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Notes on two inequalities for random variables whose probability density functions are bounded.

1. Introduction

In a previous paper (4), Hwang using an improvement of Gruss inequality to establish some inequalities for expectation and the distribution function in order to improve the corresponding inequalities established by Barnett and Dragomir in (1) or (2) by using the pre-Gruss inequality. It is a pity that there exist some errors in two main theorems (see Theorem 7 and Theorem 9 of (4)). The purpose of this paper is to provide a corrected version of these two theorems.

We will also need the following variant of Gruss inequality (see (3)). Lemma. Let f, g: [a, b] [right arrow] R be two integrable functions such that

m [less than or equal to] f(x) [less than or equal to] M; for all x [member of] [a, b],

where m, M [member of] R are constants. Then

|[[integral].sub.a.sup.b]f(x)g(x)dx - [1/[b - a]][[integral].sub.a.sup.b]f(x)dx[[integral].sub.a.sup.b]g(x)dx|[less than or equal to][[M - m]/2][[integral].sub.a.sup.b]|g(x) - [1/[b - a]][[integral].sub.a.sup.b]g(t)dt|dx. (1)

2. Main Results

Theorem 1. Let X be a random variable having the probability density function f: [a, b] [right arrow] R and there exist the constants M, m such that 0 [less than or equal to] m [less than or equal to] f(t) [less than or equal to] M [less than or equal to] 1 a.e. t on [a, b]. If

[[sigma].sub.[mu]](X): = [[[[integral].sub.a.sup.b][[(t - [mu])].sup.2]f(t)dt].sup.[1/2]],[mu][member of][a,b],

then we have the inequalities:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

where A([mu]) = [([mu] - [a+b]/2).sup.2] + [(b-a).sup.2]/12.

Proof. Put g(t) = [(t - [mu]).sup.2] in (1), and as [[integral].sub.a.sup.b] f(t) dt = 1 and

[1/[b - a]][[integral].sub.a.sup.b][[(t - [mu])].sup.2]dt = [[([mu] - [a + b]/2)].sup.2] + [[[(b - a)].sup.2]/12] = A([mu]) > 0,

we get

|[[sigma].sub.[mu].sup.2](X) - A([mu])|[less than or equal to][[M - m]/2][[integral].sub.a.sup.b]|[[(t - [mu])].sup.2] - A([mu])|dt (3)

The last integral can be calculated as follows:

For brevity, we set

p(t):=[(t-[mu]).sup.2] - A([mu]), t [member of] [a, b]

and denote [t.suib.1] = [mu] - A[([mu]).sup.1/2], [t.sub.2] = [mu] + A [([mu]).sup.1/2].

Clearly, p([mu]) = -A([mu])< 0 and

p'(t) = 2(t-[mu]),

which implies that p(t) is strictly decreasing on [a, [mu]] and strictly increasing on [[mu], b].

Moreover, we have

p(a) = [[(a - [mu])].sup.2] - A([mu]) = (b - a)([mu] - [2a + b]/3)

and

p(b) = [[(b - [mu])].sup.2] - A([mu]) = (b - a)([a + 2b]/3 - [mu]),

which imply that p(a) [less than or equal to] 0 and p(b) > 0 in case a [less than or equal to] [mu] [less than or equal to] [2a + b]/3, p(a) > 0 and p(b) > 0 in case [2a + b]/3 < [mu] < [a + 2b]/3 as well as p(a) > 0 and p(b) [less than or equal to] 0 in case [a + 2b]/3 [less than or equal to] [mu] [less than or equal to] b.

So, there are three possible cases to be determined.

(i) In case a [less than or equal to] [mu] [less than or equal to] [2a + b]/3, [t.sub.2] [member of] ([mu], b) is the unique zero of p(t) such that p(t) < 0 for t [member of] [a, [t.sub.2]) and p(t) > 0 for t [member of] ([t.sub.2], b]. We have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

(ii) In case [2a + b]/3 < [mu] < [a + 2b]/3, [t.sub.1] [member of] (a, [mu]) and [t.sub.2] [member of] ([mu] b) are two zeros of p(t) such that p(t) > 0 for t [member of] [a, [t.sub.1]) [union] ([t.sub.2], b] and p(t) < 0 for t [member of] ([t.sub.1], [t.sub.2]). We have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

(iii) In case [2a + b]/3 [less than or equal to] [mu] [less than or equal to] b, [t.sub.1] [member of] (a, [mu]) is the unique zero of p(t) such that p(t) > 0 for t [member of] [a, [t.sub.1]) and p(t) < 0 for t [member of] ([t.sub.1],b]. We have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

Consequently, the inequalities (2) follow from (3), (4), (5) and (6). The proof is completed.

Remark 1. It is not difficult to prove that the inequalities (2) are sharp in the sense that we can construct functions f to attain the equality in (2). Indeed, we may choose f such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

in case a [less than or equal to] [mu] [less than or equal to] [2a + b]/3, and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

in case a [2a + b]/3 < [mu] < [a + 2b]/3, and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

in case [a + 2b]/3 [less than or equal to] [mu] [less than or equal to] b.

It is clear that the above all f(t) satisfy the condition of Theorem 1.

Remark 2. For [mu] = [a + b]/2 in Theorem 1 and denote [[sigma].sub.0](X) = [[sigma].sub.[a + b]/2] (X), we have the inequality

|[[sigma].sub.0.sup.2](X) - [[[(b - a)].sup.2]/12]|[less than or equal to][1/[18[square root of 3]]](M - m)[[(b - a)].sup.3], (7)

which improves the inequality (2.7) in (1) and the inequality (5.10) in (2).

Notice that the inequality (7) is sharp according to Remark 1, we can conclude that the inequality

|[[sigma].sub.0.sup.2](x) - [[[(b - a)].sup.2]/12]|[less than or equal to][1/[36[square root of 3]]](M - m)[[(b - a)].sup.3]

in Corollary 8 of (4) is impossible which also implies that the inequality (2.12) in (4) is not valid.

Theorem 2. Let X and f be as above. If

[A.sub.[mu]] (X) = [[integral].sub.a.sup.b] |t - [mu]|f(t) dt, [mu] [member of] [a, b],

then we have the inequalities

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

where B([mu]) = [1/[b - a]][[[([mu] - [a + b]/2)].sup.2] + [[(b - a)].sup.2]/4].

Proof. Put g(t) = |t - [mu]| in (1), and as [[integral].sub.a.sup.b] f(t) dt = 1, and

[1/[b - a]][[integral].sub.a.sup.b]|t - [mu]|dt = [1/[b - a]][[[([mu] - [a + b]/2)].sup.2] + [[(b - a)].sup.2]/4] = B([mu])>0,

we get

|[A.sub.[mu]](X) - B([mu])|[less than or equal to][[M - m]/2][[integral].sub.a.sup.b]||t - [mu]| - B([mu])|dt = [[M - m]/2][[integral]|[mu] - t - B([mu])|dt + [[integral].sub.[mu].sup.b]|t - [mu] - B([mu])|dt]. (9)

For brevity, we set

p(t):= [mu] - t -B([mu]), q(t):= t - [mu] B([mu]), t [member of] [a, b]

and denote [t.sub.1] = [mu] - B([mu]), [t.sub.2] = [mu] + B([mu]).

Clearly,

p([mu]) = q([mu]) = -B([mu]) < 0,

and

p(a) = - [1/[b - a]][[mu] - (b - [[square root of 2]/2](b - a)][[mu] - (b + [[square root of 2]/2](b - a)], q(b) = - [1/[b - a]][[mu] - (a - [[square root of 2]/2](b - a)][[mu] - (a + [[square root of 2]/2](b - a)].

It is not difficult to find that p(a) [less than or equal to] 0 for a [less than or equal to] [mu] [less than or equal to] b - [[square root of 2]/2] (b - a) and p(a) > 0 for b - [[square root of 2]/2] (b - a) < [mu] [less than or equal to] b, q(b) [less than or equal to] 0 for a + [[square root of 2]/2] (b - a) [less than or equal to] [mu] [less than or equal to] b and q(b) > 0 for a [less than or equal to] [mu] < a + [[square root of 2]/2] (b - a).

So there are three possible cases to be determined.

(i) In case a [less than or equal to] [mu] [less than or equal to] b - [[square root of 2]/2] (b - a), p(t) [less than or equal to] 0 for t [member of] [a, [mu]] and a(b) > 0 with [t.sub.2] [member of] ([mu], b) such that a([t.sub.2]) = 0. We have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

(ii) In case b - [[square root of 2]/2] (b - a) < [mu] < a + [[square root of 2]/2] (b - a), p(a) > 0 with [t.sub.1] [member of] (a, [mu] such that p([t.sub.1] = 0 and q(b) > 0 with [t.sub.2] [member of] ([mu], b) such that q([t.sub.2]) = 0. We have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

(iii) In case a + [[square root of 2]/2] (b - a) [less than or equal to] [mu] [less than or equal to] b, p(a) > 0 with [t.sub.1] [member of] (a, [mu]) such that p([t.sub.1]) = 0 and q(t) [less than or equal to] for t [member of] [[mu], b]. We have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

Consequently, the inequalities (8) follow from (9), (10), (11) and (12). The Proof is completed.

Remark 3. It is not difficult to prove that the inequalities (8) are sharp in the sense that we can construct functions f to attain the equality in (8). Indeed, we may choose f such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

in case a [less than or equal to] [mu] [less than or equal to] b - [[square root of 2]/2] (b - a), and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

in case b - [[square root of 2]/2] (b - a), < [mu] < a + [[square root of 2]/2] (b - a), and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

in case a - [[square root of 2]/2] (b - a) [less than or equal to] [mu] [less than or equal to] b.

It is clear that the above all f(t) satisfy the condition of Theorem 2.

Remark 4. For [mu] = [[mu].sub.0] = [a + b]/2 in Theorem 2, we have the inequality

|[A.sub.[mu]0](X) - [[b - a]/4]|[less than or equal to][1/16](m - m)[[(b - a)].sup.2].

which improves the inequality (2.10) in (1) and the inequality (5.13) in (2).

It should be noted that the Theorem 9 in (4) is not valid can be simply examined by checking the case [mu] = a or [mu] = b.

References

(1) N. B. Barnett and S. S. Dragomir, Some inequalities for random variables whose probability density functions are bounded using a pre-Gruss inequality, Kyungpook Math. J., 40(2)(2000), 299-311.

(2) N. B. Barnett and S. S. Dragomir, Some inequalities for probability, expectation and variance of random variables defined over a finite interval, Comp. Math. Appl., 43(2002), 1319-1357.

(3) X. L. Cheng and J. Sun, A note on the perturbed trapezoid inequality, J. Inequal. Pure Appl. Math., 3(2002), issue 2, Article 29. (http://jipam.vu.edu.au/).

(4) D. Y. Hwang, Some inequalities for random variables whose probability density functions are bounded using an improvement of Gruss inequality, Kyungpook Math. J., 45(3)(2005), 423-431.

Institute of Applied Mathematics, School of Science University of Science and Technology Liaoning Anshan 114051, Liaoning, China

Received June 17, 2008, Accepted August 10, 2010.

* 2000 Mathematics Subject Classication. Primary 26D15.
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Author:Liu, Zheng
Publication:Tamsui Oxford Journal of Mathematical Sciences
Article Type:Report
Geographic Code:9CHIN
Date:Dec 15, 2010
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