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

Lifetime data pertain to the lifetimes of units, either industrial or biological, an industrial or a biological unit cannot be in operation forever. Such a unit cannot continue to operate in the same condition forever. Any random variable is said to be truncated if it can be observed over part of its range. Truncation occurs in various situations. For example, right truncation occurs in the study of life testing and reliability of items such as an electronic component, light bulbs, etc. Left truncation arises because, in many situations, failure of a unit is observed only if it fails after a certain period (for more on this, see [14-15] and the references therein). Unfortunately, often time in practice, the random variable which follow a N ( , 2 ) distribution do not take values that are less than or equal to zero ( ≤ 0) . As such, it naturally calls for one to truncate the in (2.1) to take care of the restriction of the random variable in the region X > 0 without alteration to the properties of the . Hence we seek for such truncated normal distribution of and then denote it by . It suffices to find a constant such that ∫ ( ) ∞ 0 = 1 , where is the so-called normalizing constant and then define ( ) = ( ) . Now, we solve for such by evaluating the integral ∫ ( ) ∞ 0 . Observe that If we take = − , then � ( ) ∞ 0 = � 1 √2 −1 2 � − � 2 ∞ 0 = � 1 √2 −1 2 2 ∞ − = Φ � − � It then follows that = 1 Φ� − � . Hence, the left truncated normal distribution of is given by ( ; , ) = 1 √2 Φ � − � −1 2 � − � 2 , ∈ + (2.2) Observe that 0 ≤ ( ; , ) ≤ 1 ∀ ∈ + ( + = (0, ∞)) and by the method of derivation of ( ; , ) , we have that ∫ ( ; , ) ∞ 0 = 1 . Thus ( ; , ) is a proper . III. D istribution associated with T runcated N ormal D istribution under A rbitrary -P ower T ransformation Let be an arbitrary but fixed point of a scalar field ℱ ( . ∈ ℱ ) and ℎ ( ) = ∀ ∈ ℱ as in equation (1.2) . There is no loss of generality if we put = ℎ ( ) and = ; ⟹ = . Hence by standard result in classical calculus [2] , the transformed function induced by ℎ on is given by ( ; , , ) = ( ; , )| � � | (3.1) Where �� �� is the absolute value of the Jacobian (determinant) of the transformation [2]. If = , then On the Generalized Power Transformation of Left Truncated Normal Distribution 1 Global Journal of Science Frontier Research Volume XXI Issue IV Year 2021 15 ( F ) © 2021 Global Journals Version I 14. M.K. Okasha and Iyad M. A. Alqanoo, ( 2014 ) Inference on The Doubly Truncated Gamma Distribution For Lifetime Data, International Journal Of Mathematics And Statistics Invention (IJMSI) E-ISSN: 2321 – 4767, Volume 2 Issue 11. R ef