0, otherwise. EX = E(X 1 + X n ) = EX j = np and. Var(X j ) = np(1 p). Var(X) = Var(X X n ) =
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1 PROBABILITY MODELS Discrete probability distributios I this sectio, we discuss several well-ow discrete probability distributios ad study some of their properties. Some of these distributios, lie the Biomial ad Geometric distributios, have appeared before i this course; others, lie the Negative Biomial distributio, have ot Biomial distributio (, p). For 1 ad 0 p 1, let X 1,..., X be idepedet, idetically distributed (i.i.d.) Beroulli radom variables with parameter p. Recall that a X 1 has the Beroulli(p) distributio if its probability mass fuctio is give by p() : p, 1 1 p, 0 0, othewise. The X : X X is said to have the Biomial distributio with parameter (, p). The mass fuctio of the Biomial distributio is obtaied by a stadard combiatorial argumet: p X () p (1 p), 0,...,. Notice that a Beroulli radom variable has the Biomial distributio with parameter (1, p). We have show previously that for X 1 Beroulli(p), EX 1 p ad Var(X) p(1 p). Therefore, by our represetatio of X Biomial(, p) as a sum X X of i.i.d. Beroulli-p radom variables, we have EX E(X 1 + X ) EX j p ad j1 Var(X) Var(X X ) Var(X j ) p(1 p) Uiform distributio (). Suppose U is uiformly distributed o {1,..., }, that is { 1/, u 1,..., p U (u) : 0, otherwise. By symmetry, we ca deduce that EU ( + 1)/2. Alteratively, let 2 : E(U + 1) 2 EU 2 (i + 1) 2 i 2 ( + 1)2 i1 j1 i1 12. Also, E(U + 1) 2 EU 2 E[(U + 1) 2 U 2 ] 2EU + 1. Puttig these two statemets together, we obtai EU ( + 1)/2. We also obtai the very importat idetity ( + 1) i. 2 We ca use the same method to compute EU 2. Put i1 3 : E(U + 1) 3 EU 3 ( + 1) ;
2 36 HARRY CRANE ad otice also that Cosequetly, we have 3 E[3U 2 + 3U + 1] 3EU 2 + 3EU + 1 3EU Here, we obtai the idetity EU ( ) 1 ( ) i1 Puttig together EU ad EU 2 gives i 2 ( ) 6 ( + 1)(2 + 1). 6 Var(U) EU 2 [EU] ( ) ( + 1) Now, suppose W is uiformly distributed over {a, a + 1,..., b}, for a < b. To fid EW ad Var(W), we ca use what we ow about EU ad Var(U) for U Uiform(). I particular, if W is uiform o {a, a + 1,..., b}, the W ca be expressed as W U + a 1, for U Uiform(b a + 1). Therefore, EW E[U + a 1] EU + a 1 b a Var(W) Var(U + a 1) Var(U) (b a + 1) a 1 b + a Hypergeometric distributio (N, m, ). We have previously ecoutered the Hypergeometric distributio whe we discussed probabilities for various evets related to lottery umbers. Suppose a ur cotais N balls, m N of which are white, N m of which are blac. We draw N balls without replacemet ad let X be the umber of white balls draw. The probability mass fuctio of X is p X () : { ( m )( N m ) / ( N ), max(0, N + m) mi(, m) 0, otherwise. If we let X 1,..., X be the outcome of the ith draw, i 1,...,, where { 1, ith draw is a white ball X i : 0, otherwise, the X ca be expressed as the sum X X X ad the X i s are exchageable, but ot idepedet. I this case, we have P{X i 1} m, i 1,...,, ad N m(m 1) P{X i X j 1}, 1 i j. N(N 1) ad
3 Clearly, EX EX 1 m/n. To compute Var(X), we ote that PROBABILITY MODELS 37 Var(X i ) m N m N N, m(m 1) EX i X j N(N 1), for i j, ad Cov(X i, X j ) EX i X j EX i EX j N 1 < 0. m N N m N Thus, Var(X) Var(X i ) + 2 Cov(X i, X j ) m N i i<j N m N m N 1 N N. N m N + 2 ( 2 ) m N N m N N 1 Writig C (N )/(N 1), we obtai Var(X) C Var(X ), where X has the Biomial distributio with parameter (, m/n). We iterpret C as the fiite populatio correctio factor for drawig without replacemet from a fiite ur with iitial proportio m/n of white to blac balls. Note that C 1 as N, ad so the Biomial distributio has the iterpretatio of drawig from a ur with ifiitely may balls, a fractio p of which are white Geometric distributio (p). Let W be the waitig time for the first head whe tossig a coi with success probability p [0, 1]. So W whe the first head appears o the th toss. Here, we ca compute both the probability mass fuctio ad the cumulative distributio fuctio i closed form: p W () : p(1 p) 1, 1, 2,..., ad P{W } p W (j) p(1 p) j 1 1 (1 p). j1 j1
4 38 HARRY CRANE We compute the expectatio of W by EW p(1 p) 1 p 1 (1 p) 1 1 p d dq 0 q p d 1 dq 1 q 1 p (1 q) 2 p/p 2 1/p. We ca also compute coditioal distributios for W, which reveals a iterestig ad uique property of the Geometric distributio. Let >, the P{W W > } P{{W } {W > }} P{W > } pq 1 1 (1 q ) pq 1. Cosequetly, the coditioal distributio of W, give W >, is Geometric(p). This property is ow as the memoryless property. The Geometric distributio is the uique distributio o the positive itegers with the memoryless property. Alteratively, we ca compute EW by coditioig o the first flip. I this case, EW 1 P{W 1} + E(W W > 1)P{W > 1} P{W 1} + E((W 1) + 1 W > 1)P{W > 1} P{W 1} + E(W + 1)P{W > 1}, where W W 1. By the memoryless property, EW EW ad we have EW p + (1 p)(1 + EW).
5 PROBABILITY MODELS 39 To compute the variace, it is easiest to compute the secod factorial momet of W: Now, ad we have EW(W 1) ( 1)pq 1 1 ( 1)pq 1 2 p/q ( + 2)( + 1)q 0 p/q d2 dq 2 0 pq d2 dq q 2pq/p 3 2q/p 2. 2q/p 2 EW(W 1) EW 2 EW EW 2 1/p; EW 2 2q/p 2 + p/p 2 q ad Var(W) 2q + p p 2 1 p 2 q/p2. Alteratively, we could defie a Geometric radom variable V to be the umber of tails before the first head. So, i our otatio, we have V W 1 ad p V (v) p W (v + 1) pq v, v 0, 1,..., EV EW 1 1/p 1 q/p, Var(V) Var(W 1) q/p Negative Biomial distributio (p, r). Cosider tossig a p-coi (0 < p < 1) repeatedly ad cosider W r, the umber of tosses util the rth tail. The probability mass fuctio of W r, r 1, is { ( 1 p Wr () : r 1) p (1 p) r, r, r + 1, r + 2,... 0, otherwise. Alteratively, for V r W r r, the umber of tails before the rth head, we have + r 1 p Vr () p Wr ( + r) p (1 p) r. r 1
6 40 HARRY CRANE This latter specificatio motivates the ame Negative Biomial: + r 1 + r 1 r 1 (r + 1)(r + 2) (r + 1)r! ( 1) ( r)( r 1) ( r + 1)! r : ( 1). Therefore, we ca write p Vr () r ( p) (1 p) r. Alteratively, we ca write W r X X r, where X 1,..., X r are i.i.d. Geometric(p). Thus, EW r E(X X r ) r/(1 p) Var(W r ) rp/(1 p) 2. We could, however, compute these quatities without oticig the represetatio of W r as a sum of r idepedet Geometric radom variables. I this case, we have 1 EW r p (1 p) r r 1 r! (r 1)!( r)! p (1 p) r r r p (1 p) r p r r } {{ } r/p. pmf of NB(p,r+1) The Negative Biomial distributio arises i a commo probabilistic theme of coupo collectig. Example 10.1 (Coupo collectig). Cosider rollig a fair 6-sided die repeatedly. How may rolls are eeded before all 6 umbers occur? Let N be the umber of rolls required to see all 6 umbers of a fair 6-sided die. Clearly, P{N > 0} 1. Now, let The, F i : {i does ot occur i the first rolls}. 6 P{N > } P i1 F i ad S 1 S 2 + S 3 S 4 + S 5 S 6,
7 by iclusio-exclusio, where We have ad so o. Therefore, S : 1 i 1 < <i 6 PROBABILITY MODELS 41 P{F i 1 F i }, 1,..., 6. P{F i } (5/6), P{F i F j } (4/6), i j, P{F i F j F } (3/6), i j, (5 ) 6 (4 ) 6 (3 ) 6 P{N > } +, ad P{N } P{N > 1} P{N > }. Alteratively, we ca write N 1 + X X 5, where X i is the umber of additioal rolls eeded to produce the (i + 1)st ew umber. I this case, X 1 Geometric(5/6), X 2 Geometric(4/6),..., X 5 Geometric(1/6), ad X i s are all idepedet. Therefore, we have EN Var(N) 6 6/i 147/10 ad i1 5 i1 1 i/6 (i/6) 2 1 i 5 6(6 i) i /100. Example 10.2 (Legth of a game of craps). The approach to the previous example ca be used to study the legth of a game of craps. Let R be the total umber of rolls ad let A 0 be the umber of additioal rolls (if ay) made after the first roll. The R A + 1, ER EA + 1 ad SD(R) SD(A). For j 2,..., 12, let T j : {first roll is j} ad let For 0, p j : P{T j } 6 7 j, j 2,..., P{A } p 2 + p 3 + p 7 + p 11 + p 12 1/3. For 1, 2,..., coditio o the result after the first roll. P{A } P{T j }P{A T j }. j4,5,6,8,9,10 If the poit is j, the the umber of additioal rolls follows the Geometric distributio with success probability p j + p 7. Writig θ j : p j + p 7, we have P{A } p j (1 θ j ) 1 θ j. j4,5,6,8,9,10
8 42 HARRY CRANE To fid EA, we ca use the coditioig rule for expectatios EA P{T j }E[A T j ] 2 2 j4,5,6,8,9,10 j4,5,6 j3,4,5 392/165. p j /θ j j Factorial momets. For a radom variable X, the quatity µ : EX is called the th momet of X ad ν : EX E[X(X 1) (X + 1)] is called the th factorial momet of X. Sometimes, computig factorial momets maes calculatio of ordiary momets easier. For example, EX µ 1 ν 1 ad Var(X) µ 2 µ 2 1 ν 2 + ν 1 ν 2 1. Let A 1,..., A be evets ad let X : j I Aj be the umber of evets that occur. We ow ν 1 EX i P[A i ] S 1. The umber of pairs of evets to occur is X X(X 1) I Ai A j i<j Therefore, ν 2 E[X(X 1)] 2!S 2, where S 2 : i<j P{A i A j }. I geeral, with S : 1 i 1 < <i P{A i1 A i }, we have ν!s Poisso distributio (µ). Example 10.3 (Modelig volume). How do we model the volume (umber of shares traded) of a stoc popular stoc o a ordiary day? To begi, we assume the umber of people i the maret is very large ad, for ow, we assume that each perso ca purchase at most oe share of stoc ad does so idepedetly of everyoe else with very small probability p > 0. If X deotes the umber of shares bought, the X Biomial(, p). We already ow that EX p ad, if p is of moderate size, we would lie to ow what P{X } is for moderate values of. If p is small ad p is moderate, the log(1 p) log(1 p) p µ; hece, (1 p) e µ. Thus, for moderate values of, we have P{X } p (1 p) 1! (p) (1 p) (1 p) 1! 1 µ e µ 1 µ e µ /!. I fact, we see that this iformal approximatio gives rise to aother distributio, ow as the Poisso distributio.
9 PROBABILITY MODELS 43 More formally, if we tae, p 0, ad p µ [0, ) i p (1 p ), we obtai the limitig distributio p (1 p) µ e µ /!. A radom variable X for which p X () µ e µ /!, 0, 1, 2,..., has the Poisso distributio with parameter µ. The Poisso distributio is ofte used to model rare evets. For the X Poisso(µ), we have EX µ e µ /! µ 0 µ e µ /( 1)! 1 µ. µ 1 e µ /( 1)! 1 } {{ } pmf of Poisso(µ) The mode (most liely value) of X Poisso(µ) is µ, the greatest iteger smaller tha µ, if µ is ot a iteger, ad both µ ad µ 1 if µ Z. To see this, we compare the ratio of successive probabilities p X () p X ( 1) µ e µ /! µ 1 e µ /( 1)! µ. Also, for j 1, 2,..., P{X j} µ j /j!: ( P{X j} µ j e µ /j! 1 + µ j µ 2 ) (j + 1)(j + 2) + µ j e µ /j! ( 1 + µ/1! + µ 2 /2! + ) µ j e µ /j!e µ µ j /j!. Theorem 10.4 (Law of rare evets). Suppose Y Biomial(, p) ad X Poisso(p), the Y D X if is large, p is small, ad p is moderate. I practice, Theorem 10.4 applies whe p 5, or so. Theorem 10.5 (A Geeral Poisso Approximatio theorem). For each, suppose A 1,,..., A, are (ot ecessarily idepedet, ot ecessarily equi-probable) evets. Let N : j1 I A j, be the
10 44 HARRY CRANE umber of evets to occur. If ad A i, s vary with i such a way that S, : P A ij, λ /! for each, the 1 i 1 < <i (18) P{N j} λ j e λ /j!, j 0, 1,.... j1 Ituitively, if you have a large umber of thigs, each with a small probability, ad they are approximately idepedet, the N is approximately Poisso. Example 10.6 (Hat matchig). Suppose there is a group of people, each with their ow hat. Everyoe throws their hat ito a pile ad the the group, oe at a time, chooses a hat radomly from the pile. Let A i, : {ith perso pics his ow hat} ad N : j1 I A j,, the umber of people who pic their ow hat i a group of. The (18) holds if for each, the A i, s are exchageable, for, P{A i, } 0 i such a way that EN P{A i, } λ, ad the A i, s are asymptotically idepedet i the sese that P{A 1, A, } j1 P{A j,} 1 as, for all 1. I this case, S, P{A 1, A, } 1! () (P{A 1, }) P{A 1, A, } j1 P{A j,} λ /!. For the hat matchig problem, we have A 1,,..., A, are exchageable, P{A 1, } 1/ 1 λ, ad P{A 1, A, }/P(A 1, ) / 1. Hece, N Poisso(1).
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