9 CONTINUOUS DISTRIBUTIONS


 Clifton Warner
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1 9 CONTINUOUS DISTIBUTIONS A rndom vrible whose vlue my fll nywhere in rnge of vlues is continuous rndom vrible nd will be ssocited with some continuous distribution. Continuous distributions re to discrete distributions s type rel is to type int in ML. Mny formule for discrete distributions cn be dpted for continuous distributions. Very often, little more is required thn the trnsltion of sigm signs into integrl signs. The min bd news is tht there is no equivlent of probbility generting functions. Adpting the P( r) Nottion In discussions which involve single discrete rndom vrible, the nottion P( r) hs been used. When required, mpping is employed to ensure tht r N. In discussing single continuous rndom vrible, will gin be used s the nme but will be used insted of r for the vlue. In probbility theory r strongly implies nonnegtive integer wheres nd my rnge from to +. There is t once problem with the nottion P( ) for the probbility is zero for ny prticulr. Even if is constrined to be in some finite rnge, such s to +, there re n infinite number of possible vlues for. Fortuntely, mny vrints of the P( ) nottion re still useful. For emple: P( <.5) P( < +) P( < b) There is n obvious difficulty with grphicl representtion of continuous rndom vrible. A plot of P( ) ginst serves no useful purpose! Nevertheless, grphicl representtions re both possible nd useful nd here is first ttempt t representing continuous rndom vrible which is distributed Uniform(,):? It is not immeditely cler wht lbel should be ttched to the verticl is but this representtion hs the right feel bout it. The height of the plot is constnt over the rnge to nd is zero outside this rnge. The constnt height is to ensure tht the totl re under the curve is nd this is the clue to much of wht follows. The ide of re corresponding to probbility ws introduced on pge.6 nd with continuous rndom vribles re is often the most convenient wy of representing probbility. 9.
2 Probbility Density Functions In the present cse, the re under the curve between nd is ( ) 8 so the probbility P( < ) 8. In generl, this clcultion will be n integrtion nd some considertion needs to be given to the function to be integrted. The function is clled probbility density function or pdf. In the cse of single rndom vrible it is often nmed f() nd this is the pproprite lbel for the verticl is: f() In the cse of the rndom vrible which is distributed Uniform(,): { f(), if <, otherwise The clcultion just undertken is more formlly written: In generl: P( < ) f() d P( < b) b d f() d [ ] Both common sense nd the ioms of probbility impose certin constrints tht hve to be met by ny probbility density function: I f() must be single vlued for ll II f() for ll III + f() d This lst is sometimes epressed s: f() d Here refers to the rnge of interest, where the probbility density function is nonzero. For the uniform distribution bove, the rnge is to. 9.
3 Continuous Distributions Informlly, discrete distribution hs been tken s lmost ny indeed set of probbilities whose sum is. The inde hs lwys been r,,,... Eqully informlly, lmost ny function f() which stisfies the three constrints cn be used s probbility density function nd will represent continuous distribution. Epecttion With discrete distributions, the generl formul for the men or epecttion of single rndom vrible is: µ E() r.p( r) r This is the first emple of formul used with discrete distributions which cn be redily dpted for continuous distributions. The men µ or epecttion E() of rndom vrible whose probbility distribution function is f() is: µ E().f() d The generl form for the epecttion of function of rndom vrible dpts too but since f is used s the nme of the generl probbility density function some other nme hs to be used for the function of the rndom vrible. In the following, h is tken s some function of the rndom vrible nd the epecttion: E ( h() ) h().f() d In the prticulr cse of the squre of, when h() : Vrince E( ).f() d The definition of vrince is ectly the sme for continuous rndom vribles s for discrete rndom vribles: Vrince σ V() E ( ( µ) ) E( ) ( E() ) Thus the vrince cn be determined by first evluting E() nd E( ). Alterntively, ( µ) cn be regrded s specil cse of the function h nd the vrince cn be directly computed thus: Vrince σ V() E ( ( µ) ) ( µ).f() d 9.
4 Illustrtion Uniform(,) Consider the rndom vrible which is distributed Uniform(,) nd whose probbility density function is: { f(), if <, otherwise The epecttion E() is: The epecttion E( ) is: The vrince V() is: E() E( ) The Generl Uniform Distribution. d [. d [ 6 ] ] V() E( ) ( E() ) In the generl cse, rndom vrible which is distributed Uniform(, b) is uniformly distributed over the rnge to b. To ensure tht the integrl of the ssocited probbility density function f() over this rnge is the function is defined s: { b f(), if < b, otherwise This function cn be represented grphiclly: f() b b By nlogy with discrete distributions, the first check is tht the integrtion over the pproprite rnge is : b [ ] b b d b 9.
5 The epecttion E() is: E() b. b d [ b ] b b. b b + This is simply confirming tht the men is hlfwy between nd b nd this ws seen erlier with the distribution Uniform(,) where the men ws. The epecttion E( ) is: E( ) b The vrince V() is:. b d [ b ] b b. b b + b + V() E( ) ( E() ) b + b + b + b + b + b + b 6b b b + (b ) With the distribution Uniform(,) nd b giving the result noted erlier. Illustrtion oulette Wheel Let be the ngle between some reference rdius on roulette wheel nd some fied direction on the csino tble. The ngle is rndom vrible which is distributed Uniform(, π). Tking the vlues nd b π, the epecttion nd vrince re: Mode nd Medin E() π π nd V() (π) π Informlly, the mode of ny distribution is the most probble vlue. This is the vlue for which f() is mimum. Clerly the Uniform distribution does not hve mode in ny useful sense. Informlly, the medin of ny distribution is the middle vlue. This is the vlue of which is such tht the re under f() to the left of is equl to the re under f() to the right of. If the vlue of the medin is M then M must be such tht: M f() d + M f() d In the cse of the Uniform distribution, the medin is the sme s the men since the hlfwy point divides the re into two equl prts. 9.5
6 Probbility Distribution Functions elted to ny probbility density function f() there is n ssocited function F () which is known s the probbility distribution function. The reltionship is: F () P( < ) f(t) dt The following figure shows the reltionship digrmmticlly. The function F () is the re under the curve from the leftmost end of the region of the distribution (which my be ) up to : f(t) t Two points stem directly from the definition of probbility distribution function. First: P( < b) F (b) F () Secondly, given tht F () is the integrl of f(), the derivtive of F () must be f(): d F () f() d It is unfortunte tht two importnt functions hve the sme initil letters. Some writers distinguish the two thus: pdf PDF stnds for probbility density function stnds for probbility distribution function Given such obvious scope for confusion, the bbrevitions will not be used. Moreover, only limited use will be mde of probbility distribution functions. 9.6
7 The Eponentil Distribution The Eponentil distribution, sometimes known s the Negtive Eponentil distribution, is relted to the Geometric nd Poisson discrete distributions. The min design criterion for this distribution is to find, for some rndom vrible, probbility density function which is such tht: P( ) e λ This is til probbility whose vlue decreses eponentilly s increses. In the contet of the Poisson distribution, imgine tht town verges one murder yer. The probbility of the town hving run of t lest yers without murder is substntilly less thn the probbility of lsting t lest one yer without murder. The first step in determining the pproprite probbility density function is to find the probbility distribution function. Given tht P( < ) + P( ) : P( ) P( < ) F () Hence: Differentite with respect to : F () e λ f() λ.e λ This is not quite suitble s probbility density function becuse the rnge hs not been specified. Clerly the rnge cnnot strt from for this would led to n infinite re under the curve. The pproprite forml specifiction of the probbility density function for the eponentil distribution is: { λ.e λ, if f(), otherwise It is simple to check tht, without ny need for scling, the integrtion over the rnge to is : [ ] λ.e λ d e λ A second check is to confirm tht the probbility density function stisfies the design criterion tht P( ) e λ : P( ) λ.e λt dt [ e λt ] e λ 9.7
8 A grphicl representtion of the Eponentil distribution is: f(t) t The probbility P( ) corresponds to the shded re. Quite clerly, the lrger the vlue of the smller this probbility. The epecttion E() is: E() λ.e λ d Let t λ so d λdt. Then: E() t.e t λ dt λ [ (t + )e t ] λ The epecttion E( ) is: E( ) λ.e λ d Let t λ so d λdt. Then: E( ) t λ t.e t λ dt λ t.e t dt [ ] λ (t + t + )e t λ The vrince V() is: V() E( ) ( E() ) λ ( ) λ λ In the cse of the Poisson distribution both the epecttion nd the vrince re λ. In the cse of the Eponentil distribution the epecttion is λ but the vrince is λ. An importnt considertion in the contet of the Eponentil distribution is tht the time you my epect to wit for No. 9 bus does not depend on when you strt witing for it. 9.8
9 Glossry The following technicl terms hve been introduced: probbility density function probbility distribution function Eercises I medin mode. The distribution of the ngle α (to the verticl) t which meteorites strike the Erth hs probbility density function: f(α) sin(α) where α π Find the epecttion nd vrince of the distribution.. Find the epecttion nd vrince of the double eponentil distribution: f() ce c. If hs the eponentil distribution show tht: P( > u + v > u) P( > v) for ll u, v > This is the lck of memory property (c.f. Eercises IV, question ). 9.9
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