Opinion
The CotX Family of Distributions with Application
Clement Boateng Ampadu, Department of Biostatistics, USA.
Received Date: August 18, 2019 Published Date: August 20, 2019
Abstract
We modify the cotangent function into a new statistical distribution and show its applicability.
Keywords: Trigonometric functions; Statistical distributions; Breast cancer; Odds TX Contents
Introduction
Statistical distributions arising from trigonometric functions have populated the literature, and for example, see [14]. On the other hand, the TX(W) family of distributions appeared in [5], and in the special case the random variable T has support [0, ∞), and the weight function is given by we get the socalled Odds TX family of distributions with the following integral representation for its CDF
where the random variable X has CDF F (x) and the random variable T has PDF
By these observations, this paper unfolds as follows. We introduce a socalled CotX family of distributions in Section 2, and in Section 3 we introduce a so called Cot Odds TX family of distributions. Section 4 and Section 5, show applicability of the new families, and the last section is devoted to the conclusions.
The New Family
The CDF of CotX is defined as
where x ∈ R, the baseline distribution has CDF F (x; ξ) and PDF f(x; ξ). By differentiating the CDF, the PDF of CotX is given as
A New Variant of the Odds TX Family of Distributions
Let T be a random variable with support [0, ∞), whose PDF and CDF are given by r(t; ξ) and R(t; ξ), respectively, with ξ being a vector of parameters in the distribution of T , and let X be a random variable with PDF f(x; β) and CDF F (x; β), where β is a vector of parameters in the distribution of X. We define the Cot Odds TX family of distributions with the following integral for its CDF
Practical Illustration of CotX
We assume X is a Dagum random variable with CDF
for x, a, b, c > 0. Now from Section 2 we have the following Proposition 6.1. The CDF of CotDagum is given by
where x, a, b, c > 0
Remark 6.2. The PDF of CotDagum can be obtained by differentiating the CDF. We write S ∼ CD(a, b, c), if S is a CotDagum random variable (Figure 1).
Practical Illustration of Cot Odds TX
We assume T is a Frechet random variable with CDF
for x, c, d > 0. We also assume that X is a Weibull random variable with CDF given by
for x, a, b > 0. Now from Section 3, we have the following
Proposition 7.1. The CDF of Cot Odds FrechetWeibull is given by
Remark 7.2. The PDF of Cot Odds FrechetWeibull can be obtained by differentiating the CDF. We write J ∼ COF W (a, b, c, d), if J is a Cot Odds FrechetWeibull random variable (Figure 2).
Concluding Remarks
In the present paper we introduced the CotX and Cot Odds TX family of distributions respectively and have shown fit to reallife data. The future interesting problem is to investigate some properties and application of these new classes of statistical distributions.
Acknowledgement
None.
Conflict of Interest
No conflict of interest.
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Clement Boateng Ampadu. The CotX Family of Distributions with Application. Annal Biostat & Biomed Appli. 3(1): 2019. ABBA. MS.ID.000551.
Trigonometric functions; Statistical distributions; Breast cancer; Odds TX Contents

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