2. The Exponential Distribution

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1 Vitual Laboatoies > 13. The Poisso Pocess > The Expoetial Distibutio Basic Theoy The Memoyless Popety The stog eewal assumptio meas that the Poisso pocess must pobabilistically estat at a fixed time s. I paticula, if the fist aival has ot occued by time s, the the time emaiig util the aival occus must have the same distibutio as the fist aival time itself. This is kow as the memoyless popety ad ca be stated i tems of a geeic iteaival time X as follows: the coditioal distibutio of X s give X > s is the same as the distibutio of X. Equivaletly, P( X > t + s X > s) = P( X > t), s 0, t 0 The Distibutio Fuctios Let G deote the ight-tail distibutio fuctio of X, so that G(t) = P( X > t) fo t Show that the memoyless popety is equivalet to the law of expoets: G(t + s) = G(s) G(t), s 0, t 0 2. Show that the oly solutios of the fuctioal equatio i Execise 1, which ae cotiuous fom the ight, ae expoetial fuctios. Let c = G(1). Successively show that G() = c fo N. G ( 1 ) = c 1/ fo N +. G( m ) = c m / fo m N ad N +. G(t) = c t fo t 0. I the cotext of Execise 2, let = l(c). The > 0 (sice 0 < c < 1) so G(t) = P( X > t) = e t, t 0 Hece X has a cotiuous distibutio with cumulative distibutio fuctio give by F(t) = P( X t) = 1 e t, t 0 The Pobability Desity Fuctio 3. Show that the pobability desity fuctio of X is f (t) = e t, t 0

2 A adom vaiable with this pobability desity fuctio is said to have the expoetial distibutio with ate paamete. The ecipocal 1 is kow as the scale paamete. 4. Show diectly that the expoetial pobability desity fuctio is a valid pobability desity fuctio. 5. I the gamma expeimet, set k = 1 so that the simulated adom vaiable has a expoetial distibutio. Vay with the scoll ba ad watch how the shape of the pobability desity fuctio chages. Fo selected values of, u the expeimet 1000 times with a update fequecy of 10, ad watch the appaet covegece of the empiical desity fuctio to the pobability desity fuctio. The Quatile Fuctio 6. Show that the quatile fuctio of X is 7. I paticula, show that F 1 ( p ) l(1 p) =, 0 < p < 1 The media of X is l(2) The fist quatile of X is The thid quatile X is l(4) The itequatile age is l(3) l(4) l(3) Costat Failue Rate Recall that if a oegative adom vaiable with a cotiuous distibutio is itepeted as the lifetime of a device, the the failue ate fuctio is h(t) = f (t) 1 F(t), t 0 whee, as usual, f deotes the pobability desity fuctio ad F the cumulative distibutio fuctio. 8. Show that the expoetial distibutio with ate paamete has costat failue ate, ad is the oly such distibutio. Momets The followig execises give the mea, vaiace, ad momet geeatig fuctio of the expoetial distibutio. 9. Show that E( X) = Show that va( X) = 1 2.

3 11. Show that E ( e u X ) = fo u <. u I the cotext of the Poisso pocess, the paamete is kow as the ate of the pocess. O aveage, thee ae 1 time uits betwee aivals, so the aivals come at a aveage ate of pe uit time. Note also that the mea ad stadad deviatio ae equal fo a expoetial distibutio, ad that the media is always smalle tha the mea. 12. I the gamma expeimet, set k = 1 so that the simulated adom vaiable has a expoetial distibutio. Vay with the scoll ba ad watch how the mea/stadad deviatio ba chages. Now set = 0.5, u the expeimet 1000 times with a update fequecy of 10, ad watch the appaet covegece of the empiical mea ad stadad deviatio to the distibutio mea ad stadad deviatio, espectively. 13. Show that E( X ) = Γ ( +1) fo > 0 whee Γ is the gamma fuctio. I paticula, E( X ) =! if N. Additioal Popeties The expoetial distibutio has a amazig umbe of iteestig mathematical popeties; some of these popeties ae satisfied oly by the expoetial distibutio, ad thus seve as chaacteizatios. Relatio to the Geometic Distibutio 14. Suppose that X has the expoetial distibutio with ate paamete. Show that the followig adom vaiables have geometic distibutios o N ad o N +, espectively, each with paamete 1 e. X, the lagest itege less tha o equal to X.. X, the smallest itege geate tha o equal to X. I may espects, the geometic distibutio is a discete vesio of the expoetial distibutio. Odeigs ad Miima 15. Suppose that X ad Y have expoetial distibutios with paametes a ad b, espectively, ad ae idepedet. Show that P( X < Y ) = a a + b 16. Suppose that ( X 1, X 2,..., X ) is a sequece of idepedet adom vaiables, ad that X i has the expoetial distibutio with ate paamete i > 0 fo each i {1, 2,..., }. Fid the distibutio fuctio ad desity fuctio of U = mi {X 1, X 2,..., X }. Fid the distibutio fuctio of V = max {X 1, X 2,..., X }. Fid the desity fuctio of V i the special case that i = fo each i {1, 2,..., }

4 Note that the miimum U i pat (a) has the expoetial distibutio with paamete I the cotext of eliability, if a seies system has idepedet compoets, each with a expoetially distibuted lifetime, the the lifetime of the system is also expoetially distibuted, ad the failue ate of the system is the sum of the compoet failue ates. I the cotext of adom pocesses, if we have idepedet Poisso pocess, the the ew pocess obtaied by combiig the adom times is also Poisso, ad the ate of the ew pocess is the sum of the ates of the idividual pocesses (we will etu to this poit latte). 17. I the ode statistic expeimet, select the expoetial distibutio. Set k = 1 (this gives the miimum U). Vay with the scoll ba ad ote the shape of the desity fuctio. Fo selected values of, u the simulatio 1000 times, updatig evey 10 us. Note the appaet covegece of the empiical desity fuctio to the tue desity fuctio. Vay with the scoll ba, set k = each time (this gives the maximum V), ad ote the shape of the desity fuctio. Fo selected values of, u the simulatio 1000 times, updatig evey 10 us. Note the appaet covegece of the empiical desity fuctio to the tue desity fuctio. We ca ow geealize Execise 15: 18. I the settig of Execise 16, show that fo i {1, 2,..., }, i P( X i < X j fo all j i) = j j =1 Fist, ote that X i < X j fo all j i if ad oly if X i < mi {X j : j i}. Note that he miimum o the ight is idepedet of X i ad, by Execise 16, has a expoetial distibutio with paamete j i j Now use Execise 15. The esults i Execise 16 ad Execise 18 ae vey impotat i the theoy of cotiuous-time Makov chais. Suppose that X i is the time util a evet of iteest occus (the aival of a custome, the failue of a device, et) fo each i; these times ae idepedet ad expoetially distibute The the fist time U that oe of the evets occus is also expoetially distibuted (Execise 16 (a)), ad the pobability that the fist evet to occu is evet i is popotioal to the ate i. The ext execise gives a adomized vesio of the memoyless popety: 19. Suppose that X ad Y ae idepedet ad that Y has the expoetial distibutio with ate paamete > 0, Show that X ad Y X ae coditioally idepedet give X < Y, ad the coditioal distibutio of Y X is also expoetial with paamete. 20. Coside agai the settig of Execise 16. Show that P( X 1 < X 2 < < X ) = i =1 i j j =i Of couse, the pobabilities of othe odeigs ca be computed by pemutig the paametes appopiately i the fomula o the ight.

5 Computatioal Execises 21. Suppose that the legth of a telephoe call (i miutes) is expoetially distibuted with ate paamete = 0.2. Fid the pobability that the call lasts betwee 2 ad 7 miutes. Fid the media, the fist ad thid quatiles, ad the itequatile age of the call legth. 22. Suppose that the lifetime of a cetai electoic compoet (i hous) is expoetially distibuted with ate paamete = Fid the pobability that the compoet lasts at least 2000 hous. Fid the media, the fist ad thid quatiles, ad the itequatile age of the lifetime. 23. Suppose that the time betwee equests to a web seve (i secods) is expoetially distibuted with ate paamete 2. Give the mea ad stadad deviatio of the time betwee equests. Fid the pobability that the time betwee equests is less that 0.5 secods. Fid the media, the fist ad thid quatiles, ad the itequatile age of the time betwee equests. 24. Suppose that the lifetime X of a fuse (i 100 hou uits) is expoetially distibuted with P( X > 10) = 0.8. Fid the ate paamete. Fid the mea ad stadad deviatio. Fid the media, the fist ad thid quatiles, ad the itequatile age of the lifetime. 25. The positio X of the fist defect o a digital tape (i cm) has the expoetial distibutio with mea 100. Fid the ate paamete. Fid the pobability that X < 200 give X > 150. Fid the stadad deviatio. Fid the media, the fist ad thid quatiles, ad the itequatile age of the positio. Vitual Laboatoies > 13. The Poisso Pocess > Cotets Applets Data Sets Biogaphies Exteal Resouces Keywods Feedback