ω R d desity fuctio f X (x). Suppose g : R R is fuctio. The Y = g(x) is rdom vrile with desity fuctio f Y (y) = f X(g (y) g (g (y)) Exmple 3.7. Suppose X is uiformly distriuted o [0, 2] d g(x) =x 2. The if Y = g(x), the desity fuctio of Y is give y f Y (y) = 2 y, sice f X (x) =/2, g (y) = y,dg (x) =/2x. 3.2 Applictio: Estimtig Defiite Itegrls Suppose we wish to estimte the qutity g(x)dx (4) for some fuctio g : R R +. I other words, we wish to clculte the re etwee the grph of g d the x xis. Oe pproch is first clculte the mximum vlue of g i the itervl [, ], d the fill the two dimesio ox [, ] [0,g mx ]with uiformly spced rdom poits. Ituitively, the proportio of these poits tht fll i the re we re tryig to estimte should e pretty close to the rtio of tht re d the re of the ox. I other words, (re eeth curve) ˆ (re of ox), 9

4 where ˆ is the umer of smple poit tht lie elow the grph of g. This suggests our first lgorithm for estimtig (4): Algorithm Clculte g mx, the mximum of g i the itervl [,]. Fill the 2-dimesiol ox [, ] [0,g mx ]with uiformly spced smple poits. Cout how my of these fll i the regio elow the grph of g. Estimte the itegrl s g(x)dx ˆ ( ) g mx. A ltertive formultio tkes dvtge of the me-vlue theorem for itegrls, which sttes tht the verge vlue of fuctio y = g(x) o itervl [, ] is or i other words, tht y vg = g(x)dx, g(x)dx = y vg ( ). This ltter formultio is useful ecuse we c estimte y vg vi rdom smplig. The simplest wy to do this is tke rdom smple poits x i i the domi [, ], evlute g t ech of these poits, d form verge of the results, i.e. form y = y i, where y i = g(x i ). i= The qutity y is pproximtio of y vg. The pproximtio of the itegrl is the y ( ). This leds to our secod lgorithm: Algorithm 2 Choose rdom smple poits x i distriuted uiformly i [, ]. Clculte y i = g(x i ) for ech i. Form y =(/)( i= y i). Estimte the itegrl s g(x)dx y i ( ) =y ( ). i= Note tht Algorithm 2 is etter th Algorithm i umer of wys: i prticulr, it requires fewer smple poits d it does t require clcultig g mx. 20

5 3.3 Error Alysis Estimtes chieved through the use of rdom qutities will ecessrily coti errors. O prcticl level, it is useful to e le to qutify these errors. For exmple, if we use 00 smple poits to estimte the itegrl vi Algorithm 2, it would e ice to e le sy somethig like This lgorithm produces swer tht is withi 0.72 of the true swer 99% of the time. Therefore, there is 99% chge tht this prticulr swer is withi 0.72 of the true swer. Elemetry proility theory llows us to mke clims like this. The followig explis how Smplig Suppose Y is rdom vrile. As strct qutity, Y c tke o y of multiple vlues. A smple of Y is cocrete outcome uder oe ru of the uderlyig rdom process. A set of smples y,,y is set of outcomes uder rus of the uderlyig rdom process. If the uderlyig process is the sme for ech i, thethey i re idepedet d ideticlly distriuted. Give smples of Y,thesmple me is the qutity y = y i d the smple vrice is s 2 = σ (y i µ Y ) 2. The smple me is just the ctul verge of the smples. The smple vrice is (siclly) the me squre distce etwee the smples d the smple me. (The resos for dividig y d ot re rce.) Our mi iterest will e with y. Note tht y is itself rdom vrile. It thus hs me d vrice. The me is esy to clculte, sice the expecttio opertor is lier: µ y = E y i = E(y i )= µ Y = µ Y. i= Note tht we use the fct tht the y i re ideticlly distriuted i settig E(y i )=µ Y for ll i. To clculte the vrice, we ote σ 2 y = E([y µ Y ] 2 )=E(y 2 ) 2E(y µ Y )+E(µ 2 Y ). It is ot hrd to show (d is left s exercise) tht if the y i re idepedet, this reduces to Cetrl Limit Theorem σ 2 y = σ2 Y. (5) We hve see tht y is good estimtor for the me of rdom vrile Y, sice its me is the correct oe d its vrice reduces s the umer of smples icreses. It turs out tht s gets lrger, the distriutio of y pproches Gussi distriutio with me µ Y d vrice σy 2 /. Wht is mzig out this sttemet is tht it does t deped o the distriutio of Y : it holds for y rdom vrile Y. This is the cotet of the cetrl limit theorem: Theorem 3. (Cetrl Limit Theorem). Let Y e y rdom vrile with me µ Y d vrice σy 2. If for i = the vriles y i represet idepedet rdom smples of Y, the s,thesmple me y =( y i )/ pproches Gussi rdom vrile with me µ Y d vrice σy 2 /. The sigificce of the cetrl limit theorem (for our purposes) is tht it llows us to mke clims like the oe sketched i the egiif of this sectio. The reso for this is tht Gussi rdom vriles hppe to stisfy the followig property: 2

6 Theorem 3.2 ( Rule for Gussis). Let X e Gussi rdom vrile with me µ d vrice σ 2. The 64% of the time, rdom smples of X will lie withi distce σ of the me µ. 95% of the time, rdom smples of X will lie withi distce 2σ of the me µ. 99.7% of the time, rdom smples of X will lie withi distce 3σ of the me µ. I other words, if X is Gussi, there is very high proility tht rdom smple will lie withi 3σ of µ. If we hve estimte which we kow is pproximtely Gussi, d we c pproximte how ig σ is, we c thus mke some pretty good clims out how ccurte the estimte is Errors i Mote-Crlo Estimte Recll from Algorithm 2 tht we chose smple poits x i uiformly i [, ], set y i = g(x i ), d formed y = y i. The y i represet smples of the rdom vrile Y = g(x). By the Me Vlue Theorem for Itegrls, the true me vlue of Y is just µ Y = g(x)dx, d thus the me vlue of y ( ) isjust µ y( ) =( )µ Y = g(x)dx. This mes tht the expected vlue of our Mote Crlo estimte is the true vlue of the itegrl. This is good. To clculte the vrice of y ( ), we ote tht the vrice of Y is just whece σ 2 Y = E([Y µ Y ] 2 )= σy( ) 2 = σ2 y( ) 2 = σ2 Y ( )2 = (g(x) µ Y ) 2 dx, (g(x) µ Y ) 2 dx. (6) I other words, the vrice of the Mote Crlo estimte decreses i iverse proportio to the umer of smples used, d is porportiol to the verge squre distce etwee g(x) d its me. Exmple 3.8. Cosider the itegrl 2 0 x 2 dx. Usig Algorithm 2 with smple poits yields estimte y ( ) whose vrice is σ 2 y( ) = (x )2 dx = With = 00, this mes tht the vrice of the estimte should e , d the stdrd devitio is = Sice = 00 is reltively lrge, the cetrl limit theorem sttes tht the estimte y( ) is roughly Gussi, whece, for this vlue of, our swer should like withi of the true swer 99.7% of the time. 22