Xidian University Liu Congfeng Page 1 of 22

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1 Rdom Sgl rocessg Chpter Expermets d robblty Chpter Expermets d robblty Cotets Expermets d robblty.... Defto of Expermet..... The Smple Spce..... The Borel Feld The robblty Mesure...3. Combed Expermets Crtes roduct of Two Expermets Crtes roduct of Expermets Coutg Expermets Selecto Combed Expermet Codtol robblty Totl robblty Theorem Byes's Theorem Rdom ots Uform Rdom ots Itervl Nouform Rdom ots Itervl....5 Summry... Xd Uversty Lu Cogfeg E-Ml:cflu@ml.xd.edu.c ge of

2 Rdom Sgl rocessg Chpter Expermets d robblty Expermets d robblty. Defto of Expermet To fully pprecte the meg of probblty d cqure strog mthemtcl foudto for lytcl wor, t s ecessry to defe precsely the cocept of expermet d smple spce mthemtclly. These deftos provde cosstet methods for the ssgmet of elemetry probbltes prdoxcl stutos, d thus llow for megful clculto of probbltes of evets other th the elemetry evets. Although t the begg ths pproch my seem stlted, t wll led to cocrete cocept of probblty d terpretto of derved probbltes. A expermet E s specfed by the three tuple S, F,., where S s fte, coutble, or ocoutble set clled the smple spce, F s Borel feld specfyg set of evets, d probblty mesure llowg clculto of probbltes of ll evets. s.. The Smple Spce The smple spce S s set of elemets clled outcomes of the expermet E d the umber of elemets could be fte, coutble, or ocoutble fte. For exmple, S could be the set cotg the sx fces of de, S f, f, f 3, f 4, f 5, f 6, or the postve tegers, :,,... or the rel vlues betwee zero d oe, S x : 0 x, respectvely. S, A evet s defed s y subset of S. O sgle trl of the expermet outcome s obted. If tht outcome s member of evet, t s sd tht the evet hs occurred. I ths wy my dfferet evets occur t ech trl of the expermet. For exmple, f f s the outcome of sgle trl of the expermet the the evets f, f, f, f, f 3,..., f, f 6,..., f, f 3, f 5,, ll occur. Evets cosstg of sgle elemets, le f, re clled elemetry evets. The mpossble evet correspods to the empty set d ever occurs, whle the cert evet, S, cots ll outcomes d thus lwys occurs o mtter wht the outcome of the trl s. Evets A d B re clled mutully exclusve or dsjot f A B, where s the ull set. Two evets A d B re clled depedet f A B A B. The evets A, A,, A re defed to be depedet f the probbltes of ll tersectos two, three,..., d evets c be wrtte s products. Ths mples for ll, j,,, tht the followg codtos must be stsfed for depedece Xd Uversty Lu Cogfeg E-Ml:cflu@ml.xd.edu.c ge of

3 Rdom Sgl rocessg Chpter Expermets d robblty A Aj A Aj A A A A A A j A A A A A... A... j (.).. The Borel Feld A feld c be defed s oempty clss of sets such tht () f d () f F d b F the b F F, the the complemet of F. Thus feld cots ll fte uos, d by vrtue of complmets d DeMorg's theorem, ll tersectos of the collecto. If we further requre tht ll fte uos d tersectos re preset the collecto, Borel feld s defed. The set of ll evets of our expermet tht wll hve probbltes ssged to them (mesurble evets) must be Borel feld to hve mthemtcl cosstecy. If A, collecto of evets, hs fte umber of elemets, Borel feld c be formed s the set of those evets plus ll possble subsets obted by uos d tersectos of those evets cludg the ull set d etre set S. If set s ocoutble, t s lttle hrder to descrbe Borel feld. The most commo Borel feld, cotg the rel umbers, s the smllest Borel feld cotg the followg tervls: x : x x for ll x rel umbers. Ths wll cot ll fte d fte closed d ope tervls of the form, d b [, b],[, b),(, b] those tervls thereof.,, where d b re rel umbers d the tersectos d uos of..3 The robblty Mesure The probblty mesure, codtos re stsfed () For y evet F, must be cosstet ssgmet of probbltes such tht the followg A, the probblty of the evet A, A, s such tht A 0 () For the cert evet, S, S. (3) If A d B re y two evets such tht B (3) If.. A the A B A B A F for,,, d A for ll j, the A j A A A A A A. Wth these codtos stsfed, the probblty of y evet A F c be clculted. How does oe go bout ssgg probbltes such tht we stsfy the codtos bove? For the cse where S s set wth fte umber of elemets the codtos bove c be stsfed Xd Uversty Lu Cogfeg E-Ml:cflu@ml.xd.edu.c ge 3 of

4 Rdom Sgl rocessg Chpter Expermets d robblty by ssgg probbltes to ll the evets wth oly sgle outcomes, e, where e S, such tht codtos () d () bove re stsfed. Ths mppg from the smple spce S to the postve rels s clled the dstrbuto fucto for the probblty mesure d equvletly specfes the probblty mesure. Whe S s set wth ocoutbly fte umber of elemets the ssgmet bove s ot useful s the probbltes of most elemetry evets wll be zero. I ths cse the ssgmet of probbltes s cosstet f the probbltes of the evets x : x x for ll x rel umbers re ssged such tht () 0 x : x x () x : x x x x x (bouded). d (3) lm s 0 : for ll x x (odecresg fucto of x ) of x : x x equls x x Ths mppg: x x x : x : x (cotuous from the rght sde)., defed for ll x, s clled the cumultve dstrbuto fucto d equvletly descrbes the probblty mesure for the ocoutble cse. From ths dstrbuto fucto we re ble to clculte ll probbltes of evets tht re members of the Borel feld F. The cumultve dstrbuto fucto could hve lso bee used to specfy the probblty mesure for the cse where S hs coutble or fte umber of elemets. A umber of exmples of expermets wll ow be preseted. They wll clude couple of co-tossg expermets d de-rollg expermet. The expermets wll be descrbed by specfyg ther smple spce, Borel feld, d probblty mesures. Exmple. Ths expermet cossts of sgle flppg of co tht results ether hed or tl showg. Gve ts descrpto by specfyg s S, F,. Soluto The possble outcomes of the expermet re ether hed or tl. Thus the smple spce c be descrbed s the set S hed, tl. The Borel feld F cossts of the elemetry evets hed d tl, the ull set, d S. To complete the descrpto of the expermet, probblty mesure must be ssged. Ths prtculr ssgmet could be bsed o prevous experece, creful expermetto, use of fvorble to totl ltertves, or y other terpretto of the cocept of probblty. There s o rght or wrog ssgmet, but cert ssgmets (models) my be more pproprte explg the results of correspodg physcl expermets. For the purpose of ths exmple, we ssume tht ths co hs bee tmpered wth for more ofte hed comes up th tl, whch we specfy by hed p d tl p. These two ssgmets comprse the dstrbuto fucto d thus the probblty mesure for the Xd Uversty Lu Cogfeg E-Ml:cflu@ml.xd.edu.c ge 4 of

5 Rdom Sgl rocessg Chpter Expermets d robblty expermet. Exmple. A more relstc ssgmet for the expermet of flppg co could be the expermet defed s follows: S hed, tl, edge wth descrbed by h S, F,, t 0. 49, e The Borel feld F s defed s the power set of S. () Idetfy ll the evets (elemets) of the Borel feld. (b) Clculte the probbltes of those evets. Soluto () The Borel feld F specfyg the mesurble evets s the power set of S (ll possble subsets of S ) gve by h, t, e, h, t, h, e, t, e, h, t e F,, Where s the mpossble evet, d h, t, e s the cert evet. The other evets cosst of ll possble proper subsets of S, sgle elemets, d combtos of two elemets. (b) By defto 0 d the probbltes of the elemetry evets re gve the specfcto of the probblty mesure of the expermet s h 0. 49, t 0. 49, d e The probbltes of the other evets c be determed by repeted pplcto of property (3) for the probblty mesure. For exmple the evets h d e re mutully exclusve therefore h, e c be foud s follows: h, e h e h e Smlrly h, t 0. 98, t, e 0. 5, d h, t, e.. Combed Expermets Combed expermets ply mportt role probblty theory pplctos. There re my wys we c combe expermets, cludg crtes products whch depedet trls of the sme or dfferet expermets c be descrbed. I some cses the probbltes of evets wll deped o the results of prevous trls of expermets or rdom selecto of dfferet expermets. A umber exmples of combed expermets re ow explored begg wth the clsscl cse of smplg wth replcemet... Crtes roduct of Two Expermets Cosder the cse of hvg two seprte expermets specfed by the followg:. E : S, F. d Xd Uversty Lu Cogfeg E-Ml:cflu@ml.xd.edu.c ge 5 of

6 Rdom Sgl rocessg Chpter Expermets d robblty. E : S, F.. The smple spces S d S re usully dfferet sets, for exmple, results of co toss d results of de roll, but they could be the sme sets represetg seprte trls of the sme expermet s repeted co-tossg expermets. We c defe ew combed expermet by usg the crtes product cocept s E E E, where the ew smple spce S S S s the crtes product of the two smple spces expressble by the ordered pr of elemets where the frst elemet s from S d the secod s from S. Exmple.4 Let expermet E : S, F, d E : S, F, be defed s follows: E s the expermet of flppg co wth outcomes hed h d tl t wth equl probblty of occurrece. E s the expermet of rdom selecto of colored bll from box wth outcomes red r, whte w, d blue E E E E b wth replcemet. Defe the ew expermet E s wth expermet E d beg performed depedetly of ech other; tht s, the outcome of expermet E o wy effects the outcome of E, d vce vers. Set up resoble model for ths ew expermet. Soluto To specfy the model t suffces to gve S, F, of the ew expermet E. The ew smple spce S s the Crtes product of the two expermets d gve by S S S. S h, t d the secod from Elemets of S re ordered prs wth the frst elemet comg from S r, w, b ; therefore h, r, h, w, h, b, t, r, t, w, t b S, The ew Borel feld F s selected s the power set of S, tht s ll possble uos d tersectos of S. Ths cludes the ull set, the etre set, d ll possble prs, trples, d so o, s show below: F The probblty mesure, h, r, h, w, h, b, t, r, t, w, t, b h, r, h, w, h, r, h, b, h, r, t, r, h, r, t, w,... h, r, h, w, h, b, h, r, h, w, t, r, h, r, h, w, t, w,... h, r, h, w, h, b, t, r, t, w, t, b c be descrbed by specfyg the probblty of the elemetry evets. Oce these re ow, the probblty of y evet c be foud by wrtg the evet s uo of those evets d usg property (3). Sce the expermets re depedet, t s resoble to ssg probbltes of the elemetry evets s product of the probbltes from ech expermet. For exmple, h r h r,. If hed d tl re eqully probble E the t s resoble for to be Xd Uversty Lu Cogfeg E-Ml:cflu@ml.xd.edu.c ge 6 of

7 Rdom Sgl rocessg Chpter Expermets d robblty descrbed by h t could be gve by r 0. 5, w 0. 3, b 0. probble E, the exmple, the probblty mesure evets (the dstrbuto fucto):, d f we hve reso to beleve tht red, whte, d blue re ot eqully. Thus, for ths c be descrbed by specfyg the probbltes of the elemetry h, r h r 0.5 t, r t r h, w h w 0.5 t, w t w h, b h b 0. t, b t b 0. The evet of hed the ew expermet s H h, r, h, w, h, b occurrece c be determed usg property (3) s H h, r, h, w, h, b h, r h, w h, b , d ts probblty of.. Crtes roduct of Expermets Cosder the cse of hvg seprte expermets specfed by Defe ew combed expermet where the ew smple spce S S S.. E : S, F, for,,,. E : S, F, s crtes product: E E E E S s the crtes product of the spces d expressble by the ordered -tuples of elemets whose frst elemet s from S d the secod s from S, the th from S. The E re, geerl, dfferet, but my cses the expermet could be formed from depedet trls of the sme expermet. Also there s mportt clss of problems where the expermets re the sme, yet they cot be thought of s depedet. A good exmple of ths s the rdom selecto of outcomes wthout replcemet. Exmples of ech type re ow preseted. Boml Dstrbuto. Cosder the expermet : S, F,. re ether flure dcted by 0 or success dcted by ; therefore S 0, probblty mesure. flure p E E E where the outcomes of the expermet. Assume tht the for the expermet s gve by success p F s the set, 0,, 0, d 0 d tht the. Defe ew expermet by E E where E : S, F,. descrbes the ew expermet. Assume tht ths represets depedet trls of the sme expermet E where the probblty of success or flure s the sme for ech trl. The S, F,. re ow descrbed for ths ew expermet. The ew S s the crtes product Xd Uversty Lu Cogfeg E-Ml:cflu@ml.xd.edu.c ge 7 of

8 Rdom Sgl rocessg Chpter Expermets d robblty S S d cossts of ll possble -tuples where the elemets re ether 0 or s show below: S S 0, 0, 0, 0 0, 0, 0, 0, 0,, 0 S (.) 0, 0,,,,, The ew probblty mesure s specfed oce the dstrbuto fucto or probbltes of the elemets of S re determed. By vrtue of the depedet expermet ssumpto, the probblty of ech elemetry evet of the ew expermet s the product of the probbltes for the elemetry evets the sgle trl. For exmple, the 0,,,0,,0,, s gve by 4 4 0,,,0,,0,, p p (.3) As mtter of fct every sequece tht hs oly four oes (successes) wll hve ths sme probblty. The totl umber of these sequeces s the combto of thgs te four t tme, sce we hve loctos d we wt oly four of them wth oes. The probblty dstrbuto fucto for the ew expermet c the be determed s follows, where the frst etry s the elemetry sequece d fter the rrow s the correspodg probblty: Outcome robblty 0,0,,0,0 0,0,,0, ,0,,0, 0,0,,0, ,0,,,0 0,0,,, ,0,,, 0,0,,, p ( p) p ( p) p ( p) p ( p) 0,,,,,,,, p ( p) The Borel feld wll be the power set ssocted wth S d specfes ll evets for whch probbltes re ssged. robbltes of dfferet type of evets for the bove exmple c be determed by usg the dstrbuto fucto descrbed. There re wde umber of pplctos s success c me ll ds of thgs. For exmple, success could be obtg ce drwg crd from stdrd dec of crds wth replcemet, or obtg successful recepto of bry symbol from rdom commucto chel. The evet of exctly successes out of depedet trls ppers frequetly physcl stutos d ts probblty wll ow be derved usg the results bove. Defe the evet A s the evet of exctly oe success out of trls. From the results bove we see tht the probblty of exctly oe success out of trls s (.4) Xd Uversty Lu Cogfeg E-Ml:cflu@ml.xd.edu.c ge 8 of

9 Rdom Sgl rocessg Chpter Expermets d robblty A exctly oe success out of trls 0,0,,0, 0,0,,,0,0,,0,0 0,0,,0, 0,0,,,0,0,,0,0 p p Smlrly the probblty of exctly successes out of trls c be foud to be A exctly success out of trls p p (.6) I some problems we my wsh to ow the probblty tht the umber of successes out of trls s wth rge of vlues. The probblty tht the umber of successes s the rge (.5) m c be obted by ddg the probbltes of exctly successes for tht rge, sce the evets A d A j re mutully exclusve evets for ll d j such tht j. Therefore J umber of successes K exctly J exctly J successes successes K K A p p J J exctly K successes (.7) Exmple.5 Cosder the expermet of tossg fr co wth the smple spce S h, t dstrbuto h 0. 6 d t 0. 4 d probblty. Suppose the expermet s performed 0 tmes depedetly to defe ew crtes product expermet. () Determe the probblty tht we get exctly 5 heds the 0 trls. (b) Determe the probblty tht we get greter th 7 heds the 0 trls. (c) Determe the probblty tht we get less th or equl 9 heds the 0 trls. (d) Determe the probblty tht the umber of tls s greter th or equl to 4 d less th or equl 5. Soluto The desred probbltes re determed for Eqs. (.6) d (.7) s follows: () A exctly 5 heds out of 0 trls (b) umber exctly exctly exctly 8 0 of heds 8 heds 9 heds 0 heds Xd Uversty Lu Cogfeg E-Ml:cflu@ml.xd.edu.c ge 9 of

10 Rdom Sgl rocessg Chpter Expermets d robblty (c) less th or equl 9 heds out of 0 trls exctly 0 heds out of 0 trls umber exctly exctly (d) 4 5 of heds 4 heds 5 heds Approxmtos for Boml robbltes. The probbltes of successes trls of Beroull expermet ws foud to be s Eq. (.6) d evlutg probbltes of rges of successes s success out of trls s plotted for vlues of gve Eq. (.7). I Fgure. the equl to 5, 0, d 50 for ll vlues of, d success The grphs show tedecy towrd hll-shped curve smlr to Guss fucto. These plots would be symmetrcl roud the pot 0.5 f p However, whe p does ot equl 0.5, s for the cses gve, the plot s ot qute symmetrcl. If the umber of trls s lrge, the clculto lod due to the fctorls s cosderble, d cert pproxmtos to these probbltes become useful. My of these pproxmtos re good provded tht the umber of trls, umber of successes, d probblty of success p stsfy gve set of codtos. Of these pproxmtos we wll preset the DeMovre d osso pproxmtos d the regos where the pproxmtos re relble. DeMovre-Lplce Approxmto. For p p p p d p p exp p( p) p p of the order of p p, (.8) osso Approxmto. For, p, d p of order, p p e ( p) p! (.9) Approxmto of Regos of Successes Beroull Trls. Sy tht we re terested pproxmtg the probblty tht the umber of successes repeted trls of Beroull expermet s the Xd Uversty Lu Cogfeg E-Ml:cflu@ml.xd.edu.c ge 0 of

11 Rdom Sgl rocessg Chpter Expermets d robblty rge. If ths rge of successes cots vlues tht stsfy the DeMovre-Lplce pproxmto, the the summto c be pproxmted by usg the error fucto or the (x) fucto s follows: p p p (.0) p p( p) p( ) p x y / where ( x) e dy The (x) c oly be determed by umercl tegrto s there s o tdervtve d t s coveet to use the tble gve some boo ppedx cotets. 0.4 ( successes) 0.3 ( successes) N=5, p=0.6, Bom (,5) N=0, p=0.6, Bom (,0) 0. ( successes) N=50, p=0.6, Bom (,50) Fgure. lot of ( successes out of trls) for p=0.6 d =5,0, 50. Xd Uversty Lu Cogfeg E-Ml:cflu@ml.xd.edu.c ge of

12 Rdom Sgl rocessg Chpter Expermets d robblty Multoml Dstrbuto. A combed expermet tht s exteso of the Beroull trl expermet just descrbed, whch resulted the boml dstrbuto, s the cse of multple occurreces of severl dfferet evets o multple trls of the sme expermet. Cosder expermet E : S, F,.. Defe set of evets A, to, cosstg of elemets of S such tht ther uo s the cert evet, they re prwse dsjot, d ther probbltes re gve by A p for to. descrbed by Defe ew expermet E by E E E E E : S, F,.. Assume tht ths represets depedet trls of the sme expermet E. Let, to, be the umber of tmes evet A occurs the ew expermet, whch s composed of repeted trls. Also ssume tht 0 d. It c be show tht the probblty tht A occurs exctly tmes the trls s A A A occurs occurs occurs exctly exctly exctly tmes, tmes,! p!!! tmes, p p (.5) Hypergeometrc Dstrbuto. I the prevous compoud expermets the trls of the expermet were cosdered to be depedet. A very mportt clss of expermetto problems s smplg doe wthout replcemet, for fte umber of elemets, d wthout depedet trls. For exmple, f s the umber of successful s elemets d b s the umber of flure f elemets d elemets re drw t rdom but ot replced, the correspodg outcomes of the expermet do ot cot ll possble -tuples of s d f, So the -tuple s, s,, s d sequeces cotg more th successes re ot possble. It c be show for ths expermet tht the probblty of successes out of trls c be determed s b ( ) successes trls, 0,, (.6) b The ext exmple s typcl of the type of problems for whch the hypergeometrc formul bove c be used. Xd Uversty Lu Cogfeg E-Ml:cflu@ml.xd.edu.c ge of

13 Rdom Sgl rocessg Chpter Expermets d robblty..3 Coutg Expermets A specl clss of compoud expermets del wth successve pplcto of expermets where the totl expermet my be stopped t y pot depedg o the results from the curret expermet. Two commo dstrbutos result from these compoud expermets. Geometrc Dstrbuto. The geometrc dstrbuto s result of other form of compoud expermet. Defe coutble umber of detcl expermets E S, F,. for,,. Ech of these : expermets wll be of the Beroull type (two possble outcomes) prevously dscussed, wth detcl probbltes of success; the result of the expermet, f performed, s depedet of prevous or future expermets. The ew expermet s descrbed s follows: erform expermet E f success s obted, the stop; f success does ot hppe, the perform expermet E ; f success occurs, the stop otherwse. Cotue ths process utl success occurs. Thus the umber of trls s ot the sme ech tme the compoud expermet s performed, d thus the smple spce s ot crtes product. If p s the probblty of success o ech expermet, the the smple spce, Borel feld d probblty mesure c be descrbed s follows: The elemetry evets re sequeces of 's (for success) d 0's for flures but wth the property tht they ed o d oly hve 0's precedg the. Thus S c be descrbed by Notce tht,0, 0,,0,,,., 0,, 0,0,,, 0,0,,0,, S (.7) etc, re ot elemets of the smple spce sce the ew expermet would stop t d ot proceed to the ext oe. Evets tht re collectos of outcomes re ot of the sme sequece legth. For exmple,, 0, s evet where success s obted less th or equl to trls. The probblty mesure. c be specfed by gvg the dstrbuto fucto for the elemetry evets. The s the probblty of gettg success o the very frst trl. Sce the probblty of success o the frst expermet s p, the probblty of gettg s lso p, p (.8) The probblty of gettg 0, equls the probblty of gettg flure o performg the frst expermet d gettg success o the performce of the secod expermet. Sce results of ech expermet re depedet f performed, the 0, c be wrtte s product:, pp 0 (.9) Smlrly the probblty of the elemetry evet cosstg of sequece of flures (0's) followed by sgle success () s Xd Uversty Lu Cogfeg E-Ml:cflu@ml.xd.edu.c ge 3 of

14 Rdom Sgl rocessg Chpter Expermets d robblty These probbltes s mesure 0,0, 0, p zeros determe the dstrbuto fucto, or equvletly the probblty.. It c be show tht p (.0) p p (.) The Borel feld s g defed s the power set of S, tht s, ll possble subsets of S. The probbltes of rbtrry evets tht re the subsets of S c the be foud by ddg up the probbltes of the elemetry elemets the evet. For exmple, defe the evet A to be the set of outcomes such tht success occurs o the performce of odd umber of expermets. Thus A c be wrtte s, 0,0,, 0,0,0,0,,, 0 A,, (.) The probblty of A c be determed by ddg up the probbltes of the elemetry evets d usg the propertes of geometrc sequece s follows. A, 0,0,, 0,0,0,0,,, 0 p p p 0,, p p (.3) Negtve Boml Dstrbuto. A exteso of the geometrc expermet s compoud expermet whch the umber of trls ecessry to obt successes s desred rther th the umber of trls eeded for sgle success. The bsc uderlyg expermet s to defe coutble umber of detcl expermets. E : S, F, for,, Ech of these expermets wll be of the Beroull type wth detcl probbltes of success, d the result of the expermet, f performed, s depedet of prevous or future expermets. The ew expermet s descrbed s follows: erform expermet E f success s obted d t s the th, the stop; f ot, cotue to the ext expermet E. Cotue ths process utl the th success occurs. Let x be the umber of trls whch the th success occurs; the the probblty of tht evet c be show to be evet tht the th success occurs o the trl x x (.4) p p x,,, The followg exmple llustrtes the type of problems tht wll use the egtve Boml dstrbuto gve bove. xth Exmple: Xd Uversty Lu Cogfeg E-Ml:cflu@ml.xd.edu.c ge 4 of

15 Rdom Sgl rocessg Chpter Expermets d robblty Suppose the probblty of gettg f o the sgle toss of de s 0.. Fd () the probblty tht the fourth f occurs o the 6th trl; (b) tht the ffth f occurs before the 7th trl. Soluto: () (b) evet tht the 4th success occurs o the 6th trl evet tht the 5th success occurs before the 7th trl evet tht the 5th success occurs o the 5th trl evet tht the 5th success occurs o the 6th trl Selecto Combed Expermet A dfferet type of combed expermet wll ow be cosdered tht s combto of three or more expermets. For purpose of llustrto oly three compoet expermets re preseted. Let the three expermets be defed s E S, F,., E : S, F,., E : S, F,. 0 : The smple spce for E 0 cossts of oly two outcomes ( e d e ). These outcomes serve to help us select whch oe of the other expermets wll be performed. If the outcome s e, the E s performed wth outcome ; f the outcome s e the E s performed wth outcome. Thus the combto expermet. E : S, F, c be descrbed s follows: The smple spce S wll cosst of ordered prs of elemets the frst ether e or e d the secod ether S or S combed expermet results outcome e c e,, where e e,e d S S.. Thus trl of the The Borel feld F wll be defed to be the set of ll possble subsets of S, d the ssocted probblty mesure c be descrbed s e, e where j 0 j j f e e j j f e e The product of the probbltes s becuse of the ssumpto tht the expermet j or E 0 s depedet of (.7) Xd Uversty Lu Cogfeg E-Ml:cflu@ml.xd.edu.c ge 5 of

16 Rdom Sgl rocessg Chpter Expermets d robblty the expermets E d E..3 Codtol robblty The probblty mesure llows us to clculte the probbltes of ll evets tht re members of the Borel feld for the defed expermet. It wll become useful to defe the cocept of codtol probblty of evets. Gve codtog evet C such tht C 0 ssumg C s defed s C, the codtol probblty of y evet A p A C A C (.8) The followg exmples cosder the clculto of codtol probbltes for dscrete d cotuous smple spces. It s see tht the fudmetl set opertos d the probblty mesure for the uderlyg expermet re used to obt codtol probbltes. Exmple. Cosder expermet defed s sgle toss of crooed de wth smple spce S f, f f, f 3, f 4, f 5, 6. The probblty mesure, mybe bsed o pst hstory, s ow to be f f f f f f Let the codtog evet C be gve s C f, f f A C where Soluto 4 5 3, A f, f, the fce s less th or equl By the defto of codtol probblty, Eq. (.8), A C s AC f, ff, f3, f5 A C C f, f3, f5 f 6 f, f, f , the fce of the de s odd, d clculte the Exmple.3 Defe expermet tht hs s outcomes, t, the set of rel umbers greter th or equl to zero d tht the outcome represets the tme utl cert devce fls. The probblty mesure for the expermet s descrbed s Xd Uversty Lu Cogfeg E-Ml:cflu@ml.xd.edu.c ge 6 of

17 Rdom Sgl rocessg Chpter Expermets d robblty x flure tme t xe for x 0 The Borel feld s gve by the smllest feld cotg the tervls t : 0 t x for ll x 0. Fd the flure tme t flure tme t. Soluto From the defto of codtol probblty the probblty tht the tme to flure s less th or equl for the devce gve tht the flure tme s greter th s Usg set operto gves flure t flure t Ad the deomtor s see to be flure t flure t flure t flure t flure t flure t flure t flure t e e Substtutg these two results to the frst equto gves us the followg result flure t flure t flure t flure t e e e e.3. Totl robblty Theorem Gve evets A, A,, A such tht A for ll j A j (mutully exclusve) A S (exhustve) (.3) The t c be show tht the totl probblty of rbtrry evet B c be wrtte terms of the followg codtol probbltes s B B A A B A A B A A B A A (.3) Xd Uversty Lu Cogfeg E-Ml:cflu@ml.xd.edu.c ge 7 of

18 Rdom Sgl rocessg Chpter Expermets d robblty.3. Byes's Theorem A very mportt theorem tht hs my pplctos s Byes's theorem whch volves the determto of codtol probbltes A B A B p uder the sme frmewor s bove. It s esly show tht A B B B A A B A A B A A B A A B A A B A A (.33) The commucto recever d rdr sgl detector re the typcl exmples tht re represettve of the type of problem tht c be solved usg Byes's theorem d the totl probblty theorem. Exmple.4 Cosder expermet volvg rdom selecto of oe of three boxes. The rdom selecto s of sgle bll from the box chose. The boxes cot red, whte, d blue blls wth specfed probbltes of selecto. Assume tht 0.5 box 0.3 box3 0. box 0.4 red box 0.5 red box3 0. whte box 0.3 whte box 0.3 whte box3 box 0.3 blue box 0. blue box box red blue () Fd the probblty of gettg red bll. (b) Fd the codtol probblty, A B the evet red bll s selected. Soluto 0., where B s the evet box selected d A s () The totl probblty theorem c be used to obt the probblty of gettg red bll s follows: red box box red boxbox red box3box3 red (b) The probblty of box gve tht the bll s red c be determed by usg Byes s theorem d the results of prt () s box red boxbox red red Xd Uversty Lu Cogfeg E-Ml:cflu@ml.xd.edu.c ge 8 of

19 Rdom Sgl rocessg Chpter Expermets d robblty.4 Rdom ots The rdom plcemet of pots tervl s mportt problem. Coceptully t s logous to rdom rrvl tmes used bsc vetory problems, d t s used s bss for shot ose commucto theory. The rdom plcemet of the pots c be uformly or ouformly dstrbuted o tervl s see the followg sectos..4. Uform Rdom ots Itervl Defe expermet E : S, F,. s the rdom plcemet of pot t somewhere the closed tervl 0,T s show Fgure.3(). The smple spce s S t : 0 t T, F s the smllest feld cotg the sets t : t t for ll t S, d. s defed by t : t t t / T, for ll t S. Thus t c be see tht the probblty tht the pot selected wll be y gve tervl s the rto of tht tervl's legth to the totl legth T. robbltes of other evets tht re uos of ooverlppg tervls c be obted by ddg up the probbltes for ech of the tervls. 0 t T 0 T t t t - t () Smple spce (b) Rdom plcemet of pots pots 0 t t T t (c) pots tervl t out of pots The purpose of ths secto s to tl bout the rdom plcemet of pots, ot just oe pot, tervl 0,T see Fgure.3(b). A coveet wy to desg such expermet s to form ew compoud expermet E S, F,. composed of ordered -tuples of tmes, t, t,, : t, obted from repetg the expermet E defed bove depedetly. Assume depedece of the trls so tht the probblty mesure c be descrbed s the product Fgure.3 Rdom tmes tervl [0, T] t x t x, t x t x t x t x,, (.34) Xd Uversty Lu Cogfeg E-Ml:cflu@ml.xd.edu.c ge 9 of

20 Rdom Sgl rocessg Chpter Expermets d robblty The Borel feld s defed to be the smllest feld cotg the evets t x, t x,, t x for ll x, x,, x [0, ]. We re terested swerg questos reltg to clcultg the T probbltes tht cert umber of pots fll gve tervl or tervls. Sy tht the probblty tht exctly of the pots fll gve tervl t,t of legth t s show Fgure.3(c) s desred. Let A be the evet tht o sgle trl the pot selected t rdom flls the gve tervl. Usg the uform probblty mesure A t t t., we gve the probblty of A by Thus, o depedet trls, the probblty of gettg pots the tervl s bomlly dstrbuted s po t, t for trls p p t t where p (.35) T Further ssume tht, the umber of trls, s very lrge,, d tht the reltve wdth of the tervl s very smll, t / T, d s of the order t / T, The resultg osso pproxmto s wrtte s exctly out of t, t t t / T pot exp trls T! (.36) where t t t I the lmtg cse ths result wll gve terpretto terms of verge umber of pots per ut tervl. If, T, d / T, the t c be show tht exctly pot t lm (.37) t 0 t Aother probblty of terest s tht of gettg exctly pots tervl t d exctly b tervl t b s dcted Fgure.4. pots b pots 0 T t t b Fgure.4 Exctly pots tervl t d exctly b tervl tb Let A, B, C equl the evets tht o sgle trl of the expermet exctly oe pot flls t, t b, d ot t or t b, respectvely. The the probblty of gettg exctly exctly b pots tervl t b, d exctly b pots ot pots tervl t, t or t b out of trls c Xd Uversty Lu Cogfeg E-Ml:cflu@ml.xd.edu.c ge 0 of

21 Rdom Sgl rocessg Chpter Expermets d robblty be obted from the multoml result s b t b tb t or! b t!! b b t! T b tb T t T tb T b (.38) The dvdul evets exctly pots t d exctly b pots t b out of trls hve probbltes s follows: t t t T tb T b t T tb T b (.39) Sce the product of the bove two probbltes does ot equl tht of Eq. (.38), the evets exctly pots t d exctly b pots t b out of trls re ot depedet evets..4. Nouform Rdom ots Itervl I cert problems the rdom pots re ot plced uformly the tervl. A commo wy to descrbe ouform rte s to ssg the probblty mesure by usg weghtg fucto the followg propertes: t 0 T 0 for t ll dt t [0, T ] t tht stsfes (.40) O sgle trl of the expermet, the rdom plcemet of sgle pot the tervl 0,T, the probblty tht the pot selected s t s gve by Thus, f t,t t t t, t ] t ( dt (.4) t hs pe t t t, t mes tht the pot selected t rdom hs hgher probblty of beg close to t th other vlues of t. If depedet trls of ths expermet re performed, the probblty of exctly pots out of trls beg the tervl t c be determed from Eq. (.6) s pot t, t p out of trls where p t t ( t, t ] t t,t p dt (.4) For the cse of ouform rte wth the ssumpto tht, the umber of trls, s very lrge,, Xd Uversty Lu Cogfeg E-Ml:cflu@ml.xd.edu.c ge of

22 Rdom Sgl rocessg Chpter Expermets d robblty d tht the reltve wdth of the tervl t,t t s very smll, t / T, d of the order t / T, the osso pproxmto results the followg probblty where p s gve s bove: exctly out pot t, t of trls exp p p! (.43).5 Summry I ths chpter the mthemtcl deftos of expermet terms of smple spce, Borel feld, d probblty mesure were gve. The cocept of expermet s bsc to the uderstdg of rdom vrbles d rdom processes to be dscussed lter chpters. The ssgmet of the probblty mesure for severl expermets obted by combg other expermets led to the boml, multol, d hypergeometrc dstrbutos. The mportt cocept of codtol probblty, the totl probblty theorem d the Byes theorem re defed. Usg ths defto of the totl probblty theorem d the Byes theorem for obtg the posteror probblty followed. These cocepts hve fudmetl role the detecto d estmto of rdom vrbles d rdom processes s wll be see Chpters, 8, d 9. The chpter cocluded wth short dscusso o the rdom plcemet of pots gve tervl. These expermets re mportt lyzg problems volvg rdom tmes of rrvl d other relted problems. It ws ot the tet of ths chpter to gve exhustve presetto o expermets d ther use but to provde bcgroud mterl tht would be used the remder of the clss. For those wshg more thorough presetto there re my excellet texts, such s the boo, <robblty, Rdom Vrbles, d Stochstc rocesses>(pouls, Athsos, McGrw Hll, 965) Ths s the ed of Chpter Xd Uversty Lu Cogfeg E-Ml:cflu@ml.xd.edu.c ge of