Some discrete distribution

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Imogen Donna Ward
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1 Some discrete distributio p Defiitio (Beroulli distributio B(p)) A Beroulli distributio takes o oly two values: 0 ad 1, with probabilities 1 p ad p, respectively. pmf: p() = p (1 p) (1 ), if =0or =1 mgf: pe t +1 p mea: p variace: p(1 p) parameter: p [0, 1] eample: toss a coi oce, p=probability that head occurs Note: If A is a evet, the the idicator radom variable I A follows the Beroulli distributio. Defiitio (Biomial distributio B(, p)) Suppose that idepedet Beroulli trials are performed, where is a fied umber. The total umber of 1 appearig i the trials follows a biomial distributio with parameters ad p. p (1 p) ( ), =0, 1,..., pmf: p() = mgf: (pe t +1 p),t R. mea: p variace: p(1 p) parameter: p [0, 1], =1, 2,... eample: #ofheads,tossacoi times p Note: (a + b) = =0 a b.

2 Note. 1. biomial distributio is a geeralizatio of beroulli distributio from 1 trial to trials 2. Let X 1,...,X be i.i.d. B(p), the Y = X X B(, p). 3. Let X i B( i,p),i =1,...,k,adX 1,...,X k are idepedet. The, Y = X X k B( k,p). p Defiitio (Geometric distributio G(p)) The geometric distributio is costructed from a ifiite sequece of idepedet Beroulli trials. Let X be the total umber of trials up to ad icludig the first appearace of 1, the X follows the geometric distributio. pmf: p() = (1 p)( 1) p, =1, 2, 3,... p cdf: F () = 1 (1 p)[], 1 [] <[]+1 0, < 1 mgf: pe t 1 (1 p)e t,t< log(1 p). mea: 1 p variace: 1 p p 2 parameter: p [0, 1] eample: lottery, # of tickets a perso must purchase up to ad icludig the first wiig ticket Note: a memoryless distributio Note: = t = t 1 t, for 1 <t<1.

3 Defiitio (Negative Biomial distributio NB(r, p)) Suppose that a sequece of idepedet Beroulli trials is performed util the appearace of the rth 1. Let X deote the total umber of trials, the X follows egative biomial distributio. p pmf: p() = 1 p r (1 p) ( r), = r, r +1,... r 1 p mgf: r e rt,t< log(1 p). [1 (1 p)e t ] r mea: r p variace: r(1 p) p 2 parameter: p [0, 1], r =1, 2,... eample: lottery, # of tickets a perso must purchase up to ad icludig the rth wiig ticket Note: +j 1 =0 j t = 1 (1 t), for 1 <t<1. Note. p egative biomial distributio is a geeralizatio of geometric distributio from 1st success to rth success 2. Let X 1,X 2,...,X r be i.i.d. G(p), the Y = X 1 + +X r NB(r, p). 3. Let X i NB(r i,p),i =1,...,k,adX 1,...,X k are idepedet. The, Y = X X k NB(r r k,p).

4 p Defiitio (Multiomial distributio Multiomial(, p1, p2,, pr)) Suppose that each of idepedet trials ca result i oe of r types of outcomes, ad that o each trial the probabilities of the r outcomes are p1, p2,..., pr. Let Xi be the total umber of outcomes of type i i the trials, i = 1,..., r. The, (X1,..., Xr ) follows a multiomial distributio. p Note: (a1 + + a k ) = k = a1 1 ak k. 1,, k

5 Defiitio (Poisso distributio P( )) Limit of biomial distributios X B(, p ), where p 0as i such a way that λ p λ. = = p (1 p ) ( ) ( 1) ( +1)! λ 1 λ ( 1) ( +1) 1! λ 1 λ = 1(1 1 ) (1 1 )λ! 1 λ 1 λ p Note: if a a, 1+ a e a. eplaatios. p if large, the pmf of B(, p) is ot easily calculated. The, we ca approimate them by pmf of P(λ), where λ = p. 2. Let X be the umber of times some evet occurs i a give time iterval I. Divide the iterval ito may small subitervals I k, k =1,...,,of equal legth. Let N k be the umber of evets occurrig i I k. Whe we ca assume N 1,...,N are idepedet ad approimately B(p), X has a distributio ear P(λ), where λ = p. pmf: p() = mgf: e λ(et 1),t R. mea: λ variace: λ parameter: λ > 0 λ! e λ, =0, 1, 2,... eample: umber of phoe calls comig ito a echage durig a uit of time.

6 p Defiitio (Hypergeometric distributio HG(r,, m)) Suppose that a ur cotais black balls ad m white balls. Let X deote the umber of black balls draw whe takig r balls without replacemet. The, X follows hypergeometric distributio. p pmf: p() = m r + m r, =0, 1,...,mi(r, ), r m mgf: No eplicit form. mea: r +m variace: rm(+m r) (+m) 2 (+m 1) parameter: r,, m, =1, 2,..., r + m eample: samplig idustrial products for defect ispectio

7 Notes. relatioship betwee hypergeometric ad biomial distributios: Let m, i such a way that p m, m+ p, where 0 <p<1. The, p m r + m r r p (1 p) r. Readig: Agresti (2002), 1.2 Readig Assigmet: Agresti (2002), 1.3, 1.4, 1.5. These are about ui-variate aalysis for data from some discrete distributio.

Probability and Statistics

Probability and Statistics robability ad Statistics rof. Zheg Zheg Radom Variable A fiite sigle valued fuctio.) that maps the set of all eperimetal outcomes ito the set of real umbers R is a r.v., if the set ) is a evet F ) for