Inequalities with applications to some k-analog random variables
- Abdur Rehman^{1},
- Shahid Mubeen^{1}Email author,
- Sana Iqbal^{1} and
- Muhammad Imran^{2}
https://doi.org/10.1186/s13660-015-0694-4
© Rehman et al. 2015
Received: 28 November 2014
Accepted: 17 May 2015
Published: 3 June 2015
Abstract
In this paper, we introduce some properties of gamma and beta probability k-distributions. We present some inequalities involving these distributions via some classical inequalities, like Chebyshev’s inequality for synchronous (asynchronous) mappings, Hölder’s and Grüss integral inequalities. Also, we discuss some inequalities involving the variance, coefficient of variation and mean deviation of the said distributions involving the parameter \(k> 0\). If \(k=1\), we get the classical results.
Keywords
random variable variance mean deviation inequalities1 Introduction
A process which generates raw data is called an experiment, and an experiment which gives different results under similar conditions, even though it is repeated a large number of times, is termed a random experiment. A variable whose values are determined by the outcomes of a random experiment is called a random variable or simply a variate. The random variables are usually denoted by capital letters X, Y and Z, while the values associated to them by corresponding small letters x, y and z. The random variables are classified into two classes, namely discrete and continuous random variables.
- (i)
Moment about any value \(x=A\) is the rth power of the deviation of variable from A and is called the rth moment of the distribution about A.
- (ii)
Moment about \(x=0\) is the rth power of the deviation of variable from 0 and is called the rth moment of the distribution about 0.
- (iii)
Moment about mean, i.e., \(x=\overline {x}\) for sample and \(x= \mu\) for population, is the rth power of the deviation of variable from mean and is called the rth moment of the distribution about mean.
Note
The moments about any number \(x=A\) and about \(x=0\) are denoted by \(\mu'_{r}\), while those about mean position by \(\mu_{r}\) and \(\mu _{0}=\mu'_{0} =1\).
Remarks
From the above discussion, we see that the first moment about the mean position is always zero, while the second moment is equal to the variance.
Note
For discrete probability distribution, all the above results and notations are the same, just replacing the integral sign by the summation sign (∑).
2 \(\Gamma_{k}\) Function and gamma k-distribution
Definition 2.1
Definition 2.2
Remarks
We can call the above function incomplete k-gamma function because, if \(k=1\), it is an incomplete gamma function tabulated in [11, 12].
Proposition 2.3
- (i)
The gamma k-distribution is a proper probability distribution.
- (ii)
The mean of the gamma k-distribution is equal to the parameter m.
- (iii)
Variance of the gamma k-distribution is equal to mk.
- (iv)
The harmonic mean of a \(\Gamma_{k}(m)\) variate in terms of k is \((m-k)\).
Proof
Parts (i), (ii) and (iii) are proved in [10].
Proposition 2.4
Proof
Remarks
When \(r=1\), we obtain \(\mu_{1,k}^{\prime}=m=\) mean, when \(r=2\), \(\mu_{2,k}^{\prime}=m(m+k)\) and hence \(\mu_{2,k}\) = \(\mu '_{2,k}-(\mu'_{1,k})^{2}\) = mk = variance of the gamma k-distribution given in Proposition 2.3.
3 Applications to the gamma k-distribution via Chebyshev’s integral inequality
In this section, we prove some inequalities which involve gamma k-distribution by using some natural inequalities [13]. The following result is well known in the literature as Chebyshev’s integral inequality for synchronous (asynchronous) functions. Here, we use this result to prove some k-analog inequalities [14] and some new inequalities.
Lemma 3.1
Definition 3.2
Theorem 3.3
Proof
Corollary 3.4
Theorem 3.5
Proof
Corollary 3.6
Theorem 3.7
Proof
Now, we discuss some estimations for the expected values of reciprocals which can be used for the harmonic mean of k-gamma random variables.
Theorem 3.8
Proof
In the following theorem, we give an inequality for the estimation of variance of the k-gamma random variable.
Theorem 3.9
Proof
Corollary 3.10
Proof
4 Some results via Holder’s integral inequality
In this section, we prove some results involving the k-gamma random variable via Hölder’s integral inequality. The mapping \(\Gamma_{k}\) is logarithmically convex proved in [18], and now we have the following theorem.
Theorem 4.1
Proof
Corollary 4.2
Theorem 4.3
Proof
Theorem 4.4
Proof
5 Some inequalities for the mean deviation
Note
When \(k\rightarrow1\), \(\beta_{k} (x, y) \rightarrow \beta(x, y)\).
Definition 5.1
Remarks
We can call the above function an incomplete k-beta function because, if \(k=1\), it is an incomplete beta function tabulated in [21].
Lemma 5.2
Now, an application of the Grüss integral inequality results in the following estimation of the mean deviation of a k-beta random variable.
Theorem 5.3
Proof
Theorem 5.4
Proof
Theorem 5.5
Proof
Declarations
Acknowledgements
The authors are grateful to the anonymous referees for their helpful comments and suggestions to improve the article.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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