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An inequality for covariance with applications
Journal of Inequalities and Applications volume 2015, Article number: 413 (2015)
Abstract
The present paper establishes a new inequality for the covariance of a random variable, which involves functions with bounded derivatives. Some Chebyshev type integral inequalities are given as applications of the new inequality.
1 Introduction
There is an important field in the theory of inequalities which involves two kinds of special inequalities. One is based on the functions with bounded derivatives or of Ostrowski type, which is successfully applied in probability theory, mathematical statistics, information theory, numerical integration, and integral operator theory. A chapter in [1] is devoted to this kind of inequalities. Another field is concerned with the inequalities with the moments of random variables; see [2–8]. By using this kind of Ostrowski type inequalities, we can get various tight bounds with the moments of random variables defined on some finite interval. There are numerous works available in the literature.
In this paper, we give an inequality for covariance involving functions with bounded derivatives. As applications of the inequality, we obtain some new inequalities similar to the Čebyšev integral inequality.
We assume throughout the paper that ξ is a random variable having the cumulative distributing function F. By Eξ we denote the expectation of ξ defined by
by Dξ the variance of ξ defined by
and by \(\operatorname {Cov}(\xi,\eta)\) the covariance of two random variables ξ, η defined by
We often use the following formula to compute \(\operatorname {Cov}(\xi,\eta)\):
2 A new random inequality
This paper gives the following new inequality for covariance involving functions with bounded derivatives.
Theorem 2.1
Assume that two functions \(f,g:[a,b]\rightarrow R\) are continuous in \([a,b]\) and differentiable in \((a,b)\) whose derivatives \(f',g':(a,b)\rightarrow R\) are bounded in \((a,b)\); if ξ is a random variable which has finite expected value Eξ and variance Dξ. Then one has
where a is a real or −∞; b is a real or +∞ and
Proof
Under the conditions of the theorem, since \(f(\xi)\) is bounded, the expected value \(\mathrm{E}f(\xi)\) exists. Applying the Lagrange mean theorem, one can get
where the parameter \(0\leq \theta \leq 1\) is not a constant but depends on x, ξ, and \(a\leq x\leq b\). Letting \(x=\xi\) in inequality (2.3) and then taking the expectation to both sides of the inequality gives
That is,
Similarly we have
Consequently,
Thus, the inequality is derived. □
3 Some applications
In the following section, we will discuss some applications as regards the inequality (2.1). In fact, if the random variable ξ in (2.1) has a certain distribution, we can derive a corresponding Čebyšev type inequality. At first, we show the famous Čebyšev integral inequality [9].
Let us consider two functions \(f,g:[a,b]\rightarrow R\) are continuous in \([a,b]\) and differentiable in \((a,b)\) whose derivatives \(f',g':(a,b)\rightarrow R\) are bounded in \((a,b)\). Then
for all \(x\in [a,b]\), where
In 1935, Grüss showed that [10]
if M, m, N, n are real numbers which satisfy \(-\infty< m\leq f(x)\leq M<+\infty\), \(-\infty< n\leq g(x)\leq N<+\infty\) for all \(x\in [a,b]\). Moreover, 1/4 is the best possible constant.
Over the years, the Čebyšev integral inequality has evoked the interest of several researchers who showed new proofs, and extended and innovated the inequality. See e.g. [9, 11–16] and the references given therein.
As the first application of the inequality (2.1), let ξ have uniform distribution in \([a,b]\), then we have the inequality as follows.
Theorem 3.1
Let \(f,g:[a,b]\rightarrow R\) be continuous in \([a,b]\) and differentiable in \((a,b)\) whose derivatives \(f',g':(a,b)\rightarrow R\) are bounded in \((a,b)\). Then
Proof
Let ξ be a random variable which possesses the uniform distribution \(u[a,b]\). So, it has the following probability density function:
Then one can have
and
Substituting (3.6) and (3.7) into (2.1) yields (3.4). Thus, the proof is complete. □
If ξ has the Gamma distribution, we can easily obtain a new inequality from (2.1).
Theorem 3.2
Let \(f,g:[0,+\infty)\rightarrow R\) be continuous in \([0,+\infty)\) and differentiable in \((0,+\infty)\) whose derivatives \(f',g':(0,+\infty)\rightarrow R\) are bounded in \((0,+\infty)\). Then for \(\alpha,\lambda>0\),
where \(\Gamma(\alpha)\) is the well-known Gamma function, defined by
Proof
Let a random variable ξ possess Gamma distribution \(\Gamma(\alpha,\lambda)\) whose probability density function is
where the parameters \(\alpha>0\), \(\lambda>0\). Then it is easy to obtain
and
Substituting (3.11) and (3.12) into (2.1) one gets (3.8). Thus, we complete the proof. □
If ξ has the Beta distribution, one has the following inequality from (2.1).
Theorem 3.3
Suppose \(f,g:[0,1]\rightarrow R\) be continuous in \([0,1]\) and differentiable in \((0,1)\) whose derivatives \(f',g':(0,1)\rightarrow R\) are bounded in \((0,1)\). Then
Proof
Let ξ be a random variable which possesses the Beta distribution \(\beta(a,b)\). So, it has the following probability density function:
where the parameters \(a>0\), \(b>0\). Then one obtains
and
Substituting (3.15) and (3.16) into (2.1) one gets (3.13). Thus, we complete the proof. □
All above results deal with continuous random variable. Finally, we give two examples of discrete random variables.
Theorem 3.4
Let \(f,g:[0,+\infty)\rightarrow R\) be continuous in \([0,+\infty)\) and differentiable in \((0,+\infty)\) whose derivatives \(f',g':(0,+\infty)\rightarrow R\) are bounded in \((0,+\infty)\). Then, for \(\lambda>0\),
Proof
Let a random variable ξ possess Poisson distribution \(P(\lambda)\). So, it has the following probability function:
where the parameters \(\lambda>0\). Then it is easy to obtain
and
Substituting (3.19) and (3.20) into (2.1) yields (3.17). The proof is complete. □
Theorem 3.5
Let \(f,g:[0,+\infty)\rightarrow R\) be continuous in \([0,+\infty)\) and differentiable in \((0,+\infty)\) whose derivatives \(f',g':(0,+\infty)\rightarrow R\) are bounded in \((0,+\infty)\). Then, for \(0< p<1\) and \(n=0,1,2,\ldots \) ,
Proof
Let a random variable ξ possess the binomial distribution \(B(n,p)\). So, it has the following probability function:
where the parameters \(0< p<1\). Then it is easy to obtain
and
Substituting (3.23) and (3.24) into (2.1) one gets (3.21). Thus, we complete the proof. □
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Acknowledgements
Supported by the National Natural Science Foundation (grant 11271057) of China.
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He, Z., Wang, M. An inequality for covariance with applications. J Inequal Appl 2015, 413 (2015). https://doi.org/10.1186/s13660-015-0942-7
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DOI: https://doi.org/10.1186/s13660-015-0942-7