3 DISCRETE DISTRIBUTIONS

Transcription

1 3 DISCRETE DISTRIBUTIONS It is always helpful when solving a poblem to be able to elate it to a poblem whose solution is aleady known and undestood. In pobability theoy, many poblems tun out to be special cases of standad examples. The most common standad examples ae the well-known distibutions. Discete Distibutions In simple tems, a distibution is an indexed set of pobabilities whose sum is. Fo the moment, discussion will be esticted to cases whee thee is a single discete andom vaiable X whose value uns fom zeo upwads and seves as the index. It is possible to think of unning fom to whee in most cases the indexed pobability is zeo. A distibution may be expessed by a table o a function o gaphically. Conside the distibution associated with a fai die. A tabula epesentation of the distibution is: X P(X = ) A functional epesentation of the distibution is:, if N P(X = ) =, othewise A gaphical epesentation of the distibution is: P(X = ) Of the thee epesentations, only the function makes it pedantically clea that unless the pobability is zeo. 3.

2 The Unifom Distibution When all the non-zeo pobabilities ae the same and ae indexed by a contiguous sequence of values of, the distibution is said to be a Unifom distibution. The behaviou of a fai die is an example of a Unifom distibution. All six non-zeo pobabilities ae the same and the index fo these pobabilities has the contiguous values, 2, 3, 4, 5 and. One can imagine a fai die with a diffeent numbe of faces. Conside a fai tetahedal die whose faces happen to be numbeed 5,, 7 and 8. The fou pobabilities ae all 4. Thee is actually a family of distibutions and the desciption: Unifom(m, n) is used to efe to the geneal case; m and n ae the stat and stop values of and ae called the paametes of the distibution. A andom vaiable whose value epesents the outcome of thowing an odinay fai die is said to be distibuted Unifom(,). If the value epesents the outcome of thowing the cuious tetahedal die the andom vaiable is distibuted Unifom(5,8). In the geneal case thee ae n m + values fo the index and, given the equipobable natue of the Unifom distibution, the pobabilities ae all /(n m + ). The functional epesentation of the geneal case is: P(X = ) = n m +, if N m n, othewise It is good pactice always to check that the pobabilities in a distibution sum to. With the geneal Unifom distibution the check is staightfowad: n =m The Tiangula Distibution n m + = n m + n m + = Many standad distibutions will be discussed but one which has aleady been noted but not until now given a name is the Tiangula distibution. The functional epesentation of an example of this distibution was given on page.7 as: {, P(X = ) = 2 if N, othewise As with all discete distibutions it satisfies the infomal equiement of being a set of indexed pobabilities whose sum is : = 2 = =

3 The Binomial Distibution If the pobability of a boy is p and the pobability of a gil is q (whee p + q = ) it has aleady been shown that the fou pobabilities fo two childen sum to as: p 2 + pq + qp + q 2 = The fou tems ae the pobabilities of BB, BG, GB and GG espectively. Since pq = qp the fou tems can conveniently be educed to thee: p 2 + 2pq + q 2 = Using Binomial coefficients, this can be witten as: ( ) 2 p 2 q + ( ) 2 p q + ( ) 2 p q 2 = 2 The middle tem is the pobability of one boy and one gil without egad to ode. With fou childen the equivalent sum is: p 4 + 4p 3 q + p 2 q 2 + 4pq 3 + q 4 = As with the pevious example, the tem which epesents all boys is fist and the tem which epesents all gils is last. Revesing the ode gives: q 4 + 4pq 3 + p 2 q 2 + 4p 3 q + p 4 = Using Binomial coefficients, this can be witten as: p q 4 + p q 3 + p 2 q p 3 q + 3 p 4 q = 4 The five tems ae, espectively, the pobabilities of having,, 2, 3 and 4 boys in a family of fou childen without egad to ode. Using the andom vaiable X to efe to the numbe of boys: ( ) 4 p q 4, if N 4 P(X = ) =, othewise This is an indexed set of pobabilities whose sum is and so is a distibution. It is an example of the Binomial distibution as it applies to 4 childen. The tem ( 4 ) p q 4 begins with ( 4 ) (the numbe of ways of thee being boys in 4 childen) and this is multiplied by p (the pobability of having boys) and q 4 (the pobability that the emaining 4 childen ae gils). 3.3

4 As a distibution it is not completely specified until a value is given fo p (and hence q) as well as saying how many childen thee ae. Taking p =.55 and q =.485, and fo once not using factions, the pobabilities may be tabulated thus: X P(X = ) It is easy to check that the five values sum to. Notice also that when thee ae fou childen of the same sex, the pobability that they ae all boys is noticeably geate than the pobability that they ae all gils. A gaphical epesentation of the distibution is:.4 P(X = ) As with the Unifom distibution, the Binomial distibution is a family of distibutions, indeed a family of families. The desciption Binomial(n, p) is used to efe to the geneal case; n and p ae the paametes. In the example just consideed, the andom vaiable X is said to be distibuted Binomial(4,.55). The (geneal) Binomial distibution applies to many cicumstances whee thee is a finite numbe of entities each of which may be one of two possibilities. A andom vaiable whose value epesents the numbe of heads which appea when 4 fai coins ae tossed is distibuted Binomial(4, 2 ). If you have 4 machines each of which has a % pobability of failing in a given time inteval, the appopiate distibution is Binomial(4, ). In geneal, whee a andom vaiable X is distibuted Binomial(n, p), the pobability P(X = ) is: ( ) n p q n, if N n P(X = ) =, othewise 3.4

5 The sum of these n + pobabilities is: p q n + p q n + p 2 q n p q n + + p n q n It is immediately clea fom the Binomial theoem that the sum is since the expession can be ewitten: n p q n = (q + p) n = = Note that (q + p) n is shown (in pefeence to (p + q) n ) since, eading fom left to ight, the tems in its expansion ae nomally witten with ascending powes of p (compae with the expansion of (x + y) n on page 2.2). The key point is that the geneal case satisfies the infomal equiement of having a set of indexed pobabilities whose sum is. A Point to Ponde In the context of childen, the paticula tem ( n ) p q n is the pobability of thee being boys and n gils in a family of n childen. The coefficient ( n ) is the numbe of ways in which n childen may divide as n boys and n gils and this coefficient multiplies the pobability of one such case. With 4 childen the pobability of thee being boy (and 3 gils) is ( 4 ) p q 3 = 4pq 3. This is eally the sum: pqqq + qpqq + qqpq + qqqp = 4p q 3 The multiplication theoem holds fo each tem because the boy-gil events ae independent and the addition ule holds oveall because the B+3G events ae mutually exclusive. Since each of the fou sepaate B+3G events has the same pobability, the addition amounts to multiplying the pobability of one of them by 4. The pevious paagaph is woth pondeing. Ae the boy-gil events eally independent? If a couple have thee gils would you eally put the same odds on the next child being a boy as you would if they wee expecting thei fist baby? Is the value of p in qqqp eally the same as the p in pqqq? Demogaphic expets geneally agee that it is. The Tinomial Distibution The Binomial distibution applies when consideing entities which have two states, boygil, heads-tails, woking-boken and so on. Thee ae cicumstances when thee states ae appopiate. Fo example a bicycle has thee pincipal states: Paked, Ridden o Pushed and taffic lights can be Red, Geen o Changing. Thee was a bief peiod when tenay computes wee thought woth exploing: voltages would have been positive, zeo o negative. Thee ae many thee-state examples in genetics. It would be fanciful to imagine that childen came in thee sexes but, if both paents have blood goup AB, then each offsping will necessaily have blood goup, AA, AB o BB and thee ae known pobabilities fo each. 3.5

6 One can easonably ask the pobability of such paents with fou childen having two childen AA, one AB and one BB. Poblems involving entities which have thee states lead natually to a discussion of the Tinomial distibution. Befoe poceeding, conside a summay of the case of a family of fou childen and the two-state boy-gil analysis: Thee ae two salient pobabilities p and q; these ae the pobabilities of a child being a boy o a gil espectively. Necessaily p + q =. If ode is taken into account, thee ae 2 4 = ways of having fou childen being GGGG, GGGB,..., BBBB. If ode is taken not into account, thee ae 4 + = 5 ways of having fou childen which can be listed as boys, boy,..., 4 boys. The pobability of thee being boys is: p q 4 = 4!! (4 )! p q 4 Suppose the thee blood goups ae labelled a, b and c and ae egaded as thee sexes: Thee ae thee salient pobabilities; call these p a, and p c and note that necessaily p a + + p c =. If ode is taken into account, thee ae 3 4 = 8 ways of having fou childen because each may be one of thee possibilities. If ode is taken not into account, the numbe of ways of having fou childen tuns out to be 5 and these ways ae most easily pesented in a tiangula aay: aaab aaaa aaac aabb aabc aacc abbb abbc abcc accc bbbb bbbc bbcc bccc cccc Thee is no diect paallel to the pobability of thee being boys because of the complication intoduced by having thee possibilities... Conside a ewite of the expession fo the pobability of thee being boys in the Binomial case: 4!! (4 )! p q 4 = 4! a! b! p a a b The boy-gil pobabilities p and q have been eplaced by p a and and the boy-gil numbes and 4 have been eplaced by a and b. Clealy p a + = and a + b = 4. 3.

7 This latte expession genealises to the Tinomial case: 4! a! b! c! p a a b p c c whee p a + + p c = and a + b + c = 4 Hee, within the total of fou childen, a, b and c ae the numbes with blood goups AA, AB and BB espectively. The only possibilities fo a, b and c ae 3 pemutations of (,,4), pemutations of (,,3), 3 pemutations of (,2,2) and 3 pemutations of (,,2) making a total of 5 possibilities. These 5 possibilities can be plugged into the expession to give the 5 pobabilities: p 4 a 4p 3 a 4p 3 ap c p 2 ap 2 b 2p 2 a p c p 2 ap 2 c 4p a p 3 b 2p a p 2 b p c 2p a p 2 c 4p a p 3 c p 4 b 4p 3 b p c p 2 b p2 c 4 p 3 c p 4 c By way of illustation, take the middle tem in the middle ow. This is the pobability that two of the childen have blood goup AA, one is blood goup AB and one is blood goup BB. Thus a = 2, b = and c =. So: 4! a! b! c! p a a b p c c = 4! 2!!! p a a b p c c = 2p a a b p c c Note that the sum of the coefficients is 8, accounting fo the 8 possibilities if ode is impotant. [Equivalently, the sum of the coefficients in q 4 + 4pq 3 + p 2 q 2 + 4p 3 q + p 4 is, accounting fo the possibilities in the binomial case if ode is impotant.] The sum of the 8 pobabilities tuns out to be: (p a + + p c ) 4 = given that p a + + p c = It is not difficult to expand this fouth powe by hand and veify that the 5 tems which esult coespond to those in the tiangle. It is an essential equiement of any distibution that the oveall total pobability is and the Tinomial distibution satisfies this. A diffeence fom the Unifom, Tiangula and Binomial distibutions is that the constituent pobabilities of the Tinomial distibution ae not indexed in a linea way. Thee is nothing special about 4 as the numbe of childen and the geneal expession fo the Tinomial distibution is: n! a! b! c! p a a b p c c whee p a + + p c = and a + b + c = n Given n childen, this is the pobability that a ae of blood goup AA, b ae of blood goup AB and c ae of blood goup BB. 3.7

8 The Multinomial Distibution If entities have k states then the Multinomial distibution may apply. The expession that should be noted is: n!! 2!... k! p p p k k whee p + p p k = and k = n Given n entities, this is the pobability that ae in state, 2 ae in state 2 and so on up to k being in state k. Expectation o Mean If you epeatedly thow a fai die you would intuitively expect the long-tem aveage of the values shown to be 3 2. On this occasion, intuition povides the ight answe but a moe fomal appoach is meited. The tems expectation (usually denoted by the lette E) and mean (usually denoted by µ) ae used to descibe the long-tem aveage. The mean may be calculated by thinking of weights and moments to detemine a cente of gavity. Including the contived zeo, the values which can esult fom thowing a die ae,, 2, 3, 4, 5 and. Imagine making these values off at unit intevals along a light beam and at each of the seven positions placing a weight whose mass is popotional to the associated pobability: µ The figue shows such an aangement with little squaes epesenting the weights. The leftmost weight has mass zeo and so is not shown. A pivot has been placed at distance µ along the beam and it is at once clea that its position would not leave the beam in balance. To achieve balance, conside the net clockwise moment about the pivot. The equied value of µ has to be such that the net moment is zeo. Accodingly, µ must satisfy: ( µ). + ( µ). + (2 µ). + (3 µ). + (4 µ). + (5 µ). + ( µ). = Conside the last tem on the left, ( µ).. The value µ is the distance of the ightmost weight fom the pivot and this is multiplied by the mass of the weight (equal to the pobability). The same consideation applies to the othe tems but notice that if a weight is to the left of the pivot, the distance (as 2 µ fo example) is negative, coectly implying that the moment is anti-clockwise. Reaange the equation: ( µ ) =

9 The item in backets is the sum of the pobabilities and this, as always, is. Accodingly: µ = = = 2 = 7 2 The value 7 2 o 3 2 comes as no supise as the long-tem aveage outcome of thowing an odinay fai die. The expession fo µ can be ewitten: µ =.P(X = ) +.P(X = ) + 2.P(X = 2) + +.P(X = ) o µ =.P(X = ) The analysis applies to any distibution which is an indexed set of pobabilities whose sum is. The geneal fomula fo the expectation o mean of a single andom vaiable is witten as: E(X) = µ =.P(X = ) (3.) = The item E(X) is ponounced the expectation of X. The sum ove is left open-ended but this is taken to efe to the ange which is appopiate. Glossay The following technical tems have been intoduced: distibution Unifom distibution paamete Tiangula distibution Binomial distibution Tinomial distibution Multinomial distibution expectation mean Execises III Wok in factions wheneve possible.. If the pobability of hitting a taget is 2 5 and five shots ae fied, what is the pobability that the taget will be hit at least twice? What is the conditional pobability that the taget will be hit at least twice, assuming that at least one hit is scoed? 2. A supemaket has 2 check-outs: 5 have A-type cash egistes and 5 have B-type. The A-type has a pobability a of beaking down duing the fist hou of tading and the B-type has a pobability b. The supeviso aives at the end of this time and leans that one egiste has boken down. Detemine the pobability that the boken egiste is (a) A-type and (b) B-type dice ae thown. What is the pobability that each face appeas twice? ( 4. Given Pascal s theoem expessed as ( ( n+ +) = n ) ( + n +) ) n pove that = = 2 n 5. Pove the Binomial Theoem (page 2.2). Hint: it may be helpful to assume that the expansion holds fo (x+y) n and to conside the effect of multiplication by one moe (x + y). 3.9

10 . Using (3.), detemine the expectation of the Tiangula distibution: {, P(X = ) = 2 if N, othewise 7. [Fom Pat IA of the Mathematical Tipos, 973] A pincess is equally likely to sleep on anything fom six to a dozen mattesses of the softest down, and beneath the lowest of these on just half the nights of the yea is placed a pea. Being a young lady of efined sensibility he sleep is invaiably distubed by the pesence of a pea beneath a mee six mattesses; with seven howeve a pea may pass unnoticed in one case out of ten, with eight it may escape detection in two cases out of ten, and so on, so that with the full twelve mattesses she slumbes on notwithstanding the offending pea as often as six times in ten. One moning, on being wakened by Bayes, he maid, she announces delightedly that she has spent the most tanquil of nights. What is the expected numbe of mattesses upon which she slept? 8. The answes to the following questions may be expessed as decimals: (a) What is the pobability of obtaining at least one six when six dice ae thown? (b) What is the pobability of obtaining at least two sixes when 2 dice ae thown? (c) What is the pobability of obtaining at least thee sixes when 8 dice ae thown? 9. A epot of a possible beakthough in the teatment of Ned s Syndome descibed a peliminay tial of a new dug. The dug was administeed to suffees all of whom wee immediately cued of the affliction. Tagically, late tials showed that % of patients teated with this dug die fom an unfotunate side-effect. Suppose that n patients take pat in a tial and that p is the pobability that a tial paticipant suffes the fatal side-effect. Let S be the pobability that at least one of the n patients dies. [Thus S is the pobability that the tial eveals the side-effect.] Now conside the following (whee, again, pobabilities may be expessed as decimals): (a) In the peliminay tial, p = and n =. What is the pobability that none of the patients dies (as was the case in the epoted tial)? (b) Again taking p =, what is the minimum value of n needed to be 9% sue that the tial eveals the side-effect? [Thus what is the minimum value of n needed to ensue that S 9?] (c) The value 9% is sometimes called the confidence. With p =, what is the minimum value of n needed to ensue a confidence of 99%? [That is S 99?] (d) Fo abitay (but known) p and abitay (but known) confidence C, what is the minimum value of n needed to ensue that S C? (e) In eal life thee is a well-known heat dug fo which the pobability of suffeing a seious side-effect is 5. The isk of using the dug is deemed acceptable because thee is a vey much geate pobability that an unteated patient will die. How lage a tial would be needed to be 99% confident that the tial eveals the side-effect? 3.