493 Β© 2025 The Author(s). Published by College of Education for Pure Science (Ibn Al-Haitham), University of Baghdad. This is an open-access article distributed under the terms of the Creative Commons Attribution 4.0 International License Right Truncated Shankar Distribution and its Properties Bayda Atiya Kalaf 1* , Feras Sh. M. Batah2 , and Ahmed Issa Abdul-Nabi3 1,3 Department of Mathematics, College of Education for Pure Science (Ibn Al Haitham), University of Baghdad, Baghdad, Iraq. 2 Department of Mathematics, College of Education for Pure Science, University of Anbar, Anbar, Iraq. * Corresponding Author. Received: 16 July 2024 Accepted:24 November 2024 Published:20 January 2025 doi.org/10.30526/38.1.4031 Abstract Since the importance of truncated distributions has increased in many scientific fields in recent years and they provide valuable insights when dealing with censored or truncated data, this paper presents the Right Truncated Shanker Distribution as a new statistical distribution developed for modelling right truncated data. The distribution is defined by specifying its probability density function and cumulative distribution function under the truncation condition, the survival function and the hazard function. In addition, some properties of the right truncated Shanker distribution are derived, such as the moments around the origin, the variance, the coefficients of skewness and kurtosis, the moment generating function, and the mean time to failure. Our statistical properties show that the new distribution has the utility and flexibility to effectively model truncated data scenarios. Keywords: Truncated distributions, Shanker distribution, probability density function, survival function, moment generating function. 1. Introduction Over the last few decades, there has been a growing interest in lifetime modeling within distribution theory, leading to the introduction of new models by statisticians. Several of these models have gained popularity and are widely used in fields such as biology, engineering, and agriculture[1]. Many new statistical distributions have presented that are more flexible in representing data-life. Lindley distribution developed by Lindley(2), and the weighted Lindley distribution introduced by Ghitany and Atieh (3). Nadarajah- Haghighi distribution (4). The Modified-Lomax distribution (5) and Komal distribution (6). The concept of Truncated Distributions (TD) provides a more accurate representation of phenomena, while still maintaining a level of generality. Therefore, truncated distributions are employed when events are restricted to values that are either higher or lower than a certain threshold, or fall within a specific rang (7) https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ https://doi.org/10.30526/38.1.3501 https://orcid.org/my-orcid?orcid=0000-0003-1136-0055 mailto:baydaa.a.k@ihcoedu.uobaghdad.edu.iq https://orcid.org/0000-0003-1254-3080 mailto:sairan.hamza@garmian.edu.krd https://orcid.org/0000-0003-1254-3080 mailto:ahmad.e.an@ihcoedu.uobaghdad.edu.iq IHJPAS. 2025, 38 (1) 494 Truncated a distribution within a specific duration is an ideal solution to the problem of the unavailability of information during that period for some reason Singh et al (8) explained that a TD arises In practical statistics when the ability to document or have knowledge of events is restricted to values that fall either above or below a given threshold or within a specified range.analyzing TD, its, parameters estimation and applications considered and studied by many authors . Najarzadegan and Alamatsaz,(9) derived truncated Weibull-G distribution. Abid and Abdulrazak (10) Presented [0,1] truncated Frechet-Weibull distribution. Akbarinasab and Arabpour (11) discussed the truncated log-logistic family of distributions. Altawil (12) proposed the [0,1] truncated Lomax – Lomax distribution. Gul et al (13) investigated Weibull-Truncated Exponential distribution. Khaleel et al (14) presented [0,1] Truncated Inverse Weibull Rayleigh distribution. Abbas (15) propose the truncated Weibull exponential distribution. Hussein (16) defined a truncated Lindley-generated family of distributions. Qasim (17) presented the Left Truncated Gumbel-Exponential distribution. (18) investigated the truncated inverse generalized Rayleigh distribution. This paper aims to precent a right truncated for Shanker distribution and discuss the statistical properties. This article is organized as follows: Section 2 discusses the Right Truncated Shanker Distribution, Section 3 finds the reliability and the hazard functions, Section 4 derives some of the statistical properties, and Section 5 presents the Mean time to failure, finally, Section 6. provides the conclusion. 2. Right Truncated Shanker Distribution Shanker (19), presented a new lifetime distribution called the "Shanker distribution", by mixture ''gamma(2,Ξ²) distribution ''with ''exponential distribution'' the pdf and the cdf of shanker distribution are respectively: 𝑓𝑆𝐻(π‘₯, πœ—) = πœ—2 πœ—2+1 (πœ— + π‘₯)π‘’βˆ’πœ—π‘₯; π‘₯ > 0, πœ— > 0 (1) 𝐹𝑆𝐻(π‘₯, πœ—) = 1 βˆ’ (πœ—2+1)+πœ—π‘₯ πœ—2+1 π‘’βˆ’πœ—π‘₯ ; π‘₯ > 0, πœ— > 0 (2) While the Survival function and hazard rate are defined as : 𝑆𝑆𝐻(π‘₯, πœ—) = (πœ—2+1)+πœ—π‘₯ πœ—2+1 π‘’βˆ’πœ—π‘₯ (3) β„Žπ‘ β„Ž = πœ—2(πœ—2+1) (πœ—2+1)+πœ—π‘₯ (4) Let the random variable 𝑋 belong to the interval [0 , 1], βˆ’βˆž < 0 ≀ π‘₯ ≀ 1 < ∞ . Then, the conditional on 0 ≀ π‘₯ ≀ 1 has a truncated distribution (20) .the pdf and the cdf for 0 ≀ π‘₯ ≀ 1 where x is followed Shanker distribution are given by 𝑀𝑅𝑇𝑆𝐻(π‘₯, πœ—) = 𝑓𝑆𝐻(π‘₯,πœ—) πΉπ‘ β„Ž(1,πœ—) (5) π‘Šπ‘…π‘‡π‘†π»(π‘₯, πœ—) = 𝐹𝑆𝐻(π‘₯,πœ—) πΉπ‘ β„Ž(1,πœ—) (6) but πΉπ‘ β„Ž(1, πœ—) = (πœ—2+1)βˆ’[(πœ—2+πœ—+1] πœ—2+1 π‘’βˆ’πœ— ; , πœ— > 0 (7) Substituting(1) and (7)in (5) gives the pdf of right truncated Shanker distribution (RTSHD). 𝑀𝑅𝑇𝑆𝐻(π‘₯, πœ—) = πœ—2(πœ—+π‘₯)π‘’βˆ’πœ—π‘₯ πœ—2+1 (πœ—2+1)βˆ’[(πœ—2+πœ—+1] πœ—2+1 π‘’βˆ’πœ— 𝑀𝑅𝑇𝑆𝐻(π‘₯, πœ—) = πœ—2(πœ—+π‘₯)π‘’βˆ’πœ—π‘₯ (πœ—2+1)βˆ’[(πœ—2+πœ—+1]π‘’βˆ’πœ— (8) IHJPAS. 2025, 38 (1) 495 And substitute (2) and (7) in (6) gives the c.d.f of RTSHD as follows: π‘Šπ‘…π‘‡π‘†π»(π‘₯, πœ—) = (πœ—2+1)+[πœ—2+πœ—π‘₯+1]π‘’βˆ’πœ—π‘₯ (πœ—2+1)βˆ’[(πœ—2+πœ—+1]π‘’βˆ’πœ— (9) 3. Reliability Function and Hazard function According to the previous equations the reliability function of (RTSHD ) can be derived as follows (21): 𝑅𝑅𝑇𝑆𝐻(π‘₯, πœ—) = 1 βˆ’π‘Šπ‘…π‘‡π‘†π»(π‘₯, πœ—) 𝑅𝑅𝑇𝑆𝐻(π‘₯, πœ—) = 1 βˆ’ (πœ—2+1)+[πœ—2+πœ—π‘₯+1]π‘’βˆ’πœ—π‘₯ (πœ—2+1)βˆ’[(πœ—2+πœ—+1]π‘’βˆ’πœ— (10) By simplifying (10) we obtain; 𝑅𝑅𝑇𝑆𝐻(π‘₯, πœ—) = βˆ’[(πœ—2+πœ—+1]π‘’βˆ’πœ—βˆ’[πœ—2+πœ—π‘₯+1]π‘’βˆ’πœ—π‘₯ (πœ—2+1)βˆ’[(πœ—2+πœ—+1]π‘’βˆ’πœ— (11) And the hazard rate function is β„Žπ‘…π‘‡π‘†π»(π‘₯, πœ—) = 𝑀𝑅𝑇𝑆𝐻(π‘₯, πœ—) 𝑆𝑅𝑇𝑆𝐻(π‘₯, πœ—) = πœ—2(πœ—+π‘₯)π‘’βˆ’πœ—π‘₯ (πœ—2+1)βˆ’[(πœ—2+πœ—+1]π‘’βˆ’πœ— Γ— (πœ—2+1)βˆ’[(πœ—2+πœ—+1]π‘’βˆ’πœ— βˆ’[(πœ—2+πœ—+1]π‘’βˆ’πœ—βˆ’[πœ—2+πœ—π‘₯+1]π‘’βˆ’πœ—π‘₯ Then β„Žπ‘…π‘‡π‘†π»(π‘₯, πœ—) = πœ—2(πœ—+π‘₯)π‘’βˆ’πœ—π‘₯ βˆ’[(πœ—2+πœ—+1]π‘’βˆ’πœ—βˆ’[πœ—2+πœ—π‘₯+1]π‘’βˆ’πœ—π‘₯ (12) 4. Statistical Properties of Right Truncated Shanker Distribution This section presents the derivation and calculation of certain statistical properties of the (RTSHD) 4.1. Moments about origin Moments about the origin can be derived as bellow (22,23): π‘€π‘˜ β€² (π‘₯) = ∫ π‘₯π‘˜ 1 0 𝑀𝑅𝑇𝑆𝐻(π‘₯, πœ—)𝑑π‘₯ = ∫ π‘₯π‘˜ 1 0 πœ—2(πœ—+π‘₯)π‘’βˆ’πœ—π‘₯ (πœ—2+1)βˆ’[(πœ—2+πœ—+1]π‘’βˆ’πœ— 𝑑π‘₯ (12) = πœ—2 (πœ—2+1)βˆ’[(πœ—2+πœ—+1]π‘’βˆ’πœ— ∫ π‘₯π‘˜ 1 0 (πœ— + π‘₯)π‘’βˆ’πœ—π‘₯𝑑π‘₯ (13) Depending on Taylor's expansion π‘’βˆ’πœ—π‘₯ = βˆ‘ (βˆ’πœ—π‘₯)𝑗 𝑗! ∞ 𝑗=0 Then Equation(13) can be represented by: π‘€π‘˜ β€² (π‘₯) = πœ—2 (πœ—2+1)βˆ’[(πœ—2+πœ—+1]π‘’βˆ’πœ— βˆ‘ (βˆ’πœ—)𝑗 𝑗! ∞ 𝑗=0 ∫ π‘₯π‘˜+𝑗 1 0 (πœ— + π‘₯)𝑑π‘₯ = πœ—2 (πœ—2+1)βˆ’[(πœ—2+πœ—+1]π‘’βˆ’πœ— βˆ‘ (βˆ’πœ—)𝑗 𝑗! ∫ (πœ—π‘₯π‘˜+𝑗 + π‘₯π‘˜+𝑗+1) 1 0 𝑑π‘₯∞ 𝑗=0 then π‘€π‘˜ β€² (π‘₯) = πœ—2 (πœ—2+1)βˆ’[(πœ—2+πœ—+1]π‘’βˆ’πœ— βˆ‘ (βˆ’πœ—)𝑗 𝑗! ∞ 𝑗=0 ( πœ— π‘˜+𝑗+1 + 1 π‘˜+𝑗+2 ) (14) Substitute k by 1,2,3,4 respectively we get: 𝑀1 β€²(π‘₯) = 𝐸(𝑋) = πœ—2 (πœ—2+1)βˆ’[(πœ—2+πœ—+1]π‘’βˆ’πœ— βˆ‘ (βˆ’πœ—)𝑗 𝑗! ∞ 𝑗=0 ( πœ— 𝑗+2 + 1 𝑗+3 ) (15) 𝑀2 β€²(π‘₯) = 𝐸(𝑋2) = πœ—2 (πœ—2+1)βˆ’[(πœ—2+πœ—+1]π‘’βˆ’πœ— βˆ‘ (βˆ’πœ—)𝑗 𝑗! ∞ 𝑗=0 ( πœ— 𝑗+3 + 1 𝑗+4 ) (16) IHJPAS. 2025, 38 (1) 496 𝑀3 β€²(π‘₯) = 𝐸(𝑋3) = πœ—2 (πœ—2+1)βˆ’[(πœ—2+πœ—+1]π‘’βˆ’πœ— βˆ‘ (βˆ’πœ—)𝑗 𝑗! ∞ 𝑗=0 ( πœ— 𝑗+4 + 1 𝑗+5 ) (17) 𝑀4 β€²(π‘₯) = 𝐸(𝑋4) = πœ—2 (πœ—2+1)βˆ’[(πœ—2+πœ—+1]π‘’βˆ’πœ— βˆ‘ (βˆ’πœ—)𝑗 𝑗! ∞ 𝑗=0 ( πœ— 𝑗+4 + 1 𝑗+5 ) (18) 4.2. Variance, Skewness and Kurtosis To discuss more properties of the truncated Shanker distribution the variance, skewness and kurtosis will be derived respectively : π‘£π‘Žπ‘Ÿ(π‘₯) = 𝑀2 β€² (π‘₯) βˆ’ (𝑀1 β€²(π‘₯)) 2 , then from Equation (16) and (15) we get π‘£π‘Žπ‘Ÿπ‘…π‘‡π‘†π»(π‘₯) = πœ—2 (πœ—2+1)βˆ’[(πœ—2+πœ—+1]π‘’βˆ’πœ— βˆ‘ (βˆ’πœ—)𝑗 𝑗! ∞ 𝑗=0 ( πœ— 𝑗+3 + 1 𝑗+4 ) βˆ’ ( πœ—2 (πœ—2+1)βˆ’[(πœ—2+πœ—+1]π‘’βˆ’πœ— βˆ‘ (βˆ’πœ—)𝑗 𝑗! ∞ 𝑗=0 ( πœ— 𝑗+2 + 1 𝑗+3 )) 2 (19) The formula of the skewness is π‘ π‘˜ = πœ‡3 (πœ‡2) 3 2 = 𝐸(π‘₯3)βˆ’3πœ‡πΈ(π‘₯2)+2πœ‡3 (𝜎2) 3 2 So from Equation (17),(15),(16) and (19) we find that: π‘ π‘˜π‘…π‘‡π‘†π» = { πœ—2 (πœ—2+1)βˆ’[(πœ—2+πœ—+1]π‘’βˆ’πœ— βˆ‘ (βˆ’πœ—)𝑗 𝑗! ∞ 𝑗=0 ( πœ— 𝑗+4 + 1 𝑗+5 )βˆ’ 3[( πœ—2 (πœ—2+1)βˆ’[(πœ—2+πœ—+1]π‘’βˆ’πœ— βˆ‘ (βˆ’πœ—)𝑗 𝑗! ∞ 𝑗=0 ( πœ— 𝑗+2 + 1 𝑗+3 ))Γ— ( πœ—2 (πœ—2+1)βˆ’[(πœ—2+πœ—+1]π‘’βˆ’πœ— βˆ‘ (βˆ’πœ—)𝑗 𝑗! ∞ 𝑗=0 ( πœ— 𝑗+3 + 1 𝑗+4 ))]+ 2( πœ—2 (πœ—2+1)βˆ’[(πœ—2+πœ—+1]π‘’βˆ’πœ— βˆ‘ (βˆ’πœ—)𝑗 𝑗! ∞ 𝑗=0 ( πœ— 𝑗+2 + 1 𝑗+3 )) 3 } [ πœ—2 (πœ—2+1)βˆ’[(πœ—2+πœ—+1]π‘’βˆ’πœ— βˆ‘ (βˆ’πœ—)𝑗 𝑗! ∞ 𝑗=0 ( πœ— 𝑗+3 + 1 𝑗+4 )βˆ’( πœ—2 (πœ—2+1)βˆ’[(πœ—2+πœ—+1]π‘’βˆ’πœ— βˆ‘ (βˆ’πœ—)𝑗 𝑗! ∞ 𝑗=0 ( πœ— 𝑗+2 + 1 𝑗+3 )) 2 ] 3 2 ) 20) The kurtosis of Right Truncated Shanker Distribution can be derived by the same way π‘˜π‘Ÿ = πœ‡3 (πœ‡2)2 βˆ’ 3 = 𝐸(π‘₯4)βˆ’4πœ‡πΈ(π‘₯3)+6πœ‡2𝐸(π‘₯2)βˆ’3πœ‡4 (𝜎2)2 βˆ’ 3 So π‘˜π‘Ÿπ‘…π‘‡π‘†π» = { πœ—2 (πœ—2+1)βˆ’[(πœ—2+πœ—+1]π‘’βˆ’πœ— βˆ‘ (βˆ’πœ—)𝑗 𝑗! ∞ 𝑗=0 ( πœ— 𝑗+4 + 1 𝑗+5 )βˆ’4( πœ—2 (πœ—2+1)βˆ’[(πœ—2+πœ—+1]π‘’βˆ’πœ— βˆ‘ (βˆ’πœ—)𝑗 𝑗! ∞ 𝑗=0 ( πœ— 𝑗+2 + 1 𝑗+3 ))Γ— ( πœ—2 (πœ—2+1)βˆ’[(πœ—2+πœ—+1]π‘’βˆ’πœ— βˆ‘ (βˆ’πœ—)𝑗 𝑗! ∞ 𝑗=0 ( πœ— 𝑗+4 + 1 𝑗+5 ))+6( πœ—2 (πœ—2+1)βˆ’[(πœ—2+πœ—+1]π‘’βˆ’πœ— βˆ‘ (βˆ’πœ—)𝑗 𝑗! ∞ 𝑗=0 ( πœ— 𝑗+2 + 1 𝑗+3 )) 2 Γ— ( πœ—2 (πœ—2+1)βˆ’[(πœ—2+πœ—+1]π‘’βˆ’πœ— βˆ‘ (βˆ’πœ—)𝑗 𝑗! ∞ 𝑗=0 ( πœ— 𝑗+3 + 1 𝑗+4 ))βˆ’3( πœ—2 (πœ—2+1)βˆ’[(πœ—2+πœ—+1]π‘’βˆ’πœ— βˆ‘ (βˆ’πœ—)𝑗 𝑗! ∞ 𝑗=0 ( πœ— 𝑗+2 + 1 𝑗+3 )) 4 } [ πœ—2 (πœ—2+1)βˆ’[(πœ—2+πœ—+1]π‘’βˆ’πœ— βˆ‘ (βˆ’πœ—)𝑗 𝑗! ∞ 𝑗=0 ( πœ— 𝑗+3 + 1 𝑗+4 )βˆ’( πœ—2 (πœ—2+1)βˆ’[(πœ—2+πœ—+1]π‘’βˆ’πœ— βˆ‘ (βˆ’πœ—)𝑗 𝑗! ∞ 𝑗=0 ( πœ— 𝑗+2 + 1 𝑗+3 )) 2 ] 2 βˆ’ 3 (21) IHJPAS. 2025, 38 (1) 497 4.3. Moment Generating Function mgf In this sub section The mgf of the Right truncated Shanker Distribution will be derived as follow: β„³x(t) = E(etx) = ∫ etx 1 0 𝑀𝑅𝑇𝑆𝐻(π‘₯, πœ—)dx ; 0 ≀ π‘₯ ≀ 1 (22) So from Equation (5) β„³x(t) = ∫ etx 1 0 πœ—2(πœ—+π‘₯)π‘’βˆ’πœ—π‘₯ (πœ—2+1)βˆ’[(πœ—2+πœ—+1]π‘’βˆ’πœ— 𝑑π‘₯ = πœ—2 (πœ—2+1)βˆ’[(πœ—2+πœ—+1]π‘’βˆ’πœ— ∫ (πœ— + π‘₯)π‘’βˆ’(πœ—βˆ’π‘‘)π‘₯ 1 0 𝑑π‘₯ (23) But π‘’βˆ’(πœ—βˆ’π‘‘)π‘₯ = βˆ‘ [βˆ’(πœ—βˆ’π‘‘)π‘₯]𝑗 𝑗! ∞ 𝑗=0 (24) Substitute Equation (24) in Equation (23) β„³x(t) = πœ—2 (πœ—2+1)βˆ’[(πœ—2+πœ—+1]π‘’βˆ’πœ— βˆ‘ [βˆ’(πœ—βˆ’π‘‘)]𝑗 𝑗! ∞ 𝑗=0 ∫ π‘₯𝑗(πœ— + π‘₯) 1 0 𝑑π‘₯ then β„³x(t) = πœ—2 (πœ—2+1)βˆ’[(πœ—2+πœ—+1]π‘’βˆ’πœ— βˆ‘ [βˆ’(πœ—βˆ’π‘‘)]𝑗 𝑗! ∞ 𝑗=0 ( πœ— 𝑗+1 + 1 𝑗+2 ) (25) 5. Mean time to failure 𝑴𝑻𝑻𝑬 Let 𝑇 denote the lifetime of a component so 𝑀𝑇𝑇𝐸 = ∫ 𝑆(𝑅𝑇𝑆𝐻)(π‘₯) 1 0 𝑑π‘₯ = ∫ βˆ’[(πœ—2+πœ—+1]π‘’βˆ’πœ—βˆ’[πœ—2+πœ—π‘₯+1]π‘’βˆ’πœ—π‘₯ (πœ—2+1)βˆ’[(πœ—2+πœ—+1]π‘’βˆ’πœ— 1 0 𝑑π‘₯ (26) = 1 (πœ—2+1)βˆ’[(πœ—2+πœ—+1]π‘’βˆ’πœ— ∫ (βˆ’[(πœ—2 + πœ— + 1]π‘’βˆ’πœ— βˆ’ [πœ—2 + πœ—π‘₯ + 1]π‘’βˆ’πœ—π‘₯)𝑑π‘₯ 1 0 = 1 (πœ—2+1)βˆ’[(πœ—2+πœ—+1]π‘’βˆ’πœ— [βˆ’[(πœ—2 + πœ— + 1]π‘’βˆ’πœ— βˆ’ ∫ (βˆ’[πœ—2 + πœ—π‘₯ + 1]π‘’βˆ’πœ—π‘₯)𝑑π‘₯ 1 0 ] (27) Recall that π‘’βˆ’πœ—π‘₯ = βˆ‘ (βˆ’πœ—π‘₯)𝑗 𝑗! ∞ 𝑗=0 ,then 𝑀𝑇𝑇𝐸 = 1 (πœ—2+1)βˆ’[(πœ—2+πœ—+1]π‘’βˆ’πœ— [βˆ’[(πœ—2 + πœ— + 1]π‘’βˆ’πœ— βˆ’ βˆ‘ (βˆ’πœ—)𝑗 𝑗! ∞ 𝑗=0 ∫ (βˆ’(π‘₯)𝑗[πœ—2 + πœ—π‘₯ + 1])𝑑π‘₯ 1 0 ] 𝑀𝑇𝑇𝐸 = 1 (πœ—2+1)βˆ’[(πœ—2+πœ—+1]π‘’βˆ’πœ— [βˆ’[(πœ—2 + πœ— + 1]π‘’βˆ’πœ— + βˆ‘ (βˆ’πœ—)𝑗 𝑗! ∞ 𝑗=0 [πœ—2+πœ—π‘₯+1] 𝑗+1 ] (28) 6. Conclusion A new lifetime truncated distribution named, Right truncated Shanker distribution was presented to study the shanker distribution when the random variable x belong to the interval [0,1] by driving the probability density function and cumulative distribution function The statistical properties including, hazard rate function, survival function, moments, variance and coefficients of skewness, and generating function, have been provided in addition to Mean time to failure to know more about the behavior of this function. Acknowledgment The authors would like to thank the referees for providing very helpful comments and IHJPAS. 2025, 38 (1) 498 suggestions that helped in improving the quality of the paper. Conflict of Interest Bayda Atiya, the manager, declared that she was the one at IHJPAS when submitting the manuscript. The editor-in-chief of IHJPAS confirms that (Bayda Atiya) was excluded from any decisions made regarding this paper. Funding There is no financial support. References 1. Shukla KK, Shanker R. Truncated akash distribution: properties and applications. Biometrics & Biostatistics International Journal. 2020;9(5):179–84. http://dx.doi.org/10.15406/bbij.2020.09.00317 2. Lindley DV. Fiducial distributions and Bayes’ Theorem. Journal of the Royal Statistical Society, Series B. 1958;20(1):102–7. https://doi.org/10.1111/j.2517-6161.1958.tb00278.x 3. Ghitany ME, Atieh B, Nadarajah S. Lindley distribution and its applications. Mathematics and Computers in Simulation. 2008;78(4):493–506. https://doi.org/10.1016/j.matcom.2007.06.007 4. Nadarajah S, Haghighi F. An extension of the exponential distribution. Statistics. 2010;45(6):543–58. https://doi.org/10.1080/02331881003678678 5. Alnssyan B. The Modified-Lomax Distribution: Properties, Estimation Methods, and Application. Symmetry. 2023;15(7):1367. https://doi.org/10.3390/sym15071367 6. Shanker R. Komal distribution with properties and application in survival analysis. Biometrics & Biostatistics International Journal. 2023;12(2):40–4. http://dx.doi.org/10.15406/bbij.2023.12.00381 7. Jawitz JW. Moments of truncated continuous univariate distributions. Advances in Water Resources. 2004;27(3):269–81. http://dx.doi.org/10.1016/j.advwatres.203.12.002 8. Singh SK, Singh U, Sharma VK. The truncated Lindley distribution: Inference and application. Journal of Statistics Applications and Probability. 2014;3(2):2019–28. http://dx.doi.org/10.12785/jsap/030212 9. Najarzadegan H, Alamatsaz MH, Saied H. Truncated Weibull-G more flexible and more reliable than beta-G distribution. International Journal of Statistics and Probability. 2017;6:1–17. http://dx.doi.org/10.5539/ijsp.v6n5p1 10. Abid S, Abdulrazak. [0, 1] truncated Frechet-Weibull and Frechet distributions. International Journal of Research in Industrial Engineering. 2018;7:106–35. https://doi.org/10.22105/riej.2018.100865.1020 11. Akbarinasab M, Arabpour AR, Mahdavi A. Truncated log-logistic family of distributions. Journal of Biostatistics and Epidemiology. 2019;5(2):137–47. https://doi.org/10.18502/jbe.v5i2.2345 12. Altawil J. [0,1] truncated lomax–lomax distribution with properties. Journal of Kufa for Mathematics and Computer. 2021;8(1):1–8. https://doi.org/10.31642/JoKMC/2018/08010 13. Gul A, Mohsin M, Adil M, Ali M. A modified truncated distribution for modeling the heavy tail, engineering and environmental sciences data. PLoS ONE. 2021;16(4). https://doi.org/10.1371/journal.pone.0249001 14. Khaleel MA, Abdulwahab MA, Gaftan AM, Abdal-hammed MK. A new [0, 1] truncated inverse Weibull Rayleigh distribution properties with application to COVID-19. International Journal of Nonlinear Analysis and Applications. 2022;13(1):2933–46. 15. Abbas S, Farooq M, Darwish JA, Shahbaz SH, Shahbaz MQ. Truncated Weibull–exponential distribution: methods and applications. Scientific Reports. 2023;13(1). https://doi.org/10.1038/s41598- 023-48288-x 16. Hussein LK, Abdullah Rasheed H, Hasan Hussein I. A Class of Exponential Rayleigh Distribution and New Modified Weighted Exponential Rayleigh Distribution with Statistical Properties. Ibn AL- Haitham Journal for Pure and Applied Sciences. 2023;36(2):390–406. https://doi.org/10.30526/36.2.3044 http://dx.doi.org/10.15406/bbij.2020.09.00317 https://doi.org/10.1111/j.2517-6161.1958.tb00278.x https://doi.org/10.1016/j.matcom.2007.06.007 https://doi.org/10.1080/02331881003678678 https://doi.org/10.3390/sym15071367 http://dx.doi.org/10.15406/bbij.2023.12.00381 http://dx.doi.org/10.1016/j.advwatres.203.12.002 http://dx.doi.org/10.12785/jsap/030212 http://dx.doi.org/10.5539/ijsp.v6n5p1 https://doi.org/10.22105/riej.2018.100865.1020 https://doi.org/10.18502/jbe.v5i2.2345 https://doi.org/10.31642/JoKMC/2018/08010 https://doi.org/10.1371/journal.pone.0249001 https://doi.org/10.1038/s41598-023-48288-x https://doi.org/10.1038/s41598-023-48288-x https://doi.org/10.30526/36.2.3044 IHJPAS. 2025, 38 (1) 499 17. Qasim BA. A new Left Truncated Gumbel-Exponential distribution: Properties and estimation. AIP Conference Proceedings. 2023. https://doi.org/10.1063/5.0118651 18. Kalaf BA, Abdul Ameer JN, Madaki UY. Truncated Inverse Generalized Rayleigh Distribution and Some Properties. Ibn AL-Haitham Journal for Pure and Applied Sciences. 2023;36(4):414–28. https://doi.org/10.30526/36.4.2977 19. Shanker R. Shanker distribution and its applications. International Journal of Statistics and Applications. 2015;5(6):338–48. http://dx.doi.org/10.5923/j.statistics.20150506.08 20. Aryuyuen S, Bodhisuwan W. The truncated power Lomax distribution: Properties and applications. Walailak Journal of Science and Technology. 2019;16(9):655–68. https://doi.org/10.48048/wjst.2019.4542 21. Mohammed MJ, Hussein IH. Some estimation methods for new mixture distribution with simulation and application. IOP Conference Series: Materials Science and Engineering. 2019;571(1). https://doi.org/10.1088/1757-899X/571/1/012014 22. Shanker R, Upadhyay R, Shukla KK. A quasi Suja distribution. Reliability: Theory & Applications. 2022;17(3(69)):162–78. 23. Ibrahim, A., Kalaf, B. Estimation of the survival function based on the log-logistic distribution. International Journal of Nonlinear Analysis and Applications, 2022; 13(1): 127-141. https://doi.org/10.22075/ijnaa.2022.5466 https://doi.org/10.1063/5.0118651 https://doi.org/10.30526/36.4.2977 http://dx.doi.org/10.5923/j.statistics.20150506.08 https://doi.org/10.48048/wjst.2019.4542 https://doi.org/10.1088/1757-899X/571/1/012014 https://doi.org/10.22075/ijnaa.2022.5466