Multinomial Allocation Model

Transcription

1 Multinomial Allocation Model Theorem 001 Suppose an experiment consists of independent trials and that every trial results in exactly one of 8 distinct outcomes (multinomial trials) Let : equal the probability of outcome on any trial and let G equal the number of times outcome occurs in these multinomial trials Then EG G G œ EG / ", á, 8 ^ ", á, ^ 8 œ where ^ßáß^ are independent Poisson random variables such that ^ :, œ ", á, 8 That is " 8 has parameter T^ œ D œ / 3 3 : 3 : D x 3 3 D 3 D3 ß "ß á Theorem 002 Let H5 equal the number of G's which equal 5, 5 ß"ßáß in the above model " with : œ for œ "ß á ß 8 Then 8 8 EG HßHß œ " EG ) / )/ á ß H ß ß ß á 8 8 ^ß^ß " á œ ) œ ) " 8x where ^ß^ßáare independent and ^ µ Poisson for œß"ßá Theorem 003 Consider a fixed set of nonnegative integers ;ßáß; " 8 If G 3 ; 3 after multinomial trials we will say outcome 3 has reached its quota (by time ) Let [ <U : represent the waiting time (ie the smallest value of ) until exactly < of the 8 different possible outcomes of each multinomial trial have reached their quota Then T [ œ <U : / T^", á, ^8 U< : œ

2 and 5 5" <U : " 8 U : < E[ œ 5 T^, á, ^ d where (1) T is the event that ^ ; (2 Ñ is the event that at least 8<" of the (independent) events T,, T occur and U< : " á 8 (3) ^ ßáß^ are independent, ^ µ Poisson : and : á : œ "Þ " 8 " 8 This result can be generalized to what are known in the literature as sooner and later waiting time problems Consider an experiment consisting of repeated multinomial trials, each consisting of 8= distinct outcomes We will assume that 8of the outcomes have been classified as type E outcomes and that the remaining = outcomes as type F outcomes Let ; ßáß; ß; ßáß; be a fixed set of nonnegative integers Let G œg equal the number of times outcome 3 occurs in the first trials If G 3 ; 3 after multinomial trials we will say outcome 3 has reached its quota (by time ) " 8 8" 8= 3 3 Let [ < < U represent the waiting time until exactly < " of the type E outcomes exactly < # of the type F outcomes have reached their quota ( ie a sooner waiting time) " #: or Let [ < < U represent the waiting time until exactly < " of the type E outcomes " #: and exactly < # of the type F outcomes have reached their quota ( ie a later waiting time) Then T [ œ < < U / T^ á ^ U < < " # : ",, 8= : " # œ

3 5 < < : U " 8= U : < < 5" E[ œ 5 T^, á, ^ d " # " # T [ œ < < U / T^ á ^ U < < " # : ",, 8= : " # œ and 5 < < : U " 8= U : < < 5" E[ œ 5 T^, á, ^ d " # " # where (1) T is the event that ^ ; (2) U :< < is the event that at least 8< " " of the (independent) events " # T, á, T occur or at least =< " of the (independent) events T, á, T occur " 8 # 8" 8= (3) U :< < is the event that at least 8< " " of the (independent) events " # T", á, T8 occur and at least =< # " of the (independent) events T, á, T occur and 8" 8= () ^ ßáß^ are independent, ^ µ Poisson : and : á : œ "Þ " 8= " 8 These results could be further extended in an obvious way to allow for the possibility of splitting the set of possible outcomes into more than two types Theorem 00 Suppose an experiment consists of repeated identical and independent trials where each trial results in exactly one of 8" distinct outcomes E ßE" ßáßE 8 Let : œte 2 for œ ß "ß á ß 8 Assume the trials continue until the 5 occurrence of outcome E

4 Let G, "ßáß8 equal the number of times outcome E is observed before the trials stops Then " 5" ) EGG", á, G8 œ EG^", á, ^ 8) / ) 5"x where ^ßáß^ are independent, ^ µ Poisson " 8 Note: In addition to its direct use in waiting time problems where the stopping rule depends on one distinctive outcome achieving its quota, Theorem 00 can also be used when the stopping rule depends on multiple outcomes achieving their quotas by partitioning the problem according to which outcome occurs last Example 00 illustrates such an application ): : Theorem 005 Suppose we perform multinomial trials with 8< distinct possible outcomes We will assume that 8of the outcomes have been classified as type E outcomes and that the remaining < outcomes as type F outcomes We will assume such that all type E outcomes occur with probability :" and all the type F outcomes occur with probability : # Let G equal the number of type E outcomes which occur exactly times in the trials and let H equal the number of type F outcomes which occur exactly times in the trials, ß "ß á ß Then, EGG ßH, G ßH ßá, G ßH ß ß ß ß ßá " " 8 < )/ œ / : " 7/ : # EG [ ß^ [ ß^ ß 8 <, " " á ) 7 œ )7 œ œ ) : 7: x x " # where [ µ Poisson ", ^ µ Poisson #, 8: <: œ " and where all random variables are independent Theorem 006 Suppose we perform multinomial trials with 8< distinct possible outcomes We will assume that 8of the outcomes have been classified as type E outcomes and that the remaining < outcomes as type F outcomes We will assume such that all type E

5 outcomes occur with probability :" and all the type F outcomes occur with probability : # Let G equal the number of type E outcomes which occur exactly times in the trials 2 and let ], "ß á ß < equal the number of times the type F outcome in the first trials Then, EGG, G ßá, G ßßßá, ] ß á, ] " " < 8 )/ œ / : " 7: [ [ ß [ ßßßá ^ ß ^ 8 " " < ) # E G, á,, á, œ ) œ ) : x # " # " where [ µ Poisson, ^ µ Poisson :, 8: <: œ " and where all random variables are independent Examples for Section 00 Example 001 Joint moments of count size distribution in multinomial trials Suppose an experiment consists of independent trials and that every trial results in exactly one of 8 distinct outcomes (multinomial trials) Let : equal the probability of outcome on any trial and let G equal the number of times outcome occurs in these multinomial trials Then E G â G œ x α " α 8 α : á x â : " 8 α α " 8 " α8 " 8 This result is given in Johnson and Kotz [0] Proof If we take GG, á, G œ G âg " 8 " α 8 α " 8

6 in Theorem 001 then EGG", á, G 8 œ / E^ â^ " α" 8 α8 œ / E^ âe^ " α" 8 α8 using the independence of the ^3's where ^ 3 µ Poisson : 3 We note that E\ + œ $ for \µ Poisson $ and the final form follows on using this result and extracting the coefficient of œ œ + Example 002 Fixed number of multinomial outcomes selected a fixed number of times within both of two types Suppose we perform multinomial trials with 8< distinct possible outcomes We will assume that 8of the outcomes have been classified as type E outcomes and that the remaining < outcomes as type F outcomes We will assume that all type E outcomes occur with constant probability :" and that all type F outcomes occur with constant probability : so that 8: <: œ " # " # Let G equal the number of type E outcomes which occur exactly times in the trials and let H equal the number of type F outcomes which occur exactly times in the trials, ß "ß á ß Then, x TG 3 œ 5 œ " 8 : ": 8 3x 38 x " " œ and EG âg H âh α α " " 8x <x x œ 8 αx < " x E E x x â x α " α " α " E E E E " # " # : : ": α : " α " α "

7 where α œ α á α, " œ" á ", Eα œ α á α, and E " œ " á " The first result is given in Charalambides [0] David and Barton [0] give the first result for the special case 3œ Charalambides [0] gives the second result for the special case α œ@, α œ for 3Á5and " œ for all 5 We note that \ " by definition ÐÑ Proof For the first result let Z equal the number of balls in urn and let GZ Z œ " exactly 5 of Z " ßáßZ 8 equal 3 ", á, 8< else By Theorem 001 TG œ 5 œ EG Z, á, Z 3 " 8< œ / T 5 ^ ßáß^ 3 exactly of " 8 equal œ œ / T^ œ 3 "T^ œ 3 5 œ where ^µ Poisson : " The final form follows on substituting for T^œ3 and extracting the coefficient of For the joint factorial moment result, we have by Theorem 005 that EG âg H âh α α " " 8 < )/ œ / : " 7/ : # E[ â [ ^ â^ ) 8 7< α α " " œ )7 œ œ 8 < )/ œ / : " 7/ : # E[ â E[ E^ âe^ ) 8 7< α α " " œ )7 œ œ

8 using the independence of the random variables [ß[ßâß^ß^ßâ where " " 3 3 ) : " 7: # [ 3 µ Poisson and ^ 3 µ Poisson 3x 3x + We note that E\ + œ $ for \ µ Poisson $ and the final form follows on using this < 8 result and extracting the coefficient of 7 ) Example 003 A selection of waiting time problems In each of the following parts of this example let [ <U : represent the waiting time until exactly < of the 8 different possible outcomes of each multinomial trial have reached their quota (a) An urn contains 7 distinct solid colored balls distinct striped balls (so all 7@ balls in the urn are distinct) You continue to randomly sample with replacement from this urn until you observe some = of the 7 different solid colored balls at least once each This is a variation of the classical coupon collector' s problem In this case, 7 5 : œ7<" 5 5 7<" 7 " " E[ œ 5x7@ <U " 7< This result is given in Charalambides [0] (b) Newman and Shepp [0] consider the same model as in (a) except that sampling continues until one observes some = of the 7 different solid colored balls at least twice each Newman and Shepp note that this situation occurs when trading is allowed between two cooperative collectors Newman and Shepp's solution is only for the case 5œ" and is in a complicated form An easier form to work with for general 5is, E [ <U : œ57@ 7<" 7 " œ7<" 3œ " " 53" x 7< 3 53 (c) Suppose a multinomial experiment consists of 7@ distinct outcomes and of these outcomes, 7 can be classified as type T outcomes with probabilities :", á,: 7 and quotas ;, á,; Assume that these multinomial trials continue until any one of the type " 7

9 T outcomes has met its quota This is a generalization of the classical birthday problem In this case, ;" ; " : œ œ " 7 " 7 E " 7 5 " á 7 5 " x : " â : 7 ["U œ 5 â " xâ7x " á75 : á : " 7 This result was derived by Klamkin and Newman [0] for 5œ" and extended to general 5 " by Dwass [0] The case 7 œ œ, :" œ :# œ # and ;" œ ;# œ ; is often referred to as Banach' s match box problem (see Feller [0] and Holst [0]) (d) The game of one person Knock' m Down is described in full in Benjamin, Fluet and Huber [0] but the basic idea of the game is to choose ;", á,; 8 for a fixed value of ;" á ; 8 that minimizes E[ 8: U, that is the expected waiting time for all outcomes to meet their quota In this case 7 ; " ; " " " 8U : ) œ" œ E[ œ â Ð" " á : â " xâ x : á : á@ " " where ; <, á,; < are the nonzero quotas amongst ;", á,; 8 and is the set of all " 7 element subsets of <ßáß< " 7 This result allows for the calculation of E [ 8: U for any given quota vector Ubut does not aid in finding that U which minimizes E[ 8U : for a fixed value of ;" á ; 8 short of calculating E[ 8: U for all possible UÞ The paper by Benjamin, Fluet and Huber [0] gives some results for reducing the class of feasible U Benjamin and Fluet [0] posed but left open the question of how to choose a quota vector Uthat maximizes TUfinishes before U w for a given opponent quota vector U w in a two person version of Knock' m Down We point out but do not further consider here that an algorithm for computing TUfinishes before U w can be devised by generalizing the ideas given in Thorp [0] The nontransitive probabilities that arise in two person Knock' m Down makes the game especially interesting Proof For part (a) we apply Theorem 003 with 8 œ and U œ "ß áß", ß áß In this case we have

10 E [ 5 <U : œ 5 5" Tat least 7@<" of the 7@ events ] "á],, "], á],, are true d " 7 7" 7@ œ5 Tat least 7<" of events ]" ", á,] 7 " are trued 5" 5" 7<" " 7 œ5 " T ] œ < d 7 7 œ7<" where ]µ Poisson 7@ For part (b) we apply Theorem 003 with 8œ7@ and U œ #ß áß#, ß áß In this case we have E [ 5 <U : œ 5 5" Tat least 7@<" of the 7@ events ] #á],, #], á],, are true d " 7 7" 7@ œ5 Tat least 7<" of events ]" #, á,] 7 # are trued 5" 5" 7<" " 7 œ5 " T ] Ÿ" < d 7 7 œ7<" where ]µ Poisson 7@ In both parts (a) and (b) the final form follows on simplification

11 For part (c) we apply Theorem 003 with and Uœ ;ß " áß; 7, ßáß In this case we have E [ 5 "U : 5" œ 5 Tat least 7@ of the 7@ events ] ;, á,] ;, ], á,] are true d " " 7 7 7" 7@ œ5 Tat least 7of the 7 events ]" ; ", á,] 7 ; 7 are trued 5" 5" œ5 T ] 3 ; 3d 7 3œ" where ] 3 µ Poisson: 3 The final form follows on simplification Part (d) is a direct application of Theorem 003 Example 00 Some more involved waiting time problems Suppose identical balls are independently distributed into 8 equally likely boxes ( ie a multinomial model or classical allocation scheme) until any 5 of the 8 boxes have at least 7balls each Let denote the number of boxes containing balls when the distribution of balls into boxes stops Let ^µpoisson ) TQ@ " œ = œ 8x 7"x85x=x5="x T^ Ÿ 7" 85 = 5=" 7" ) T^œ7 T^ 7" ) / ) for = ß á ß 5 " and equals else In the œ 7

12 EQ < œ T^ 85 < ) T^œ@ ) / ) Analogous results for the be worked out but are not given here The special simplifies to 5=" 5=" TQ" " œ = œ " 3 œ 3œ 8x = 3x " 85x=x5=" x 85=" =3" and 5<" 5<" EQ" " < œ " œ 8x<x " 85 x 5<" x 85<" <" Holst [0], [0] considers the special and additionally assumes 5œ8but the form of the solutions given in his papers is complicated Proof TQ " œ = œ T 2 Q " œ = and box is last box to receive a 8 œ" œ8tq "œ= and œ8tt " = box is last box to receive a ball box is last box to receive a ball where T is the event that exactly 5" of boxes 2 through 8have at least 7balls and exactly = of boxes 2 through 8have exactly 7 balls

13 Stated in this way we can view the problem as asking for the probability of event T = where the stopping rule is to stop when the " box has exactly 7balls The purpose of casting the problem in this way is to make the application of Theorem 00 appropriate Let G equal the number of balls distributed into box "ß "ßáß8", before the trials stop and define GG G œ " ", á, 8" exactly 5" of G", á, G8" are 7 and exactly = of G, á, G equal 7 else " 8" It then follows from Theorem 00 that TQ@ "œ= œ8 " exactly 5" of ^ ", á, ^8" are 7 5" ) T ) / ) 7" x and exactly = of ^, á, ^ equal 7 " 8" where ^ßáß^ " 8 are iid Poisson ) The final form follows on recognizing that exactly 5" of ^", á, ^8" are at least 7 and exactly = of ^", á, ^8" equal 7if and only if exactly 85of ^", á, ^8" are less than 7, exactly = of ^", á, ^8" equal 7, and exactly 5=", á, ^ exceed 7 of ^" 8" The factorial moment result follows from definition We have EQ " œ = TQ " œ < 5" =œ< 8x œ 7" x 85 x T ^ Ÿ 7" ) / 5" 85 7" ) " = 5=" T ^œ7 T ^ 7" ) =< x 5=" x =œ< and the final form follows on applying the binomial theorem to remove the sum The results for the special follow immediately on simplification Example 005 Sooner and later waiting time problems : how many children? Consider parents who continue to have children until they get ;" boys or ;# girls, whichever comes first Let Y equal the number of children they will have

14 Alternatively, consider parents who continue to have children until they get ;" and ; girls Let Z equal the number of children they will have Then, # boys ;" ;" " # 5 EY œ 5x 53"x 3 : ": 3xx5 " x 3œ œ and E Z 5 œ 5x 5; " " 5x 5; # " E : 5 ": 5 Y where :œt boy and ":œtgirl Ebneshahrashood and Sobel [0] derive expressions for both of the above expectations for 5œ" and 5œ# However their solutions are in a complex form involving G and H incomplete Dirichlet integrals of type 2 as defined in Sobel, Uppuluri and Frankowski [0] Ling [0] derives a recurrence relation for both of the above expectations for the case 5œ" Proof Both parts of this problem are applications of T heorem 003 with 8œ" Type E outcome (boys), =œ" Type F outcome (girls), :œt boy, ":œtgirl In the first part we are waiting for < " œ" of the 8œ" Type E outcomes or < # œ" of the =œ" Type Foutcomes By Theorem < < : U " 8= U : < < E 5 5" Y œ E[ œ 5 T^, á, ^ d " # " # 5" œ 5 T^, á, ^ d " 8= U :< < œ 5 T^" ;", ^# ;# d 5" " # where ^ " µ Poisson : and ^# µ Poisson ": The final form follows on using the independence of the ^" and ^# to solve for T^" ;", ^# ;# and carrying out the integration

15 In the second part we are waiting for < " œ" of the 8œ" Type E outcomes and < # œ" of the =œ" Type Foutcomes By Theorem < < : U " 8= U : < < E 5 5" Z œ E[ œ 5 T^, á, ^ d " # " # œ 5 T^" ;" or ^# ;# d 5" œ 5 T^" ;" T^# ;# T^" ;", ^# ;# d 5" where ^ µ Poisson : and ^ µ Poisson ": The solution in the form " # ;" ;" " # 5x 53" 5x 5" Y : E ": 3œ œ 5 follows directly on integrating The final form follows on using identity (19) of Gould [0]