On characterizing some mixtures of probability distributions

Transcription

1 Journal of Appled Mathematcs & Bonformatcs, vol3, no3, 03, 53-7 ISSN: (prnt), (onlne) Scenpress Ltd, 03 On characterzng some mxtures of probablt dstrbutons Al AA-Rahman Abstract The concept of recurrence relatons s used to characterze mxtures of two exponental famles, two st tpe Ouang's dstrbutons and two nd tpe Ouang's dstrbutons Our results are used to deduce conclusons concernng some mxtures of some well nown dstrbutons le Burr, Pareto, Power Webull, st tpe Pearsonan and Ferguson's dstrbutons Furthermore, some characterzatons related to some recentl dstrbutons le, mxtures of two exponentated Webull, exponentated Pareto and generalzed exponental dstrbutons are derved from our results In addton, some well-nown results follow from our results as specal cases Mathematcs Subject Classfcaton: 6E0 Kewords: Mxtures of dstrbutons; truncated moments; falure rate; reversed falure rate; recurrence relatons; exponentated Burr; exponentated Webull; generalzed exponental; Ouang's dstrbuons and Ferguson's dstrbutons Insttute of Statstcal Studes and Research, Caro Unverst, Egpt Artcle Info: Receved : Ma 30, 03 Revsed : Jul 9, 03 Publshed onlne : September 5, 03

2 54 On characterzng some mxtures of probablt dstrbutons Introducton Theor of characterzatons s a ver vtal and nterestng branch of scence whch plas an mportant role n man felds such as mathematcal statstcs, relablt, statstcal nference, theor of mxtures and actuaral scence It concerns wth the characterstc propertes of the probablt dstrbutons so that t helps researchers to dstngush them and n most cases determne dstrbutons unquel Some excellent references are Azlarov and Volodn [], Kagan, Lnn and Rao [] and Galambos and Kotz [5], among others Several deas and concepts have been used to dentf dfferent tpes of the probablt dstrbutons In man practcal stuatons, researchers ma have reasons to assume the nowledge of the expected value of the random varable under stud Ths has motvated several authors and scentsts to characterze dstrbutons usng the concept of condtonal expectatons Gupta [9], Ouang [6], Talwaler [0] and Elbatal et al[3] have used the concept of rght truncated moments to characterze varous tpes of dstrbutons le Webull, gamma, beta, exponental, Burr, Pareto and Power dstrbutons Actuall characterzatons va rght truncated moments are ver mportant n practce snce, n some stuatons some measurng devces ma be unable to record values greater than tme t On the other hand, there are some measurng devces whch ma not be able (or fal) to record values smaller than tme t Ths has encouraged several authors to deal wth the problem of characterzng dstrbutons b means of left truncated moments, see, eg, Gla nzel [],Gupta [9], Ahmed [4], Dma and Xeala [7] and Fahr [8] The dea of recurrence relatons s another mportant concept that has been used man tmes to characterze dfferent tpes of dstrbutons In fact, a recurrence relaton s a relaton n whch the functon of nterest, f n, s defned n terms of a smaller value of n Recurrence relatons ma be used to mnmze the number of steps requred to obtan a general form for the functon under consderaton Furthermore, recurrence relatons together wth some ntal

3 Al AA-Rahman 55 condtons defne functons unquel Ths has motvated several authors to use ths concept n characterzng dstrbutons see, eg, Khan et al [3], Ln [], Fahr [8], Ahmad [], and Al-Hussan et al [0] In ths paper, we are nterested n characterzng mxtures of some classes of probablt dstrbutons usng the concepts of dfferental equatons, recurrence relatons and truncated moments of some functon h(x) of the random varable X Man Results Mxtures of probablt dstrbutdons pla a vtal role n statstcs, relablt, lfe testng, fsheres research, economcs, medcne and pscholog, among others Some excellent references are Evertt and Hand [5], Ttterngton et al[9], Lndsa [6], Maclachlan and Peel [] and Bo hnng [8] The followng Theorem characterzes a mxture of two exponental famles usng the concept of rght truncated moment Theorem Let X be a contnuous random varable wth denst functon f ( ), cdf F ( ), falure rate ω( ) and reversed falure rate µ( ) such that F( ) has non vanshng frst order dervatve on (, ) so that n partcular 0 F( for all xassume that h( ) s a dfferentable functon defned on (, ) such that: (a) E(h (X)) < for ever natural number (b) h( 0, x (, ) (c) lm h( = h( ) x + (d) lm h( = x Then the followng statements are equvalent: h( h( ) () F( = = ρ exp, x (, ) θ

4 56 On characterzng some mxtures of probablt dstrbutons Where ρ and ρ are postve real numbers such that θθ F ( x ) F ( ) ( x ) + θ + θ + F x = h ( h ( h ( () ( ) ( ) = ρ =, x (, ) (3) m = Eh ( ( X) X< µ µ = h + θθ [ h ] + ( θ + θ ) m ω h Proof [ h ] F ( ) θθ m + h θθ F h We have from () the followng results: (a) (b) Therefore, F ( ρ h( h = exp [ ] h ( θ θ = F ( = h ( h ( ρ hx ( ) h exp [ ] θ = θ θθ F ( F ( + ( θ + θ ) + F( h ( h ( h ( ρ ( ) ρ ( ) = θθ exp [ ] + exp [ ] 3 hx h hx h + ( θ θ) = θ θ = θ θ hx ( ) h + ρ[exp [ ] = θ = We have, b defnton: (4) m = E(h (X) X < = = h ( F( h ( df( h ( h ( F( dx F(

5 Al AA-Rahman 57 Let J = h ( h ( F( dx Elmnatng F(, usng equaton (), we have: F ( h ( h ( dx θ+ θh ( F ( dx θθ h ( dx h ( J = ( ) Integratng b parts and mang use of the defnton of m, one gets: F ( J = h h ( θ + θ) m F θθ h ( dx h( F = h h ( θ+ θ) m F θθ h h ( ) + θθ h + θθ F ( ) ( ) h ( h = h ( θ + θ ) m F θθ h F ( dx F h ( ) ( ) ( ) F + θθ m Fh h θθ h Substtutng these results n equaton (4), recallng that: One gets: F (a) ω = (b) F ( ) F µ = F µ µ m = h + θθ h + ( θ + θ ) m ω h 3 h F ( ) θθ m + [ h θθ ] F h Equaton (3) can be wrtten n ntegral form as follows: h ( df( [ F] h h ( ) h ( df( F = + θθ + θ+ θ h F ( ) θθ h df( h θθ h h

6 58 On characterzng some mxtures of probablt dstrbutons Dfferentatng both sdes of the last equaton wth respect to, addng to both sdes the term " h F ", cancellng out the term " ( ) θ θ h ( F ) " from the rght sde and dvdng the result b " h h ", one gets: θ θ F ( h h F ( h (5) + ( θ + θ ) + F( = Ths s a non lnear dfferental equaton of the second order wth varable coeffcent To solve t put z = h( h(), then df df( z) dz F = = = h F ( z) d dz d F ( = h d d F ( df ( z) df ( z) = = h d dz Substtutng these results n equaton (5), we get: z dz d (6) θ θ F z) + ( θ + θ ) F ( z) + F( ) Set ( = F(z) = G(z), equaton (6) becomes: ( θ + θ ) G ( z) + G( ) 0 = h F ( z) θ θ G ( z) + z = The soluton of ths dfferental equaton s nown to be (see, Ross [7]): Therefore G( z) = = z ρ exp θ h( h( ) F( = ρ exp = θ The fact that lm F( = and the condtons mposed on the functon h( mpl x that: ρ + ρ, whle the fact that 0 F( mples that 0 ρ, for {,} =

7 Al AA-Rahman 59 Remars () If we set h(x) = X b, b > 0, = 0, =, we obtan characterzatons concernng a mxture of two Webull dstrbutons wth respectve parameters b, θ and b, θ For b =, the results reduce to that of a mxture of two exponental dstrbutons For b =, = and θ = θ = θ, result (3) reduces to that of Talwaler [0], concernng the exponental dstrbuton, namel, E(X X < = (F( + θ, ff X follows the exponental F( dstrbuton wth parameter θ () If we set h(x) = ln ( + X b ), b > 0, θ =, { 0,}, {,}, =0 and =, we obtan characterzatons concernng a mxture of two Burr dstrbutons wth respectve parameters b, and b, For b =, we have characterzatons concernng a mxture of two Pareto dstrbutons For = =, we have characterzatons concernng Burr dstrbuton (3) If we set h(x) = ln X, θ =, {,}, we obtan characterzatons concernng a mxture of two st tpe Pearsonan dstrbutons wth respectve parameters,, and,, For =, = 0, we have results concernng a mxture of two beta dstrbutons wth respectve parameters, and, Now, we use the concept of rght truncated moments to dentf a mxture of two nd tpe Ouang's dstrbutons [6] wth parameters d, c and d, c Theorem Let X be a contnuous random varable wth denst functon f( ),cdf F( ), falure rate ω( ) and reversed falure rate functon µ( ) such that F ( > 0 for all x (, ) so that n partcular 0 F( for all x Assume that h( ) s a dfferentable functon defned on (, ) such that: (a) E(h (X)) < for ever natural number (b) h ( 0 x (, )

8 60 On characterzng some mxtures of probablt dstrbutons (c) (d) lm h( = d + x lm h( = d x Then the followng statements are equvalent: (7) Fx ( ) = ρ ( hx ( ) + d) = c, x (, ), c { } where d and c are parameters such that 0, (8) (9) hx+ d F ( ( c+ c ) F x [ hx ( ) + d ] + Fx ( ) = h ( h ( h ( [ ( ) ] ( ) m = Eh ( ( X) X< ( ) ( ) h ( ) ( ) F [ ( ) h ( ) ] ( ) h ( ) µ µ = δ h + h + d h ω d c + c + m d m h + F ( ) where δ = + ( c + c + ) Proof From equaton (7), we have the followng results: (a) C F ( = h ( ρ c[ h( + d], and = F h ( ( c [ ( ) + ] h x d (b) = h ( ρ c ( c ) c = =, therefore [ hx ( ) + d] F ( ( c+ c ) F ( [ hx ( ) + d ] + F( h ( h ( h ( ρ c c = ( c + c c )( hx ( ) + d ) + ρ [ ( hx ( ) + d ) ] =

9 Al AA-Rahman 6 3 B defnton we have h ( df( [ h( ] ( 0) m = = h F( ) h ( F( dx F( Mang use of equaton (8) to elmnate F(, the second part of equaton (0) becomes: + = + + F F [ hx ( )] h( c ( c ) J dx [ h( d][ h( ] F ( dx F ( [ ( ) ] [ ( )] F + h ( hx d hx dx Integratng b parts and usng the defnton of m as well as the assumptons mposed on the functon h( ), one gets: h h () J = + [ m + dm ] F ( ) = c c F( F ( [ hx ( ) + d] [ h( ] F ( c + c ) F ( h ( x Let Q [ h x d] h x dx ( ) + [ )] ) ( dx h ( Integratng b parts, tang nto consderaton, the defnton of m and the condtons mposed on the functon h( ), one gets: () µ F Q = [ h ( ) + d] [ h ( )] [ h] h F h ( ) [ m + dm ] [ m + dm + dm Substtutng these results n equaton (0) and collectng smlar terms, we get: F µ c ( + c+ ) = + + ( ) ( ) m h [ h d] [ h ] m F h d ( ) d [ h ] F ( c + c + ) m m + [ h ] F h ( ) ]

10 6 On characterzng some mxtures of probablt dstrbutons Solvng the last equaton for m, we get µ µ m = δ h + [ h + d] h d( c + c + ) m ω h [ h] F ( ) d m + [ h ], F( h where δ = + ( c + c + ) 3 Equaton (9) can be wrtten n ntegral form as follows: (3) [ + ( c + c + )] h ( df( = ( F) h + [ h + d] [ h ( )] ( ) F h [ ] [ ( )] ( ) h d c + c + h x df x F ( ) d [ h( ] df( + c c h ( ) [ ( )] h Dfferentatng equaton (3) wth respect to, cancellng out the terms h F from both sdes, removng the term ( ) d[ h ( )] F from the rght sde and collectng smlar terms together then dvdng the result b h h, one gets: [ ] (4) ( ) ( c+ c ) [ h ( ) + d ] + F ( ) = h h h [ h ( ) + d] F F To solve ths equaton, set z = h( + d, then df df( z) dz F = = = h F ( z) d dz d F ( d F ( df ( z) df ( z) dz and = = = = h F ( z) h d h d dz d Substtutng these results nto equaton (4), we get:

11 Al AA-Rahman 63 z c + c (5) F ( z) zf ( z) + F( z) = c c Set z = e x, then df( z) df( dx F ( F ( z) = = = dz dx dz z df ( z) d F ( d F ( F ( F ( F ( z) = = = F ( = dz dz z z dz z z Substtutng these results n equaton (5), we get: F ( c + c + = F ( F( Puttng Fx ( ) = Gx ( ), the last equaton becomes: G ( + G ( + Gx ( ) = 0 c c Therefore G( = ρexp( cx ) + ρexp( cx ) Hence, c F ( ) = ρ [ ( ) ] h+ d = The fact that lm F( = as well as the condtons mposed on the functon h( gve ρ + ρ, whle the fact that 0 F( mples that 0 ρ for = {,} Ths completes the proof Remars () If we set h(x) = X a, d =, c = b, where a and b are postve parameters such that b { 0,}, {,}, = 0 and =, we obtan characterzatons concernng a mxture of two Burr dstrbutons wth respectve parameters a, b and a, b For a =, we obtan characterzatons concernng a mxture of two Pareto dstrbutons Forb = b = b, we have characterzatons concernng Burr dstrbuton wth parameters a, b

12 64 On characterzng some mxtures of probablt dstrbutons () If we set h(x) = X, d =, c = θ > 0, characterzatons {,}, we have concernng a mxture of two frst tpe Pearsonan dstrbutons wth respectve parameters,, θ and,, θ For = and = 0, we have characterzatons concernng a mxture of two beta dstrbutons wth respectve parameters,θ and, θ (3) If we set h(x) = exp X b, d = 0, c = θ, = 0 and = where b and θ are postve parameters, we obtan characterzatons concernng a mxture of two Webull dstrbutons wth respectve postve parameters b, θ and b, θ For =, we have a mxture of two Ralegh dstrbutons wth respectve parameters θ and θ, For θ = θ = θ, =, result (9) reduces to Ouang's result [6] concernng Webull dstrbuton, namel, E e Xb X < = (eb θ + ), ff X follows the Webull dstrbuton wth θ+ postve parameters b and θ For b =, we have characterzatons concernng a mxture of two exponental dstrbutons wth respectve parameters θ and θ If n addton, we have θ = θ = θ, result (9), reduces to Talwaler's result [0] (4) If we set h(x) = c = c = m(n) m(n) Z(X) Z()+g(n) [m(n)], d = g(n) [m(n)] Z()+g(n) [m(n)] and, where m( ) and g( ) are fnte real valued functons of n and Z( s a dfferentable functon defned on (, ) such that lm Z( = Z( ) x + and g( n) lm Z( = x m( n) concernng a class of dstrbuton defned b Talwaler [0], we obtan characterzatons The followng Theorem characterzes a mxture of two st dstrbutons wth respectve parameters d, c and d, c tpe Ouang's

13 Al AA-Rahman 65 Theorem 3 Let X be a contnuous random varable wth denst functon f( ), cdf F( ), falure rate functon ω( ) and reversed falure rate functon µ( ) such that F( s a twce dfferentable functon on (, ) wth F ( 0 for all x so that n partcular 0 F( Assume that h( ) s a dfferentable functon defned on (, ) such that: (a) E(h (X)) < for ever natural number, x ( ) (b) h( 0 (c) lm hx ( ) = d x + (d) lm h( = d x, Then the followng statements are equvalent: (9) c Fx ( ) = ρ [ ( )] d hx, x (, ), = where d and c are parameters such that c { 0, } for {,} (0) [ d h( ] F x ( c + c ) d h x F x h ( h( h ( ( ) F ( + + = ( ( ) ) () m = E h X X > ( ) ( ) ( ) ( ) ω d h ω ( ) [ ( ) ] = δ h h ( ) d m µ h where δ = + ( c + c + ) Proof 0 ( ) ( ) [ h ] F + d ( c+ c + ) m + [c ch ( ) + ], F h It eas to see that (usng (9)): F ( = h ( ρ c [ d h( ] = c

14 66 On characterzng some mxtures of probablt dstrbutons and F h Therefore 3 ( = h ( ( )[ ( )] c c c d h x ( = ρ d h x F x ( c + c ) F x + [ d hx ( )] + F h ( h ( h ( c c c = ρc[ d hx ( )] + ρ[ d hx ( )] = 0 = = [ ( )] ( ) ( ) B defnton we have: m = h ( df( F Integratng b parts, recallng that lm F( = F µ = F, one gets: x F ω = F and, [ ] () m ( ) ω [ h( ] ( ) ( ) h h x F x dx = h F µ F Let [ ( )] ( ) ( ) J = h x h x F x dx Elmnatng F(, usng equaton (0), one gets: h ( [ d hx ( )] [ hx ( )] F( c+ c J = dx [ d h( ][ h( ] F ( dx Integratng b parts, recallng that lm [ d h( ] = and lm F( =, one gets: x [ h( )] F ( ) F J= + [ dh ( )] [ h ( )] h ( ) h x [ ( )] [ ( )] c + c + ( ) [ ( )][ ( )] ( ) + d hx hx F xdx d hx hx F xdx Substtutng ths result n equaton (), mang use of the defnton of m and performng some elementar computaton, one gets:

15 Al AA-Rahman 67 h ( ) ω m = h [ h ( )] [ dh ( )] F µ [ F] ( ) c ( + c+ ) [ d m dm + m ] + [ dm m ] h F + [ F( ] h ( ) [ ( )] ( ) Solvng the last equaton for m, we get: F h ω dh ω m = δ { c c h h + d c + c + m µ h ( ) [ ( )] ( ) ( ) [ ( )] ( ) [ h( )] F ( ) ( ) d m + [ h ( ) + ], F h ( ) where δ = + ( c + c + ) 3 Multplng both sdes of equaton () b δ then wrtng t n ntegral form as follows: [ + ( c + c + )] h ( df( =Fh ( ) h [ ( )] [ d h ( )] + dc ( + c+ ) [ hx ( )] dfx ( ) F h d h x df x + h c c h + h F ( ) ( ) [ ( )] ( ) [ ( )] [ ( ) ] ( ) Dfferentatng both sdes of the last equaton wth respect to, cancellng out the term h F from both sdes and dvdng the result b h [ ( )], one gets: ( c+ c + ) h F = h h F + ( ) d h F F h d h F + h d [ h h + d c + c + h F d F ( ) ( ) ( ) ( ) ( )

16 68 On characterzng some mxtures of probablt dstrbutons Collectng smlar terms, removng the term ( ) d F from the rght sde and dvdng the result b h h, one gets: (3) d h F ( c + c ) F + [ d h ( )] + F ( ) = 0 h h h [ ( )] ( ) ( ) Ths s a homogenous second order dfferental equaton wth varable coeffcent To solve t set z d h = ln[ ] Then df dz df( z) h F = = = F ( z), d d dz d h df ( z) [ d h ( )] + h ( F ) ( z) F d F d F ( z) d = = = h d h d d h dh [ ] dz df ( z) [ d h ( )] + h ( F ) ( z) d dz h = = [ F ( z) F ( z)] [ d h] [ d h] Substtutng these results n equaton (3), one gets: Therefore F ( z) + F ( z) + F( z) = 0 c c F( z) = ρ exp( c z) + ρ exp( c z) c c Hence, F ( ) = ρ [ d h ( )] + ρ [ d h ( )] The fact that lm F( = and the assumpton that lm h( = d show that ρ + ρ =, whle the fact that 0 F( mples that 0 ρ, {,} Ths completes the proof Remars 3 () If we set h(x) = X rb b, d =, = b, = r, c rb = θ > 0, {,}, we have characterzatons concernng a mxture of two Ferguson' dstrbutons of

17 Al AA-Rahman 69 the frst tpe wth respectve parameters b, r, θ and b, r, θ For b = 0 and r = we have characterzatons concernng a mxture of two Power dstrbutons wth parameters θ and θ For b = 0, r = and θ = θ = θ, we obtan characterzatons concernng a Power dstrbuton wth parameter θ, f, n addton, we set θ =, we obtan results concernng the unform dstrbuton () If we set h(x) = rb rx, d = 0, =, = b, c = θ > 0, {,}, we obtan characterzatons concernng a mxture of two Ferguson's dstrbutons of the second tpe wth respectve parameters r, b, θ and r, b, θ (3) If we set h(x) = exp(x b), d = 0, =, = b and c = θ > 0, {,}, we obtan characterzatons concernng a mxture of two Ferguson's dstrbutons of the thrd tpe (4) If we set h(x) = exp X b a, d =, c = θ, {,}, where a and θ are postve parameters, =0 and =, we obtan characterzatons concernng a mxture of two exponentated Webull dstrbutons wth respectve postve parameters a, b, θ and a, b, θ For θ = θ = θ, we have results concernng the exponentated Webull dstrbuton For θ =, we obtan characterzatons concernng the Webull dstrbuton wth parameters b, a If n addton a =, we have characterzatons concernng the exponental dstrbuton wth parameter b (5) If we set h(x) = b X a, d =, c = θ, {,}, = b and =, we obtan characterzatons concernng a mxture of two exponentated Pareto dstrbutons of the frst tpe wth respectve parameters b, a, θ and b, a, θ For θ = θ = θ, we have characterzatons concernng exponentated Pareto dstrbuton of the frst tpe wth parameters b, a (6) If we set h(x) = [ + X] a, a > 0, d =, c = θ, {,}, = 0 and =, we obtan characterzatons concernng a mxture of two exponentated Pareto

18 70 On characterzng some mxtures of probablt dstrbutons dstrbutons of the second tpe wth respectve parameters a, θ and a, θ For θ = θ = θ, we have characterzatons concernng the second tpe exponentated Pareto dstrbuton (7) If we set h(x) = e γx, γ > 0,d =, c = θ, ε{,}, = 0 and =, we obtan characterzatons concernng a mxture of two generalzed exponental dstrbutons wth respectve postve parameters γ, θ and γ, θ For θ = θ = θ, we have characterzatons concernng the generalzed exponental dstrbuton wth parameters γ and θ If n addton, we set, θ=, we obtan characterzatons concernng the ordnar exponental dstrbuton wth parameter γ (8) If we set h(x) = m(n) m(n) Z(X) Z()g(n) [m(n)], d = g(n) [m(n)] Z()g(n) [m(n)], c = c =, where m( ) and g( ) are fnte real valued functons of n and Z( s a dfferentable functon defned on (, ) such that g( n) lm Z( = + x m( n) and lm Z( = Z( ) x, we obtan characterzatons concernng a class of dstrbutons defned b Talwaler [0] References [] AA Ahmad, Recurrence relatons for sngle and product moments of record values from Burr tpe XII dstrbuton and a characterzaton, J Appled Statstcal Scence, 7, (998), 7-5 [] A Kagan, K Lnn and C Rao, Characterzaton Problems In Mathematcal Statstcs, John Wle and Sons, New Yor, 973 [3] A Khan, M Yaqub and S Parvez, Recurrence Relatons between moments of order statstcs, Naval Res Logst Quart, 30, (983), 49-44

19 Al AA-Rahman 7 [4] AN Ahmed, Characterzaton of beta, bnomal and Posson dstrbutons, IEEE Transactons on Relablt, 40, (99), [5] BS Evertt and DJ Hand, Fnte Mxture Dstrbutons, Chapman and Hall, 98 [6] BS Lndsa, Mxtures Models: Theor, Geometr and Applcatons, Insttute of Mathematcal Statstcs, Haward CA, 995 [7] C Dma and E Xeala, Towards a unfcaton of certan characterzatons b condtonal expectatons, Ann Inst Statstcal Mathematcs, 48, (996), [8] D Bo hnng, Computer-Asssted Analss Of Mxtures And Applcatons, Chapman Hall, CRC, 000 [9] DM Ttterngton, AF Smth and U A Maov, Statstcal Analss of Fnte Mxture Dstrbutons, Wle, London, 985 [0] EK Al-Hussan, A A Ahmed and M A Al-Kashf, Recurrence relatons for moment and condtonal moment generatng functons of generalzed order statstcs, Metra, 6, (005), 99-0 [] G Ln, Characterzaton of dstrbutons va relatonshps between two moments of order statstcs, Journal of Statstcal Plannng Inferences, 9, (988), [] G Maclachan and D Peel, Fnte Mxture Models, Wle, New Yor, 000 [3] I Elbatal, AN Ahmed and M Ahsanullah, Characterzatons of contnuous dstrbutons b ther mean nactvt tmes, Pastan J Stast, 8, (0), 79-9 [4] I Ferguson, On characterzng dstrbutons b propertes of order statstcs, Sanha, Ser A, 9, (967), [5] J Galambos and S Kotz, Characterzaton of Probablt Dstrbutons, Sprnger-Verlag Berln, 978 [6] L Ouang, On characterzaton of dstrbutons b condtonal expectatons, Tamang J Management Scence, 4, (983), 3-

20 7 On characterzng some mxtures of probablt dstrbutons [7] L Ross, Introducton to Ordnar Dfferental Equatons, John Wle and Sons, 989 [8] ME Fahr, Two characterzatons of mxtures of Webull dstrbutons, The Annual Conference, Insttute of Statstcal Studes and Research, Caro Unverst, Egpt, 8, (993), 3-38 [9] P Gupta, Some characterzatons of dstrbutons b truncated moments, Statstcs, 6, (985), [0] S Talwaler, A note on characterzaton b condtonal expectaton, Metra, 4, (977), 9-36 [] TA Azlarov and NA Volodn, Characterzaton Problems Assocated wth The Exponental Dstrbuton, Sprnger-Verlag Berln, 986 [] W Gl a nzel, Some concequences of characterzaton theorem based on truncated moments Statstcs,, (990), 63-68