A Generalized Ostrowski Type Inequality For A Random Variable Whose Probability Density Function Belongs To L p [a, b]

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1 Applied Mathematics E-Notes, 8008), c ISSN Available free at mirror sites of amen/ A Generalized Ostrowski Type Ineuality For A Random Variable Whose Probability Density Function Belongs To L p [a, b] Arif Rafi, Nazir Ahmad Mir, Fiza Zafar Received 3 June 007 Abstract We establish here an ineuality of Ostrowski type for a random variable whose probability density function belongs to L p[a, b], in terms of the cumulative distribution function and expectation. The ineuality is then applied to generalized beta random variable. Introduction The following theorem describes an ineuality which is known in literature as Ostrowski ineuality [7]. THEOREM. Let f : I R R be a differentiable mapping in I 0 interior of I), and let a, b I 0 with a < b. If f : a, b) R is bounded on a, b), i.e., f := sup f t) <, then we have t a,b) f x) b a f t)dt [ 4 + x a+b ) ) ] ) f, ) for all x [a, b]. The constant 4 is sharp in the sense that it cannot be replaced by a smaller one. In [], N. S. Barnett and S. S. Dragomir established the following version of Ostrowski type ineuality for cumulative and probability density functions. Mathematics Subject Classifications: 6D0, 6D5, 60E5. Department of Mathematics, COMSATS Institute of Information Technology, Plot # 30, Sector H-8/, Islamabad 44000, Pakistan. Department of Mathematics, COMSATS Institute of Information Technology, Plot # 30, Sector H-8/, Islamabad 44000, Pakistan. Centre for Advanced Studies in Pure and Applied Mathematics, B. Z. University, Multan 60800, Pakistan. 46

2 Rafi et al. 47 THEOREM. Let X be a random variable with probability density function f : [a, b] R R + and with cumulative distribution function Fx) = PrX x). If f L [a, b] and f := sup f t) <, then we have the ineuality: t [a,b] b EX) PrX x) for all x [a, b]. Euivalently, EX) a PrX x) [ 4 + [ 4 + x a+b ) x a+b ) ) ) ] ] ) f, ) ) f. 3) The constant 4 in ) and 3) is sharp. In [4], S. S. Dragomir, N. S. Barnett and S. Wang developed Ostrowski type ineuality for a random variable whose probability density function belongs to L p [a, b] in terms of the cumulative distribution function and expectation. The ineuality is given in the form of the following theorem: THEOREM 3. Let X be a random variable with the probability density function f : [a, b] R R + and with cumulative distribution function Fx) = PrX x). If f L p [a, b], p >, then we have the ineuality: [ x )+ ] )+ b EX) PrX x) + ) a b x + + b a), 4) for all x [a, b], where p + =. In [8], we may find the following theorem: THEOREM 4. Let f : [a, b] R be continuous, differentiable on [a, b] and f L p [a, b] for some p >. Then [ ] f a) + f b) b ) h)f x) + h f t)dt a [ ) + ) + h ) h ) + x a + ) ) ] + h ) f + b x p, 5) where = p p, h [0, ] and a + h b a x b h b a. The main aim of this paper is to develop an Ostrowski type ineuality for random variables whose probability density functions are in L p [a, b] based on 5). Applications for a generalized beta random variable are also given.

3 48 Ostrowski Type Ineuality Main Results The following theorem holds. THEOREM 5. Let X and F be as defined above. Then from Theorem 4 we have h)f x) + h b F t)dt a [ ) + ) + h ) h ) + x a ) + ) ] ) + h ) + b x f p, 6) where f is the probability density function associated with the cumulative distribution function F. Euivalently, h)prx x) + h b EX) ) + ) + b x [ h ) ) + + x a ) + h ) ] ) + h ) f p, 7) for all x [a + h b a, b h b a ] and h [0, ]. The proof is obvious. Hence, the details are omitted. We now give some corollaries of the above theorem for the expectations of the variable X. COROLLARY. Under the above assumptions, we have the double ineuality b h ) + ) E X) a + h ) + + ), h)) +, h)) +, 8) for h [0, ] and, h) = h ) + ) ) + h ) ) +. 9)

4 Rafi et al. 49 PROOF. It is known that a E X) b. If x = a in 7), we obtain implies h b E X) ), h) f p, + b h ) + ) E X) b h ) +, h)) + + ), h)) +. 0) The left hand estimate of the ineuality 0) is euivalent to first ineuality in 8). Also, if x = b in 7), then which reduces to E X) a a + h ) h + + ) E X) a + h ) + ), h) f p,, h)) + + ), h)) +. ) The right hand side of the ineuality ) proves the second ineuality of 8). REMARK. As for the probability density function f associated with the random variable X, implies = b a f t)dt,. ) If we suppose that f is not too large so that ) + ) h ), h), ) then from the double ineuality 8), it can be verified that and a + h ) + + ), h)) + b, b h ), h)) + f + ) p a,

5 50 Ostrowski Type Ineuality when ) holds. It shows that 8) gives a much tighter estimate of the expected value of the random variable X. COROLLARY. With the above assumptions, E X) a + b [ b ) ] a ), h) f p h, 3) + where, h) is defined by 9). PROOF. From the ineuality 8), which is exactly 3). ) h) + ) E X) a + b ) h) + + ), h)) +, h)) + This corollary provides the mechanism for finding a sufficient condition, in terms of, for the expectation E X) to be close to the midpoint of the interval, a+b. COROLLARY 3. Let X and f be as above and ε > 0. If then h) + ) + ) ε +, 4), h)), h)) + E X) a + b ε The following corollary of Theorem 5 also holds. COROLLARY 4. Let X and F be as above, then h)pr X a + b h + + h) +) + ) h) ) f p + ), h) f p h). 5) + PROOF. If we choose x = a+b in 7), then we get h)pr X a + b ) + h b E X) h + + h) +) ) f p, +

6 Rafi et al. 5 which may be rewritten in the following form h)pr X a + b h + + h) +) Using the triangular ineuality, we get h)pr X a + b ) h)pr X a + b ) + E X) a + b gives the desired result. ) + h + ) f p. + + h + + h + A similar ineuality holds for Pr X a+b ). E X) a + b ) E X) a + b ) E X) a + b ) E X) a + b 3 Applications for Generalized Beta Random Variable If X is a beta random variable with parameters β 3 >, β 4 > and for β > 0 and any β, the generalized beta random variable Y = β + β X, is said to have a generalized beta distribution [6] and the probability density function of the generalized beta distribution of beta random variable is, ) f x) = { x β) β 3β +β x) β 4 Bβ 3+,β 4+)β β 3 +β 4 +) for β < x < β + β 0 otherwise., where B l, m) is the beta function with l, m > 0 and is defined as B l, m) = For s, t > 0 and h [0, ), we choose 0 x l x) m dx. β = h, β = h), β 3 = s, β 4 = t. Then, the probability density function associated with generalized beta random variable Y = h + h)x, takes the form f x) = { h x ) s h x)t h Bs,t) h) s+t < x < h 0 otherwise..

7 5 Ostrowski Type Ineuality Now, and provided = E Y ) = h h xf x)dx = h) s s + t + h, 6) h) p B s, t) B p p s ) +, p t ) + ), 7) s > p, t > p, for p >. Then, by Theorem 5, we may state the following. PROPOSITION. Let X be a beta random variable with parameters s, t). Then for generalized beta random variable Y = h + h)x, we have the ineuality PrY x) t s + t h ) + ) + x h + ) + x h + ) h) p B s, t) + B p p s ) +, p t ) + ), 8) for all x [ h, h ]. In particular, Y Pr ) t s + t h) p B s, t) h + + h) + + ) B p p s ) +, p t ) + ). 9) REMARK. For h = 0 in 8), we have the ineuality Pr X x) t s + t ) x + + x) + B p p s ) +, p t ) + ), 0) + B s, t) for all x [0, ], and particularly, X Pr ) t s + t + ) B p p s ) +, p t ) + ). B s, t) Acknowledgment. The authors are thankful to the referee for giving valuable comments and suggestions for the preparation of the final version of this paper.

8 Rafi et al. 53 References [] N. S. Barnett, P. Cerone and S. S. Dragomir, Ineualities for Random Variables Over a Finite Interval, Nova Science Publisher, in press, Preprint. [] N. S. Barnett and S. S. Dragomir, An Ostrowski type ineuality for a random variable whose probability density function belongs to L [a, b], Nonlinear Anal. Forum, 5000),5 35. [3] N. S. Barnett and S. S. Dragomir, An ineuality of Ostrowski s type for cumulative distribution functions, Kyungpook Math. J., 39)999), [4] S. S. Dragomir, N. S. Barnet and S. Wang, An Ostrowski s type ineuality for a random variable whose probability density function belongs to L p [a, b], p >, Math. Ineual. Appl., 4)999), [5] S. S. Dragomir, P. Cerone, J. Roumeliotis, A new generalization of Ostrowski s intergral ineuality for mappings whose derivatives are bounded and applications in numerical integration and for special means, Appl. Math. Lett., 3000), 9 5. [6] Z. A. Karian, E. J. Dudewicz, Fitting Statistical Distributions, The Generalized Lambda Distribution and Generalized Bootstrap Methods, CRC Press, 000). [7] A. Ostrowski, Über die Absolutabweichung einer differentienbaren Funktionen von ihren Integralmittelwert, Comment. Math. Helv., 0938), 6 7. [8] X. J. Yang, Refinement of Hölder ineuality and application to Ostrowski ineuality, Appl. Math. Comput., 38003),