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

2) If = 1 , then ℎ ( . . ℎ 1 ) reduces to a identity function so that ( ) = ( ) ∀ ∈ X(Ω) . Hence in this research, we require that ≠ 0 , as such we consider the following propositions: Proposition 1.2 If ≤ −1 , then is an inverse -power transform of . Proof. This easily follows from the fact that ℎ ( ) = 1 ∀ ≥ 1 . Proposition 1.3 If ≥ 1 , then is an -power transform of . Proof. This easily follows from the fact that ℎ ( ) = ∀ ≥ 1 . Proposition 1.4 If 0 < < 1 , then there exist a positive constant such that is a ( + 1) ℎ power transform of . Proof. If 0 < < 1 , then it follows that 1 > 1;⟹ 1 = 1 + , for some > 0; ⟹ = 1 1+ , for some > 0, so that ℎ ( ) = 1 1+ ∀ > 0 which is as stated. Proposition 1.5 If −1 < < 0 , then there exist a positive constant such that is an inverse ( + 1) ℎ power transform of . Proof. If −1 < < 0 , then it follows that 0 < − < 1;⟹ 0 < < 1 , where =– . Thus by proposition 1.4 = 1 1+ , for some > 0; ⟹ = −1 1+ , for some > 0 which is as stated. Remark 1.6 Now, observe in particular; 1) In proposition 1.2, if = −1, − 2 , then is an inverse, inverse square, transform of respectively. 2) In proposition 1.3, if = 1, 2 , then is the identity, square, transform of respectively. 3) In proposition 1.4, if = 1, ⟹ = 1 2 , then is a square root transform of . 4) In proposition 1.5, if = 1, ⟹ = − 1 2 , then is an inverse square root transform of . II. The Left Truncated Normal Distribution Definition 2.1 Let be a random variable that follow a normal distribution with ( ≠ 0) and variance 2 ( 2 > 0)( . . ~( , 2 )) then the probability distribution function ( ) [4] is given by ( ; , ) = 1 √2 −1 2 � − � 2 , ∈ (2.1) On the Generalized Power Transformation of Left Truncated Normal Distribution © 2021 Global Journals 1 Global Journal of Science Frontier Research Volume XXI Issue IV Year 2021 14 ( F ) Version I N otes