Global Journal of Science Frontier Research, F: Mathematics and Decision Science, Volume 21 Issue 4

On the Generalized Power Transformation of Left Truncated Normal Distribution Okoli Christian O. α , Nwosu Dozie F. σ , Osuji George A. ρ & Nsiegbe Nelson A. Ѡ Abstract- The purpose of this study is to establish a unified approach to the transformation problems of certain type of random variable and their associated probability density functions in the generalized setting. The results presented in this research, trivialized the results obtained by several researchers in the literature, in particular for a random variable that follows a left-truncated normal distribution. Keywords: truncated distribution, normal distribution, transformation, moments. I. I ntrodution and P reliminary Let be an element of an appropriate non-empty sample space Ω and : Ω ⟶ R( = (−∞, ∞)) a real-valued function (random variable) defined on Ω . To each element of the event Γ = { ∈ Ω: X( ) = x} ∈ 2 Ω (1.1) is associated with a probability measure ∶ 2 Ω ⟶ [0,1] in the measure space (Ω, 2 Ω , ) and then denotes the probability density function (pdf) associated with the real- valued function (random variable) by ( ) . Where : X(Ω) ⟶ [0,1] . Let be an arbitrary but fixed point of a scalar field ℱ ( . ∈ ℱ ), then we consider a continuous bijective function or transformation ℎ : ( Ω) ⟶ R define by ℎ ( ) = ∀ ∈ (1.2) If ℎ is the function induced by ℎ on , then we denoted the probability density function (pdf) associated with the real-valued function (random variable) ℎ by ℎ ( ) ; ℎ is the probability density function induced by ℎ on such that : X(Ω) = ℎ : X(Ω) = : ℎ �X(Ω)� ⟶ [0,1] (1.3) Remark 1.1 1) If = 0 , then ℎ ( . . ℎ 0 ) reduces to a constant function. Hence at this point the domain of reduces to a singleton set which is not of interest (in terms of data transformations). 1 Global Journal of Science Frontier Research Volume XXI Issue IV Year 2021 13 ( F ) © 2021 Global Journals Version I Author α Ѡ : Department of Mathematics, Chukwuemeka Odumegwu Ojukwu University Uli, Nigeria. e-mail: fedocon2003@gmail.com Author σ : Department of Mathematics & Statistics, Federal Polythenic Nekede, Owerri, Imo State Nigeria. Author ρ : Department of Statistics, Nnamdi Azikiwe University, Awka Nigeria. N otes