414 This work is licensed under a Creative Commons Attribution 4.0 International License *Corresponding Author: naji198887@yahoo.com Abstract Truncated distributions arise naturally in many practical situations. It is a conditional distribution that develops when the parent distribution's domain is constrained to a smaller area. The distribution of a right truncated is one of the types of single truncated that is restricted within a specific field and usually occurs when the specified period for the study is complete. Hence, this paper introduces the Right Truncated Inverse Generalized Rayleigh Distribution (RTIGRD) with two parameters. Then, provided some properties such as probability density function, cumulative distribution function (CDF), survival function, hazard function, rth moment, mean, variance, Moment Generating Function, Skewness, kurtosis, Median, and Mode for Right Truncated Inverse Generalized Rayleigh Distribution on [0, 1]. Keywords: Hazard Function, Inverse Generalized Rayleigh Distribution, Right Truncated, Survival Function. 1. Introduction A Truncated Distribution is a conditional distribution on a specific range that restricts the full range. The purpose of truncated distributions is to get better results. When a distribution is truncated, the domain of the truncated random variable is restricted based on the doi.org/10.30526/36.4.2977 Article history: Received 14 August 2022, Accepted 1 November 2022, Published in October 2023 Ibn Al-Haitham Journal for Pure and Applied Sciences Journal homepage: jih.uobaghdad.edu.iq Truncated Inverse Generalized Rayleigh Distribution and Some Properties Noor Abdul Ameer Jabbar * Department of Mathematics, College of Education for Pure Sciences Ibn AL-Haitham, University of Baghdad, Baghdad, Iraq. Bayda Atiya Kalaf Department of Mathematics, College of Education for Pure Sciences Ibn AL-Haitham, University of Baghdad, Baghdad, Iraq. Umar Yusuf Madaki Department of Mathematics and Statistics, Faculty of Science, Yobe State University Damaturu, Nigeria https://creativecommons.org/licenses/by/4.0/ mailto:naji198887@yahoo.com mailto:naji198887@yahoo.com mailto:baydaa.a.k@ihcoedu.uobaghdad.edu.iq mailto:turkyilm@hacettepe.edu.tr 415 truncation points of interest, and thus the shape of the distribution changes. In addition, it happens when we are unable to detect or record events that take place inside or outside of a predetermined range or below or above a given threshold. The truncation can be from the left side, the right side, or both sides [1-3]. Galton et al. introduced truncated distribution in 1898 [4] . Then, [5] provided forms for the probability density function, cumulative distribution function, hazard function, characteristic function, mean, mode, median, variance, skewness, and kurtosis of doubly truncated Fréchet distributions. [6] introduced [0, 1]; Truncated Fréchet Gamma and truncated Fréchet inverted Gamma distributions are discussed as special cases. [7] Proposed a new truncated Weibull-G (TW-G). [8] introduced [0, 1] Truncated Fréchet distributions and [0, 1] Truncated Fréchet Weibull as special cases. The cumulative distribution function, rth moment, mean, variance, skewness, kurtosis, mode, median, characteristic function, reliability function, and hazard function. [9] Introduced Truncated Weibull power Lomax distribution with four parameters. [10] introduced [0,1] Truncated Gompertz Exponential distribution and [0,1] truncated Gompertz-G family distribution and then discussed as cases: probability density function (PDF), Cumulative distribution function (CDF), Hazard rate function (HF), Survival function (SF), moments, the mean μ, variance σ2, Moment Generating Function (M.G.F. ), Median M, kurtosis KR, and Skewness SK. [11] Introduced the Zero Truncated Discrete Transmuted Generalized Inverse Weibull Distribution (ZT-DTGIW). Jumana introduce [0,1] Truncated Lomax – Lomax ([0,1]TLD) distribution. Some properties of the ([0,1] TLLD) distribution were derived [12]. Therefore, in this study, the Right Truncated Inverse Generalized Raleigh Distribution was derived and some of its statistical and mathematical properties were studied on [0,1] (Probability density function, cumulative distribution function, Survival function, hazard rate function, rth moment, variance, Moment Generating Functions, kurtosis, skewness, median, and mode). 2.Truncated Inverse Generalized Rayleigh Distribution The Rayleigh distribution derives from the Weibull distribution with two parameters and is a suitable model for life-testing studies [13]; 𝑓(𝑥) = 2 𝜆 𝑥𝑒− 𝑥2 𝜆 (1) 𝐹(𝑥) = 1 − 𝑒− 𝑥2 𝜆 (2) Due to the practical significance of the Rayleigh distribution, numerous extended forms of the Rayleigh distribution have been proposed. For example, [14-23] However, the generalized inverted scale family distributions were introduced by [24]. These newly developed models were formulated by introducing a new shape parameter to the scale family of distributions. These models give major flexibility in modelling complex data and the results drawn from them seem genuine and quite sound [25-27] . Mudholkar and Srivastava [28] suggested a new method for generalization different distributions 416 dependent on c.d.f, which we will be used to generalize the Rayleigh distribution in this paper as follows: 𝐺(𝑥) = [𝐹(𝑥)]𝜃 = [1 − 𝑒− 𝑥2 𝜆 ]𝜃 (3) 𝑔(𝑥) = 𝜃 [1 − 𝑒− 𝑥2 𝜆 ] 𝜃−1 2𝑥 𝜆 𝑒− 𝑥2 𝜆 (4) The Generalized Raleigh Distribution can be shown by the transformation of a random variable. If the random variable 𝑇 has Generalized Raleigh Distribution, then the r. v. X = ( 1 T ) has an Inverse Generalized Raleigh Distribution (IGRD). Suppose 𝑇 is a random variable following Inverse Generalized Rayleigh distribution with two parameters θ and λ. Then p.d.f and c.d.f functions of inverse Generalized Rayleigh distribution are given for equations (3) and (4), respectively, by [29]; 𝑔(𝑡) = [1 − 𝑒 − 1 𝜆𝑡2] 𝜃−1 2𝜃 𝜆𝑡3 𝑒 − 1 𝜆𝑡2 (5) 𝐺(𝑡) = 1 − [1 − 𝑒 − 1 𝜆𝑡2] 𝜃 (6) when 0 < t < ∞ and 𝑔 (𝑡) = 0 o.w Hance for truncation for the Inverse Generalized Rayleigh Distribution, Right-Side Truncation for the Inverse Generalized Rayleigh Distribution to called Right Truncated Inverse Generalized Rayleigh distribution (RTIGRD) on [0, 1] by using 𝑓𝑅𝑇𝐼𝐺𝑅𝐷(𝑡) = 𝑔(𝑡) 𝐺(1) [30], When t=1 in equation ( 6 ) 𝐺(1) = 1 − [1 − 𝑒− 1 𝜆] 𝜃 𝑓𝑅𝑇𝐼𝐺𝑅𝐷(𝑡) = 𝑔(𝑡) 𝐺(1) The p.d.f of RTIGRD is 𝑓𝑅𝑇𝐼𝐺𝑅𝐷(𝑡) = [1−𝑒 − 1 𝜆𝑡2] 𝜃−1 2𝜃 𝜆𝑡3 𝑒 − 1 𝜆𝑡2 1−[1−𝑒 − 1 𝜆] 𝜃 , 0 ≤ t ≤ 1 The c.d.f of RTIGRD is 𝐹𝑅𝑇𝐼𝐺𝑅𝐷(𝑡) = ∫ [1−𝑒 − 1 𝜆𝑡2] 𝜃−1 2𝜃 𝜆𝑡3 𝑒 − 1 𝜆𝑡2 1−[1−𝑒 − 1 𝜆] 𝜃 𝑡 0 𝑑𝑡 417 Therefore, 𝐹𝑅𝑇𝐼𝐺𝑅𝐷(𝑡) = 1−[1−𝑒 − 1 𝜆𝑡2] 𝜃 1−[1−𝑒 − 1 𝜆] 𝜃 The Survival Function of RTIGRD is 𝑆𝑅𝑇𝐼𝐺𝑅𝐷(𝑡) = 1 − 𝐹𝑅𝑇𝐼𝐺𝑅𝐷(𝑡) = 1 − 1−[1−𝑒 − 1 𝜆𝑡2] 𝜃 1−[1−𝑒 − 1 𝜆] 𝜃 = 1−[1−𝑒 − 1 𝜆] 𝜃 −1+[1−𝑒 − 1 𝜆𝑡2] 𝜃 1−[1−𝑒 − 1 𝜆] 𝜃 𝑆𝑅𝑇𝐼𝐺𝑅𝐷(𝑡) = [1−𝑒 − 1 𝜆𝑡2] 𝜃 −[1−𝑒 − 1 𝜆] 𝜃 1−[1−𝑒 − 1 𝜆] 𝜃 The Hazard Function of RTIGRD is 𝐻𝑅𝑇𝐼𝐺𝑅𝐷(𝑡) = 𝑓𝑅𝑇𝐼𝐺𝑅𝐷(𝑡) 𝑆𝑅𝑇𝐼𝐺𝑅𝐷(𝑡) 𝐻𝑅𝑇𝐼𝐺𝑅𝐷(𝑡) = [1−𝑒 − 1 𝜆𝑡2] 𝜃−1 2𝜃 𝜆𝑡3 𝑒 − 1 𝜆𝑡2 1−[1−𝑒 − 1 𝜆] 𝜃 [1−𝑒 − 1 𝜆𝑡2] 𝜃 −[1−𝑒 − 1 𝜆] 𝜃 1−[1−𝑒 − 1 𝜆] 𝜃 𝐻𝑅𝑇𝐼𝐺𝑅𝐷(𝑡) = [1−𝑒 − 1 𝜆𝑡2] 𝜃−1 2𝜃 𝜆𝑡3 𝑒 − 1 𝜆𝑡2 [1−𝑒 − 1 𝜆𝑡2] 𝜃 −[1−𝑒 − 1 𝜆] 𝜃 Where, t: is a value of random variable and 0 < 𝑡 < 1 . 𝜃: Shape parameter and 𝜃 > 0. 𝜆: Scale parameter and 𝜆 > 0. Figures (1),(2),(3) and (4) plot the p.d.f , c.d.f , SF and HF for the RTIGRD for some cases of 𝜃 and 𝜆 418 Figure 1. probability density function Figure 2. cumulative distribution function Figure 3. Survival function Figure 4. Hazard function IHJPAS. 36 (4) 2023 419 3.Some properties of Right Truncated Inverse Generalized Rayleigh Distribution In this section, some properties are given for RTGRD. However, some properties are complicated to solve. For this reason, use numerical analysis to find it. We made some simplifications for the p.d.f. by using the Binomial theorem and Tyler series (𝑎 ∓ 𝑥)𝑛 =∑ (𝑛 𝑗 ) (∓𝑥)𝑗𝑎𝑛−𝑗 𝑛 𝑗=0 [1 − 𝑒 − 1 𝜆𝑡2] 𝜃−1 =∑ (𝜃−1 𝑗 ) (−𝑒 − 1 𝜆𝑡2)𝑗 𝜃−1 𝑗=0 Thus, 𝑓𝑅𝑇𝐼𝐺𝑅𝐷(𝑡) = ∑ (𝜃−1𝑗 )(−1)𝑗 𝜃−1 𝑗=0 𝑒 − 𝑗 𝜆𝑡2 2θ 𝜆𝑡3 𝑒 − 1 𝜆𝑡2 1−[1−𝑒 − 1 𝜆] 𝜃 𝑓𝑅𝑇𝐼𝐺𝑅𝐷(𝑡) = ∑ (𝜃−1𝑗 )(−1)𝑗 𝜃−1 𝑗=0 2θ 𝜆𝑡3 𝑒 − 𝑗+1 𝜆𝑡2 1−[1−𝑒 − 1 𝜆] 𝜃 let 𝑒 − 𝑗+1 𝜆𝑡2 =∑ (−1)𝑘 𝑘! ∞ 𝑘=0 (𝑗+1)𝑘 (𝜆𝑡2)𝑘 𝑓𝑅𝑇𝐼𝐺𝑅𝐷(𝑡) = ∑ (𝜃−1𝑗 )(−1)𝑗 𝜃−1 𝑗=0 2θ 𝜆𝑡3 ∑ (−1)𝑘 𝑘! ∞ 𝑘=0 (𝑗+1)𝑘 (𝜆𝑡2)𝑘 1−[1−𝑒 − 1 𝜆] 𝜃 = ∑ ∑ (−1)𝑘+𝑗 𝑘! (𝜃−1𝑗 ) 𝜃−1 𝑗=0 ∞ 𝑘=0 (𝑗+1)𝑘 (𝜆)𝑘 𝑡2𝑘 2θ 𝜆𝑡3 1−[1−𝑒 − 1 𝜆] 𝜃 = ∑ ∑ (−1)𝑘+𝑗 𝑘! (𝜃−1𝑗 ) 𝜃−1 𝑗=0 ∞ 𝑘=0 2θ(𝑗+1)𝑘 (𝜆)𝑘+1 𝑡2𝑘+3 1−[1−𝑒 − 1 𝜆] 𝜃 𝑓𝑅𝑇𝐼𝐺𝑅𝐷(𝑡) = ∑ ∑ (−1)𝑘+𝑗 𝑘! (𝜃−1𝑗 ) 𝜃−1 𝑗=0 ∞ 𝑘=0 2θ(𝑗+1)𝑘 (𝜆)𝑘+1 𝑡−2𝑘−3 1−[1−𝑒 − 1 𝜆] 𝜃 , 0 ≤ t ≤ 1 IHJPAS. 36 (4) 2023 420 3.1 rth moment: The rth moment can be derived as follow: 𝐸(𝑡𝑟) = ∫ 𝑡𝑟 1 0 𝑓𝑅𝑇𝐼𝐺𝑅𝐷(𝑡) 𝑑𝑡 = ∫ 𝑡𝑟 ∑ ∑ (−1)𝑘+𝑗 𝑘! (𝜃−1𝑗 ) 𝜃−1 𝑗=0 ∞ 𝑘=0 2θ(𝑗+1)𝑘 (𝜆)𝑘+1 𝑡−2𝑘−3 1−[1−𝑒 − 1 𝜆] 𝜃 1 0 𝑑𝑡 = ∑ ∑ (−1)𝑘+𝑗 𝑘! (𝜃−1𝑗 ) 𝜃−1 𝑗=0 ∞ 𝑘=0 2θ(𝑗+1)𝑘 (𝜆)𝑘+1 1−[1−𝑒 − 1 𝜆] 𝜃 ∫ 𝑡𝑟−2𝑘−3 1 0 𝑑𝑡 = ∑ ∑ (−1)𝑘+𝑗 𝑘! (𝜃−1𝑗 ) 𝜃−1 𝑗=0 ∞ 𝑘=0 2θ(𝑗+1)𝑘 (𝜆)𝑘+1 1−[1−𝑒 − 1 𝜆] 𝜃 𝑡𝑟−2𝑘−2 𝑟−2𝑘−2 |0 1 𝐸(𝑡𝑟) = ∑ ∑ (−1)𝑘+𝑗 𝑘! (𝜃−1𝑗 ) 𝜃−1 𝑗=0 ∞ 𝑘=0 2θ(𝑗+1)𝑘 (𝜆)𝑘+1 (𝑟−2𝑘−2) 1−[1−𝑒 − 1 𝜆] 𝜃 When r = 1, the mean of RTIGRD equal to 𝜇 = 𝐸(𝑡) = ∑ ∑ (−1)𝑘+𝑗 𝑘! (𝜃−1𝑗 ) 𝜃−1 𝑗=0 ∞ 𝑘=0 2θ(𝑗+1)𝑘 (𝜆)𝑘+1 (1−2𝑘−2) 1−[1−𝑒 − 1 𝜆] 𝜃 𝜇 = 𝐸(𝑡) = ∑ ∑ (−1)𝑘+𝑗 𝑘! (𝜃−1𝑗 ) 𝜃−1 𝑗=0 ∞ 𝑘=0 2θ(𝑗+1)𝑘 (𝜆)𝑘+1 (−2𝑘−1) 1−[1−𝑒 − 1 𝜆] 𝜃 When r=2, we will get 𝐸(𝑡2) E(𝑡2) = ∑ ∑ (−1)𝑘+𝑗 𝑘! (𝜃−1𝑗 ) 𝜃−1 𝑗=0 ∞ 𝑘=0 2θ(𝑗+1)𝑘 (𝜆)𝑘+1 (−2𝑘) 1−[1−𝑒 − 1 𝜆] 𝜃 IHJPAS. 36 (4) 2023 421 E(𝑡2) = ∑ ∑ (−1)𝑘+𝑗 𝑘! (𝜃−1𝑗 ) 𝜃−1 𝑗=0 ∞ 𝑘=0 θ(𝑗+1)𝑘 (𝜆)𝑘+1 (−𝑘) 1−[1−𝑒 − 1 𝜆] 𝜃 when r=3 E(𝑡3) = ∑ ∑ (−1)𝑘+𝑗 𝑘! (𝜃−1𝑗 ) 𝜃−1 𝑗=0 ∞ 𝑘=0 θ(𝑗+1)𝑘 (𝜆)𝑘+1 (1−2𝑘) 1−[1−𝑒 − 1 𝜆] 𝜃 When r=4 E(𝑡4) = ∑ ∑ (−1)𝑘+𝑗 𝑘! (𝜃−1𝑗 ) 𝜃−1 𝑗=0 ∞ 𝑘=0 θ(𝑗+1)𝑘 (𝜆)𝑘+1 (2−2𝑘) 1−[1−𝑒 − 1 𝜆] 𝜃 3.2 Variance The Variance (𝑉𝑎𝑟) of RTIGRD can be found as follows: 𝜎2 = 𝑉𝑎𝑟(𝑡) = 𝐸(𝑡2) − [𝐸(𝑡)]2 𝑉𝑎𝑟(𝑡) = ∑ ∑ (−1)𝑘+𝑗 𝑘! (𝜃−1𝑗 ) 𝜃−1 𝑗=0 ∞ 𝑘=0 θ(𝑗+1)𝑘 (𝜆)𝑘+1 (−𝑘) 1−[1−𝑒 − 1 𝜆] 𝜃 − [ ∑ ∑ (−1)𝑘+𝑗 𝑘! (𝜃−1𝑗 ) 𝜃−1 𝑗=0 ∞ 𝑘=0 2θ(𝑗+1)𝑘 (𝜆)𝑘+1 (−2𝑘−1) 1−[1−𝑒 − 1 𝜆] 𝜃 ] 2 = [1−[1−𝑒 − 1 𝜆] 𝜃 ]∑ ∑ (−1)𝑘+𝑗 𝑘! (𝜃−1𝑗 ) 𝜃−1 𝑗=0 ∞ 𝑘=0 θ(𝑗+1)𝑘 (𝜆)𝑘+1 (−𝑘) −[∑ ∑ (−1)𝑘+𝑗 𝑘! (𝜃−1𝑗 ) 𝜃−1 𝑗=0 ∞ 𝑘=0 2θ(𝑗+1)𝑘 (𝜆)𝑘+1 (−2𝑘−1) ] 2 [1−[1−𝑒 − 1 𝜆] 𝜃 ] 2 IHJPAS. 36 (4) 2023 422 = ∑ ∑ (−1)𝑘+𝑗 𝑘! (𝜃−1𝑗 ) 𝜃−1 𝑗=0 ∞ 𝑘=0 2θ(𝑗+1)𝑘 (𝜆)𝑘+1 [ 1 (−2𝑘) [1−[1−𝑒 − 1 𝜆] 𝜃 ] −∑ ∑ (−1)𝑘+𝑗 𝑘! (𝜃−1𝑗 ) 𝜃−1 𝑗=0 ∞ 𝑘=0 2θ(𝑗+1)𝑘 (𝜆)𝑘+1 (−2𝑘−1) ] [1−[1−𝑒 − 1 𝜆] 𝜃 ] 2 3.3 Moment Generating Function The Moment Generating Function of RTIGRD can be derived as follow: ℳ𝑡(ŧ) = 𝐸(𝑒ŧ𝑡) = ∫ 𝑒ŧ𝑡 1 0 𝑓𝑅𝑇𝐼𝐺𝑅𝐷(𝑡) 𝑑𝑡 ℳ𝑡(ŧ) = ∫ 𝑒ŧ𝑡 ∑ ∑ (−1)𝑘+𝑗 𝑘! (𝜃−1𝑗 ) 𝜃−1 𝑗=0 ∞ 𝑘=0 2θ(𝑗+1)𝑘 (𝜆)𝑘+1 𝑡−2𝑘−3 1−[1−𝑒 − 1 𝜆] 𝜃 1 0 𝑑𝑡 = ∑ ∑ (−1)𝑘+𝑗 𝑘! (𝜃−1𝑗 ) 𝜃−1 𝑗=0 ∞ 𝑘=0 2θ(𝑗+1)𝑘 (𝜆)𝑘+1 1−[1−𝑒 − 1 𝜆] 𝜃 ∫ 𝑒ŧ𝑡 𝑡−2𝑘−3 1 0 𝑑𝑡 Use, 𝑒ŧ𝑡 = ∑ (ŧ𝑡)𝑛 𝑛! ∞ 𝑛=0 = ∑ ∑ (−1)𝑘+𝑗 𝑘! (𝜃−1𝑗 ) 𝜃−1 𝑗=0 ∞ 𝑘=0 2θ(𝑗+1)𝑘 (𝜆)𝑘+1 1−[1−𝑒 − 1 𝜆] 𝜃 ∫ ∑ (ŧ𝑡)𝑛 𝑛! ∞ 𝑛=0 𝑡−2𝑘−3 𝑑𝑡 1 0 = ∑ ∑ ∑ (−1)𝑘+𝑗 (ŧ)𝑛 𝑘! 𝑛! (𝜃−1𝑗 ) 2θ(𝑗+1)𝑘 (𝜆)𝑘+1 𝜃−1 𝑗=0 ∞ 𝑘=0 ∞ 𝑛=0 1−[1−𝑒 − 1 𝜆] 𝜃 ∫ 𝑡𝑛−2𝑘−3 1 0 𝑑𝑡 = ∑ ∑ ∑ (−1)𝑘+𝑗 (ŧ)𝑛 𝑘! 𝑛! (𝜃−1𝑗 ) 2θ(𝑗+1)𝑘 (𝜆)𝑘+1 𝜃−1 𝑗=0 ∞ 𝑘=0 ∞ 𝑛=0 1−[1−𝑒 − 1 𝜆] 𝜃 𝑡𝑛−2𝑘−2 𝑛−2𝑘−2 |0 1 ℳ𝑡(ŧ) = ∑ ∑ ∑ (−1)𝑘+𝑗 (ŧ)𝑛 𝑘! 𝑛! (𝜃−1𝑗 ) 2θ(𝑗+1)𝑘 (𝜆)𝑘+1 (𝑛−2𝑘−2) 𝜃−1 𝑗=0 ∞ 𝑘=0 ∞ 𝑛=0 1−[1−𝑒 − 1 𝜆] 𝜃 IHJPAS. 36 (4) 2023 423 3.4 Kurtosis The kurtosis of RTIGRD can be found as follows: kurtosis of RTIGRD can be found as follows: 𝑘𝑟 = 𝜇3 (𝜇2)2 − 3 = 𝐸(𝑡4)−4𝜇𝐸(𝑡3)+6𝜇2𝐸(𝑡2)−3𝜇4 (𝜎2)2 − 3 kr = { ∑ ∑ (−1)𝑘+𝑗 𝑘! (𝜃−1𝑗 ) 𝜃−1 𝑗=0 ∞ 𝑘=0 θ(𝑗+1)𝑘 (𝜆)𝑘+1 (2−2𝑘) 1−[1−𝑒 − 1 𝜆] 𝜃 } −4 { ∑ ∑ (−1)𝑘+𝑗 𝑘! (𝜃−1𝑗 ) 𝜃−1 𝑗=0 ∞ 𝑘=0 2θ(𝑗+1)𝑘 (𝜆)𝑘+1 (−2𝑘−1) 1−[1−𝑒 − 1 𝜆] 𝜃 } { ∑ ∑ (−1)𝑘+𝑗 𝑘! ( 𝜃−1 𝑗 ) 𝜃−1 𝑗=0 ∞ 𝑘=0 θ(𝑗+1)𝑘 (𝜆)𝑘+1 (1−2𝑘) 1−[1−𝑒 − 1 𝜆] 𝜃 } +6 { ∑ ∑ (−1)𝑘+𝑗 𝑘! ( 𝜃−1 𝑗 ) 𝜃−1 𝑗=0 ∞ 𝑘=0 2θ(𝑗+1)𝑘 (𝜆)𝑘+1 (−2𝑘−1) 1−[1−𝑒 − 1 𝜆] 𝜃 } 2 { ∑ ∑ (−1)𝑘+𝑗 𝑘! ( 𝜃−1 𝑗 ) 𝜃−1 𝑗=0 ∞ 𝑘=0 θ(𝑗+1)𝑘 (𝜆)𝑘+1 (−𝑘) 1−[1−𝑒 − 1 𝜆] 𝜃 } −3 { ∑ ∑ (−1)𝑘+𝑗 𝑘! ( 𝜃−1 𝑗 ) 𝜃−1 𝑗=0 ∞ 𝑘=0 2θ(𝑗+1)𝑘 (𝜆)𝑘+1 (−2𝑘−1) 1−[1−𝑒 − 1 𝜆] 𝜃 } 4 ( ∑ ∑ (−1)𝑘+𝑗 𝑘! ( 𝜃−1 𝑗 ) 𝜃−1 𝑗=0 ∞ 𝑘=0 2θ(𝑗+1)𝑘 (𝜆)𝑘+1 [ 1 (−2𝑘) [1−[1−𝑒 − 1 𝜆] 𝜃 ] −∑ ∑ (−1)𝑘+𝑗 𝑘! ( 𝜃−1 𝑗 ) 𝜃−1 𝑗=0 ∞ 𝑘=0 2θ(𝑗+1)𝑘 (𝜆)𝑘+1 (−2𝑘−1) ] [1−[1−𝑒 − 1 𝜆] 𝜃 ] 2 ) 2 − 3 3.5 Skewness The Skewness of RTIGRD can be found as follows: 𝑠𝑘 = 𝜇3 (𝜇2) 3 2 = 𝐸(𝑡3)−3𝜇𝐸(𝑡2)+2𝜇3 (𝜎2) 3 2 IHJPAS. 36 (4) 2023 424 sk = ∑ ∑ (−1)𝑘+𝑗 𝑘! (𝜃−1𝑗 ) 𝜃−1 𝑗=0 ∞ 𝑘=0 θ(𝑗+1)𝑘 (𝜆)𝑘+1 (1−2𝑘) 1−[1−𝑒 − 1 𝜆] 𝜃 −3 { ∑ ∑ (−1)𝑘+𝑗 𝑘! (𝜃−1𝑗 ) 𝜃−1 𝑗=0 ∞ 𝑘=0 2θ(𝑗+1)𝑘 (𝜆)𝑘+1 (−2𝑘−1) 1−[1−𝑒 − 1 𝜆] 𝜃 } { ∑ ∑ (−1)𝑘+𝑗 𝑘! ( 𝜃−1 𝑗 ) 𝜃−1 𝑗=0 ∞ 𝑘=0 θ(𝑗+1)𝑘 (𝜆)𝑘+1 (−𝑘) 1−[1−𝑒 − 1 𝜆] 𝜃 } +2 { ∑ ∑ (−1)𝑘+𝑗 𝑘! ( 𝜃−1 𝑗 ) 𝜃−1 𝑗=0 ∞ 𝑘=0 2θ(𝑗+1)𝑘 (𝜆)𝑘+1 (−2𝑘−1) 1−[1−𝑒 − 1 𝜆] 𝜃 } 3 { ∑ ∑ (−1)𝑘+𝑗 𝑘! (𝜃−1𝑗 ) 𝜃−1 𝑗=0 ∞ 𝑘=0 2θ(𝑗+1)𝑘 (𝜆)𝑘+1 [ 1 (−2𝑘) [1−[1−𝑒 − 1 𝜆] 𝜃 ] −∑ ∑ (−1)𝑘+𝑗 𝑘! (𝜃−1𝑗 ) 𝜃−1 𝑗=0 ∞ 𝑘=0 2θ(𝑗+1)𝑘 (𝜆)𝑘+1 (−2𝑘−1) ] [1−[1−𝑒 − 1 𝜆] 𝜃 ] 2 } 3 2 3.6 Median The Median of RTIGRD can be found as follows: 𝐹𝑅𝑇𝐼𝐺𝑅𝐷(𝑡) = 1 2 1−[1−𝑒 − 1 𝜆𝑡2] 𝜃 1−[1−𝑒 − 1 𝜆] 𝜃 = 1 2 1 − [1 − 𝑒 − 1 𝜆𝑡2] 𝜃 = 1−[1−𝑒 − 1 𝜆] 𝜃 2 2 − 2 [1 − 𝑒 − 1 𝜆𝑡2] 𝜃 = 1 − [1 − 𝑒− 1 𝜆] 𝜃 1 − 2 [1 − 𝑒 − 1 𝜆𝑡2] 𝜃 = −[1 − 𝑒− 1 𝜆] 𝜃 [1 − 𝑒 − 1 𝜆𝑡2] 𝜃 = 1+[1−𝑒 − 1 𝜆] 𝜃 2 1 − 𝑒 − 1 𝜆𝑡2 = [ 1+[1−𝑒 − 1 𝜆] 𝜃 2 ] 1 𝜃 𝑒 − 1 𝜆𝑡2 = 1 − [ 1+[1−𝑒 − 1 𝜆] 𝜃 2 ] 1 𝜃 IHJPAS. 36 (4) 2023 425 𝑙𝑛 𝑒 − 1 𝜆𝑡2 = 𝑙𝑛 [ 1 − [ 1+[1−𝑒 − 1 𝜆] 𝜃 2 ] 1 𝜃 ] − 1 𝜆𝑡2 = 𝑙𝑛 [ 1 − [ 1+[1−𝑒 − 1 𝜆] 𝜃 2 ] 1 𝜃 ] 1 𝜆𝑡2 = 𝑙𝑛 [ 1 − [ 1+[1−𝑒 − 1 𝜆] 𝜃 2 ] 1 𝜃 ] −1 𝑡2 = 1 𝜆𝑙𝑛 [ 1− [ 1+[1−𝑒 − 1 𝜆] 𝜃 2 ] 1 𝜃 ] −1 𝑡𝑀𝑒𝑑𝑖𝑎𝑛 = √ 1 𝜆𝑙𝑛 [ 1− [ 1+[1−𝑒 − 1 𝜆] 𝜃 2 ] 1 𝜃 ] −1 3.7 Mode The Mode of RTIGRD can be found as follows: 𝑓𝑅𝑇𝐼𝐺𝑅𝐷(𝑡) = [1−𝑒 − 1 𝜆𝑡2] 𝜃−1 2𝜃 𝜆𝑡3 𝑒 − 1 𝜆𝑡2 1−[1−𝑒 − 1 𝜆] 𝜃 , 0 ≤ t≤ 1 𝑓𝑅𝑇𝐼𝐺𝑅𝐷(𝑡) = ∑ ( 𝜃−1 𝑗 )(−1)𝑗 𝑒 − 𝑗 𝜆𝑡2 2𝜃 𝜆𝑡3 𝑒 − 1 𝜆𝑡2𝜃−1 𝑗=0 1−[1−𝑒 − 1 𝜆] 𝜃 𝑓𝑅𝑇𝐼𝐺𝑅𝐷(𝑡) = ∑ ( 𝜃−1 𝑗 )(−1)𝑗 2𝜃 𝜆𝑡3 𝑒 − 𝑗+1 𝜆𝑡2𝜃−1 𝑗=0 1−[1−𝑒 − 1 𝜆] 𝜃 𝑓𝑅𝑇𝐼𝐺𝑅𝐷(𝑡) = 𝑀𝑗 𝑡3 𝑒 − 𝑗+1 𝜆𝑡2 IHJPAS. 36 (4) 2023 426 , where 𝑀𝑗 = ∑ ( 𝜃−1 𝑗 )(−1)𝑗 2𝜃 𝜆 𝜃−1 𝑗=0 1−[1−𝑒 − 1 𝜆] 𝜃 ∴ 𝑑𝑓𝑅𝑇𝐼𝐺𝑅𝐷(𝑡) 𝑑𝑡 = 2𝑀𝑗𝑡 3𝑗+1 𝜆𝑡3 𝑒 − 𝑗+1 𝜆𝑡2−3𝑀𝑗𝑒 − 𝑗+1 𝜆𝑡2 𝑡2 𝑡6 = 0 2∑ 𝑗+1 𝜆 − 3 𝜃−1 𝑗=0 𝑡2 = 0 2∑ 𝑗+1 𝜆 𝜃−1 𝑗=0 = 3 𝑡2 2∑ 𝑗+1 3𝜆 𝜃−1 𝑗=0 = 𝑡2 𝑡𝑀𝑜𝑑𝑒 = √2∑ 𝑗+1 3𝜆 𝜃−1 𝑗=0 4. 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