Journal of Inequalities in Pure and Applied Mathematics SOME P.D.F.-FREE UPPER BOUNDS FOR THE DISPERSION σ (X) AND THE QUANTITY σ 2 (X) + (x − EX)2 volume 7, issue 5, article 186, 2006. N. K. AGBEKO Institute of Mathematics University of Miskolc H-3515 Miskolc–Egyetemváros Hungary Received 22 April, 2006; accepted 11 December, 2006. Communicated by: C.E.M. Pearce EMail: matagbek@uni-miskolc.hu Abstract Contents JJ J II I Home Page Go Back Close c 2000 Victoria University ISSN (electronic): 1443-5756 118-06 Quit Abstract In comparison with Theorems 2.1 and 2.4 in [1], we provide some p.d.f.-free upper bounds for the dispersion σ (X) and the quantity σ2 (X) + (x − EX)2 taking only into account the endpoints of the given finite interval. 2000 Mathematics Subject Classification: 60E15, 26D15. Key words: Dispersion, P.D.F.s. Some p.d.f.-free Upper Bounds for the Dispersion σ (X) and the Quantity σ 2 (X) + (x − EX)2 Contents 1 Introduction and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References 3 N. K. Agbeko Title Page Contents JJ J II I Go Back Close Quit Page 2 of 7 J. Ineq. Pure and Appl. Math. 7(5) Art. 186, 2006 http://jipam.vu.edu.au 1. Introduction and Results Let f : [a, b] ⊂ R → [0, ∞) be the p.d.f. (probability density function) of a random variable X whose expectation and dispersion are respectively given by Z b EX = tf (t) dt a and s Z σ (X) = a b (t − EX)2 f (t) dt = s Z b t2 f (t) dt − (EX)2 . a In [1], Theorems 2.1 and 2.4, the following upper bounds were obtained for the dispersion σ (X) √ 3 (b − a)2 kf k∞ if f ∈ L∞ [a, b] 6 √ −1 2 (b − a)1+q if f ∈ Lp [a, b] , p > 1, 2 kf kp q σ (X) ≤ 2 [(q + 1) (2q + 1)] 1 + 1q = 1 p √ 2 (b − a) if f ∈ L1 [a, b] 2 Some p.d.f.-free Upper Bounds for the Dispersion σ (X) and the Quantity σ 2 (X) + (x − EX)2 N. K. Agbeko Title Page Contents JJ J II I Go Back Close Quit Page 3 of 7 J. Ineq. Pure and Appl. Math. 7(5) Art. 186, 2006 http://jipam.vu.edu.au and the quantity σ 2 (X) + (x − EX)2 σ 2 (X) + (x − EX)2 h ip (b−a)2 b+a 2 + x − kf k∞ if f ∈ L∞ [a, b] (b − a) 12 2 h (b−x)2q+1 +(x−a)2q+1 i 2q1 q kf kp if f ∈ Lp [a, b] , 2q+1 ≤ p > 1, p1 + 1q = 1 b−a + x − b+a 2 if f ∈ L1 [a, b] 2 2 for all x ∈ [a, b]. In this communication we intend to make free from the p.d.f. the above upper bounds for the dispersion σ (X) and the quantity σ 2 (X) + (x − EX)2 taking only into account the endpoints of the given finite interval. Theorem 1.1. Under the above restriction on the p.d.f. we have σ (X) ≤ min {max {|a| , |b|} , b − a} . Proof. First, for any number t ∈ [a, b] we note (via f (t) ≥ 0) that af (t) ≤ tf (t) ≤ bf (t) leading to a ≤ EX ≤ b. Consequently, (1.1) 0 ≤ EX − a ≤ b − a and 0 ≤ b − EX ≤ b − a. Some p.d.f.-free Upper Bounds for the Dispersion σ (X) and the Quantity σ 2 (X) + (x − EX)2 N. K. Agbeko Title Page Contents JJ J II I Go Back Close Quit Page 4 of 7 We point out that the function g : [a, b] → [0, ∞), defined by g (t) = (t − EX)2 , is a bounded convex function which assumes the minimum at point (EX, 0). J. Ineq. Pure and Appl. Math. 7(5) Art. 186, 2006 http://jipam.vu.edu.au Thus β := sup (t − EX)2 : t ∈ [a, b] = max (a − EX)2 , (b − EX)2 ≤ (b − a)2 , by taking into consideration (1.1). Now, it can be easily seen that s s Z b Z b p 2 σ (X) = (t − EX) f (t) dt ≤ β f (t) dt = β ≤ b − a. a a Next, using the facts that function h (t) = t2 decreases on (−∞, 0) and increases on (0, ∞) on the one hand and, s s Z b Z b 2 σ (X) = t2 f (t) dt − (EX) ≤ t2 f (t) dt a a on the other, we can easily check that 2 b if a ≥ 0 Z b max {a2 , b2 } if a < 0 and b > 0 σ 2 (X) ≤ t2 f (t) dt ≤ a 2 a if b ≤ 0, Some p.d.f.-free Upper Bounds for the Dispersion σ (X) and the Quantity σ 2 (X) + (x − EX)2 N. K. Agbeko Title Page Contents JJ J II I Go Back Close Quit Page 5 of 7 so that σ 2 (X) ≤ max {a2 , b2 }. Therefore, we can conclude on the validity of the argument. J. Ineq. Pure and Appl. Math. 7(5) Art. 186, 2006 http://jipam.vu.edu.au Theorem 1.2. Under the above restriction on the p.d.f. we have q σ 2 (X) + (x − EX)2 ≤ 2 min {max {|a| , |b|} , b − a} for all x ∈ [a, b]. Proof. We recall the identity 2 2 Z σ (X) + (x − EX) = b (t − x)2 f (t) dt, x ∈ [a, b] , a from the proof of Theorem 2.4 in [1]. Clearly, Z b (t − x)2 f (t) dt ≤ max (t − x)2 : t, x ∈ [a, b] , Some p.d.f.-free Upper Bounds for the Dispersion σ (X) and the Quantity σ 2 (X) + (x − EX)2 N. K. Agbeko a so that q Title Page σ 2 (X) + (x − EX)2 ≤ max {|t − x| : t, x ∈ [a, b]} . It is obvious that 0 ≤ t − a ≤ b − a and 0 ≤ x − a ≤ b − a, since t, x ∈ [a, b]. We note that we can estimate from above the quantity |t − x| in two ways: |t − x| ≤ |t − a| + |a − x| ≤ 2 (b − a) Contents JJ J II I Go Back and |t − x| ≤ |t| + |x| ≤ 2 max {|a| , |b|} . Consequently, max {|t − x| : t, x ∈ [a, b]} ≤ 2 min {max {|a| , |b|} , b − a} . This leads to the desired result. Close Quit Page 6 of 7 J. Ineq. Pure and Appl. Math. 7(5) Art. 186, 2006 http://jipam.vu.edu.au References [1] N.S.BARNETT, P. CERONE, S.S. DRAGOMIR AND J. ROUMELIOTIS, Some inequalities for the dispersion of a random variable whose pdf is defined on a finite interval, J. Inequal. Pure and Appl. Math., 2(1) (2001), Art. 1. [ONLINE: http://jipam.vu.edu.au/article. php?sid=117]. Some p.d.f.-free Upper Bounds for the Dispersion σ (X) and the Quantity σ 2 (X) + (x − EX)2 N. K. Agbeko Title Page Contents JJ J II I Go Back Close Quit Page 7 of 7 J. Ineq. Pure and Appl. Math. 7(5) Art. 186, 2006 http://jipam.vu.edu.au