Journl of Inequlities in Pure nd Applied Mthemtics MOMENTS INEQUALITIES OF A RANDOM VARIABLE DEFINED OVER A FINITE INTERVAL PRANESH KUMAR Deprtment of Mthemtics & Computer Science University of Northern British Columbi Prince George BC V2N 4Z9, Cnd. EMil: kumrp@unbc.c volume 3, issue 3, rticle 41, 2002. Received 01 October, 2001; ccepted 03 April, 2002. Communicted by: C.E.M. Perce Abstrct Home Pge c 2000 Victori University ISSN (electronic): 1443-5756 069-01
Abstrct Some estimtions nd inequlities re given for the higher order centrl moments of rndom vrible tking vlues on finite intervl. An ppliction is considered for estimting the moments of truncted exponentil distribution. 2000 Mthemtics Subject Clssifiction: 60 E15, 26D15. Key words: Rndom vrible, Finite intervl, Centrl moments, Hölder s inequlity, Grüss inequlity. This reserch ws supported by grnt from the Nturl Sciences nd Engineering Reserch Council of Cnd. Thnks re due to the referee nd Prof. Sever Drgomir for their vluble comments tht helped in improving the pper. 1 Introduction......................................... 3 2 Results Involving Higher Moments...................... 4 3 Some Estimtions for the Centrl Moments............... 6 3.1 Bounds for the Second Centrl Moment M 2 (Vrince)........................................ 8 3.2 Bounds for the Third Centrl Moment M 3......... 9 3.3 Bounds for the Fourth Centrl Moment M 4........ 10 4 Results Bsed on the Grüss Type Inequlity.............. 11 5 Results Bsed on the Hölder s Integrl Inequlity.......... 16 6 Appliction to the Truncted Exponentil Distribution...... 20 References Pge 2 of 24
1. Introduction Distribution functions nd density functions provide complete descriptions of the distribution of probbility for given rndom vrible. However they do not llow us to esily mke comprisons between two different distributions. The set of moments tht uniquely chrcterizes the distribution under resonble conditions re useful in mking comprisons. Knowing the probbility function, we cn determine the moments, if they exist. There re, however, pplictions wherein the exct forms of probbility distributions re not known or re mthemticlly intrctble so tht the moments cn not be clculted. As n exmple, n ppliction in insurnce in connection with the insurer s pyout on given contrct or group of contrcts follows mixture or compound probbility distribution tht my not be known explicitly. It is this problem tht motivtes to find lterntive estimtions for the moments of probbility distribution. Bsed on the mthemticl inequlities, we develop some estimtions of the moments of rndom vrible tking its vlues on finite intervl. Set X to denote rndom vrible whose probbility function is f : [, b] R R + nd its ssocited distribution function F : [, b] [0, 1]. Denote by M r the r th centrl moment of the rndom vrible X defined s (1.1) M r = (t µ) r df, r = 0, 1, 2,..., where µ is the men of the rndom vrible X. It my be noted tht M 0 = 1, M 1 = 0 nd M 2 = σ 2, the vrince of the rndom vrible X. When reference is mde to the r th moment of prticulr distribution, we ssume tht the pproprite integrl (1.1) converges for tht distribution. Pge 3 of 24
2. Results Involving Higher Moments We first prove the following theorem for the higher centrl moments of the rndom vrible X. Theorem 2.1. For the rndom vrible X with distribution function F : [, b] [0, 1], (2.1) = (b t)(t ) m df m k=0 ( ) m (µ ) k [(b µ)m m k M m k+1 ], m = 1, 2, 3,.... k Proof. Expressing the left hnd side of (2.1) s (b t)(t ) m df = nd using the binomil expnsion we get [(t µ) + (µ )] m = (b t)(t ) m df = [(b µ) (t µ)][(t µ) + (µ )] m df, m k=0 [(b µ) (t µ)] ( ) m (µ ) k (t µ) m k, k [ m k=0 ( ] m )(µ ) k (t µ) m k df k Pge 4 of 24
= m ( m k k=0 m ( m k k=0 nd hence the theorem. ) (b µ)(µ ) k ) (µ ) k (t µ) m k df (t µ) m k+1 df, In prctice numericl moments of order higher thn the fourth re rrely considered, therefore, we now derive the results for the first four centrl moments of the rndom vrible X bsed on Theorem 2.1. Corollry 2.2. For m = 1, k = 0, 1 in (2.1), we hve (2.2) (b t)(t )df = (b µ)(µ ) M 2. This is result in Theorem 1 by Brnett nd Drgomir [1]. Corollry 2.3. For m = 2, k = 0, 1, 2 in (2.1), (2.3) (b t)(t ) 2 df = (b µ)(µ ) 2 + [(b µ) 2(µ )]M 2 M 3. Corollry 2.4. For m = 3, k = 0, 1, 2, 3, we hve from (2.1) (2.4) (b t)(t ) 3 df = (b µ)(µ ) 3 + 3(µ )[(b µ) (µ )]M 2 + [(b µ) 3(µ )]M 3 M 4. Pge 5 of 24
3. Some Estimtions for the Centrl Moments We pply Hölder s inequlity [4] nd results of Brnett nd Drgomir [1] to derive the bounds for the centrl moments of the rndom vrible X. Theorem 3.1. For the rndom vrible X with distribution function F : [, b] [0, 1], we hve (b ) r+s+1 Γ(r + 1)Γ(s + 1) f, (3.1) (b t) r (t ) s Γ(r + s + 2) df (b ) 2+ 1 q [B(rq + 1, sq + 1)] f p, for p > 1, 1 p + 1 q = 1, r, s 0. Proof. Let t = (1 u) + bu. Then (b t) r (t ) s dt = (b ) r+s+1 Since 1 0 us (1 u) r du = Γ(r+1)Γ(s+1) Γ(r+s+2), (b t) r (t ) s dt = (b ) r+s+1 Using the property of definite integrl, (3.2) 1 0 (1 u) r u s du. Γ(r + 1)Γ(s + 1). Γ(r + s + 2) (b t) s (t ) r df 0, for r, s 0, Pge 6 of 24
we get, (b t) s (t ) r df f (b t) s (t ) r dt, = (b ) r+s+1 Γ(r + 1)Γ(s + 1) Γ(r + s + 2) the first inequlity in (3.1). Now pplying the Hölder s integrl inequlity, (b t) s (t ) r df [ the second inequlity in (3.1). f for r, s 0, ] 1 [ p b ] 1 q f p (t)dt (b t) sq (t ) rq dt = (b ) 2+ 1 q [B(rq + 1, sq + 1)] f p, Theorem 3.2. For the rndom vrible X with distribution function F : [, b] [0, 1], (3.3) m(b ) r+s+1 Γ(r + 1)Γ(s + 1) Γ(r + s + 2) (b t) s (t ) r df Pge 7 of 24
M(b ) r+s+1 Proof. Noting tht if m f M,.e. on [, b], then Γ(r + 1)Γ(s + 1), r, s 0. Γ(r + s + 2) m(b t) s (t ) r (b t) s (t ) r f M(b t) s (t ) r,.e. on [, b] nd by integrting over [, b], we prove the theorem. 3.1. Bounds for the Second Centrl Moment M 2 (Vrince) It is seen from (2.2) nd (3.2) tht the upper bound for M 2, vrince of the rndom vrible X, is (3.4) M 2 (b µ)(µ ). Considering x = (b µ) nd y = (µ ) in the elementry result we hve xy (x + y)2, x, y R, 4 (3.5) M 2 nd thus, (b )2, 4 Pge 8 of 24 (3.6) 0 M 2 (b µ)(µ ) (b )2. 4
From (2.2) nd (3.1), we get (b )3 (b µ)(µ ) M 2 f, 6 (b µ)(µ ) M 2 f p (b ) 2+ 1 1 q [B(q + 1, q + 1)], p > 1, p + 1 q = 1. Other estimtions for M 2 from (2.2) nd (3.1) re resulting in (b )3 m 6 (b µ)(µ ) M 2 M (3.7) M 2 (b µ)(µ ) m (b )3, m f M, 6 (b )3, m f M. 6 3.2. Bounds for the Third Centrl Moment M 3 From (2.3) nd (3.2), the upper bound for M 3 M 3 (b µ)(µ ) 2 + [(b µ) 2(µ )]M 2. Further we obtin from (2.3) nd (3.4), (3.8) M 3 (b µ)(µ )( + b 2µ), from (2.3) nd (3.5), Pge 9 of 24 (3.9) M 3 1 4 [(b µ)3 + (b µ)(µ ) 2 2(µ ) 3 ],
nd from (2.3) nd (3.7), (3.10) M 3 (b µ)(µ )( + b 2µ) m(b )3 (b + µ 2). 6 3.3. Bounds for the Fourth Centrl Moment M 4 The upper bounds for M 4 from (2.4) nd (3.2) M 4 (b µ)(µ ) 3 + 3(µ )[(b µ) (µ )]M 2 + [(b µ) 3(µ )]M 3. Using (2.4), (3.4) nd (3.8), we hve (3.11) M 4 (b µ)(µ )[(b ) 2 3(b µ)(µ )], from (2.4), (3.5) nd (3.9), (3.12) M 4 1 4 [ (b µ) 4 + 4(b µ) 2 (µ ) 2 4(b µ)(µ ) 3 + 3(µ ) 4], nd from (2.4), (3.7) nd (3.10), (3.13) M 4 (b µ)(µ )[(µ ) 2 + ( + b 2µ)( + b 4µ) + 3(b µ)( + b 2µ)] m(b )3 ( + b 2µ)(b 2 µ). 6 Pge 10 of 24
4. Results Bsed on the Grüss Type Inequlity We prove the following theorem bsed on the pre-grüss inequlity: Theorem 4.1. For the rndom vrible X with distribution function F : [, b] [0, 1], (4.1) (b t) r (t ) s f(t)dt (b ) r+s Γ(r + 1)Γ(s + 1) Γ(r + s + 2) 1 (M m)(b )r+s+1 2 [ ( ) ] 1 2 2 Γ(2r + 1)Γ(2s + 1) Γ(r + 1)Γ(s + 1), Γ(2r + 2s + 2) Γ(r + s + 2) where m f M.e. on [, b] nd r, s 0. Proof. We pply the following pre-grüss inequlity [4]: (4.2) h(t)g(t)dt 1 1 h(t) dt b b [ 1 2 (φ γ) 1 b g 2 (t)dt b g(t)dt ( 1 b ) 2 ] 1 2 g(t)dt, provided the mppings h, g : [, b] R re mesurble, ll integrls involved exist nd re finite nd γ h φ.e. on [, b]. Pge 11 of 24
(4.3) Let h(t) = f(t), g(t) = (b t) r (t ) s in (4.2). Then (b t) r (t ) s f(t)dt 1 b 1 f(t)dt b 1 2 (M m) [ 1 b (b t) r (t ) s dt {(b t) r (t ) s } 2 dt ( 1 b where m f M.e. on [.b]. On substituting from (3.2) into (4.3), we prove the theorem. Corollry 4.2. For r = s = 1 in (4.2), (b )2 (b t)(t )f(t)dt 6 result (2.7) in Theorem 1 by Brnett nd Drgomir [1]. ) 2 ] 1 2 (b t) r (t ) s dt, (M m)(b )3 12, 5 We hve the following lemm bsed on the pre-grüss inequlity: Lemm 4.3. For the rndom vrible X with distribution function F : [, b] Pge 12 of 24
[0, 1], (4.4) (b t) r (t ) s r+s Γ(r + 1)Γ(s + 1) f(t)dt (b ) Γ(r + s + 2) 1 2 (M m) [(b ) f 2 (t)dt 1 ] 1 2, where m f M.e. on [, b] nd r, s 0. Proof. We choose h(t) = (b t) r (t ) s, g(t) = f(t) in the pre-grüss inequlity (4.2) to prove this lemm. We now prove the following theorems bsed on Lemm 4.3: Theorem 4.4. For the rndom vrible X with distribution function F : [, b] [0, 1], (4.5) (b t) r (t ) s r+s Γ(r + 1)Γ(s + 1) f(t)dt (b ) Γ(r + s + 2) where m f M.e. on [, b] nd r, s 0. 1 4 (b )(M m)2, Proof. Brnett nd Drgomir [3] estblished the following identity: (4.6) 1 b f(t)g(t)dt = p + ( ) 2 1 b f(t) dt b g(t) dt, Pge 13 of 24
where p 1 (Γ γ)(φ φ), nd Γ < f < γ, Φ < g < φ. 4 By tking g = f in (4.6), we get (4.7) 1 b f 2 (t)dt ( ) 2 1 = p +, where p 1 (M m), M < f < m. b 4 Thus, (4.4) nd (4.7) prove the theorem. Another inequlity bsed on result from Brnett nd Drgomir [3] follows: Theorem 4.5. For the rndom vrible X with distribution function F : [, b] [0, 1], (4.8) (b t) r (t ) s f(t)dt (b ) r+s Γ(r + 1)Γ(s + 1) Γ(r + s + 2) where m f M.e. on [, b] nd r, s 0. 1 M(M m)(b ), 4 Proof. Brnett nd Drgomir [3] hve estblished the following inequlity: (4.9) 1 b ( n 1 f n (t)dt b b ) [ ] Γ 2 Γ n 1 (b ) n 1 1, 4(b ) n 2 Γ (b ) 1 Pge 14 of 24
where γ < f < Γ. From (4.9), we get [ 1 b f 2 (t)dt ( ) 2 1 b ] 1 2 M 2, m f M. nd substituting in (4.4) proves the theorem. Pge 15 of 24
5. Results Bsed on the Hölder s Integrl Inequlity 1 We consider the Hölder s integrl inequlity [4] nd for t [, b], + 1 = p q 1, p > 1, t (5.1) (t u) n f (n+1) (u)du ( t ) 1 ( f (n+1) p t (u) du [ (t ) f (n+1) nq+1 p nq + 1 On pplying (5.1),we hve the theorem: ] 1 q. (t u) nq du Theorem 5.1. For the rndom vrible X with distribution function F : [, b] [0, 1], suppose tht the density function f : [, b] is n times differentible nd f (n) (n 0) is bsolutely continuous on [, b]. Then, (5.2) (t ) r (b t) s f(t)dt n (b ) r+s+k+1 k=0 Γ(s + 1)Γ(r + k + 1) Γ(r + s + k + 2) ) 1 q Pge 16 of 24
f (n+1) n + 1 (b ) r+s+n+2 Γ(r + n + 2)Γ(s + 1), Γ(r + s + n + 3) if f (n+1) L [, b], 1 n! f (n+1) p (nq + 1) 1 (b )r+s+n+ q +1 Γ(r + n + 1 + 1)Γ(s + 1) q 1/q Γ(r + s + n + 1 + 2), q if f (n+1) L p [, b], p > 1, f (n+1) 1 (b ) r+s+n+1 Γ(r + n + 1)Γ(s + 1), Γ(r + s + n + 2) if f (n+1) L 1 [, b], where. p (1 p ) re the Lebesgue norms on [, b], i.e., ( ) 1 p g := ess sup g(t), nd g p := g(t) p dt, (p 1). t [,b] Proof. Using the Tylor s expnsion of f bout : n (t ) k f(t) = f k () + 1 t (t u) n f (n+1) (u)du, t [, b], k! n! we hve (5.3) k=0 n [ (t ) r (b t) s f(t)dt = (t ) r+k (b t) s dt f ] k () k=0 k! [ 1 b ( t ) ] + (t ) r (b t) s (t u) n f (n+1) (u)du dt. n! Pge 17 of 24
Applying the trnsformtion t = (1 x) + xb, we hve (5.4) (t ) r+k (b t) s dt = (b ) r+s+k+1 Γ(s + 1)Γ(r + k + 1). Γ(r + s + k + 2) For t [, b],it my be seen tht t (5.5) (t u) n f (n+1) (u)du Further, for t [, b], t (5.6) (t u) n f (n+1) (u)du Let (5.7) M(, b) := 1 n! t (t u) n f (n+1) (u) du sup f (n+1) (u) u [,b] f (n+1) t (t ) r (b t) s ( t t (t )n+1. n + 1 (t u) n du (t u) n f (n+1) (u) du t (t ) n f (n+1) (u) du f (n+1) (t ) n. ) (t u) n f (n+1) (u)du dt. Pge 18 of 24
Then (5.1) nd (5.5) to (5.7) result in (5.8) M(, b) f (n+1) (t n+1 )r+n+1 (b t) s dt, n! 1 f (n+1) p (nq+1) 1/q if f (n+1) L [, b], (t )r+n+ 1 q (b t) s dt, if f (n+1) L p [, b], p > 1, f (n+1) 1 (t )r+n (b t) s dt, Using (5.3), (5.4) nd (5.8), we prove the theorem. if f (n+1) L 1 [, b]. Corollry 5.2. Considering r = s = 1, the inequlity (5.8) leds to M(, b) f (n+1) (b ) n+4 (n + 1) (n + 3)(n + 4), if f (n+1) L [, b], 1 n! f (n+1) p (nq + 1) 1 q f (n+1) 1 (b ) n+ 1 q +3 (n + 1q + 2 ) (n + 1q + 3 ), if f (n+1) L p [, b], p > 1, (b ) n+3 (n + 2)(n + 3), if f (n+1) L 1 [, b], which is Theorem 3 of Brnett nd Drgomir [1]., Pge 19 of 24
6. Appliction to the Truncted Exponentil Distribution The truncted exponentil distribution rises frequently in pplictions prticulrly in insurnce contrcts with cps nd deductible nd in the field of lifetesting. A rndom vrible X with distribution function 1 e λx for 0 x < c, F (x) = 1 for x c, is truncted exponentil distribution with prmeters λ nd c. The density function for X : λe λx for 0 x < c f(x) = + e λc δ c (x), 0 for x c where δ c is the delt function t x = c. This distribution is therefore mixed with continuous distribution f(x) = λe λx on the intervl 0 x < c nd point mss of size e λc t x = c. The moment generting function for the rndom vrible X: M X (t) = = c e tx λe λx dx + e tc e λc 0 λ te c(λ t), for t λ, λ t λc + 1, for t = λ. Pge 20 of 24
For further clcultions in wht follows, we ssume t λ. From the moment generting function M X (t), we hve: E(X) = 1 e λc, λ E(X 2 ) = 2[1 (1 + λc)e λc ], λ 2 E(X 3 ) = 3[2 (2 + 2λc + λ2 c 2 )e λc ] λ 3, E(X 4 ) = 4[6 (6 + 6λc + 3λ2 c 2 + λ 3 c 3 )e λc ] λ 4. The higher order centrl moments re: in prticulr, M k = k i=0 M 2 = 1 2λce λc e 2λc λ 2, ( ) k E(X i ) µ k i, for k = 2, 3, 4,..., i M 3 = 16 3e λc (10 + 4λc + λ 2 c 2 ) + 6e 2λc (3 + λc) 4e 3λc λ 3, M 4 = 65 4e λc (32 + 15λc + 6λ 2 c 2 + λ 3 c 3 ) λ 4 + 3e 2λc (30 + 16λc + 4λ 2 c 2 ) 4e 3λc (8 + 3λc) + 5e 4λc λ 4. Pge 21 of 24
Using the moment-estimtion inequlity (3.6), the upper bound for M 2, in terms of the prmeters λ nd c of the distribution: The upper bounds for M 3 using (3.8) ˆM 2 (1 e λc )(λc 1 + e λc ) λ 2. ˆM 3 (2 3λc + λ2 c 2 ) e λc (6 6λc + λ 2 c 2 ) + 3e 2λc (2 λc) 2e 3λc λ 3, nd using (3.9) ˆM 3 ( 3 + 4λc 3λ2 c 2 + λ 3 c 3 ) 4λ 3 The upper bounds for M 4 using (3.11) + e λc (9 8λc + 3λ 2 c 2 ) e 2λc (9 4λc) + 3e 3λc 4λ 3. ˆM 4 ( 3 + 6λc 4λ2 c 2 + λ 3 c 3 ) + e λc (12 18λc + 8λ 2 c 2 λ 3 c 3 ) λ 4 nd from (3.12), 2e 2λc (9 + 9λc + 2λ 2 c 2 ) 6e 3λc (2 λc) + 3e 4λc λ 4, ˆM 4 (12 16λc + 103λ2 c 2 4λ 3 c 3 + λ 4 c 4 ) 4λ 4 Pge 22 of 24
4e λc (12 12λc + 5λ 2 c 2 λ 3 c 3 ) 4λ 4 + 2e 2λc (36 + 24λc + 5λ 2 c 2 ) 16e 3λc (3 λc) + 12e 4λc 4λ 4. Pge 23 of 24
References [1] N.S.BARNETT AND S.S.DRAGOMIR, Some elementry inequlities for the expecttion nd vrince of rndom vrible whose pdf is defined on finite intervl, RGMIA Res. Rep. Coll., 2(7) (1999). [ONLINE] http://rgmi.vu.edu.u/v1n2.html [2] N.S.BARNETT AND S.S.DRAGOMIR, Some further inequlities for univrite moments nd some new ones for the covrince, RGMIA Res. Rep. Coll., 3(4) (2000). [ONLINE] http://rgmi.vu.edu.u/v3n4.html [3] N.S. BARNETT, P. CERONE, S.S. DRAGOMIR AND J. ROUMELIOTIS, Some inequlities for the dispersion of rndom vrible whose pdf is defined on finite intervl, J. Ineq. Pure & Appl. Mth., 2(1) (2001), 1 18. [ONLINE] /v2n1.html [4] J.E. PEČARIĆ, F. PROSCHAN AND Y.L. TONG, Convex Functions, Prtil Orderings nd Sttisticl Applictions, Acdemic Press, 1992. Pge 24 of 24