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

If we denote this mean and variance of the truncated normal distribution ( ; , ) by and 2 (i.e. = 1 ( , , 1) and 2 = 2 ( , , 1) ). It is well known that the shape of ( ; , ) varies as the value of 2 varies (consequently as varies since 2 depend on ), hence is also the shape parameter for ( ; , ) . Also recall that ( ; , , ) , the generalized power transformation of ( ; , ) , which is given by ( ; , , ) = ⎩⎪⎨ ⎪⎧ 1 −1 | | √2 Φ � − � −1 2 � 1 − � 2 , ∈ + , ∈ ℱ . . 0 ℎ Is normal distribution in the region > 0 with mean 1 ( , , ) and variance 2 ( , , ) , where 1 ( , , ) = −1 2 2 ∑ 2 r+jn +1 2 r ! =⌊− ⌋ Γ � r + j + 1 2 � 2 +2 √2 Φ � − � 2 ( , , ) = �(−1) 2− � 2 2 − � 2− 2 =0 −1 2 2 ∑ 2 r+n +1 2 r ! =⌊− ⌋ Γ � r + + 1 2 � 2 n +2 √2 Φ � − � If we denote this mean and variance of the generalized -power transform of ( ; , ) by ( ) and 2 ( ) (i.e. ( ) = 1 ( , , ) and 2 ( ) = 2 ( , , )) . It follows that for every fixed ∈ , the shape of ( ; , , ) varies as the value of 2 ( ) varies (consequently as varies since 2 ( ) depend on ), hence is also the shape parameter for ( ; , , ) . Observe that. (1) = 1 ( , , 1) = and 2 (1) = 2 ( , , 1) = 2 . Now, we observe that 2 ( ) (and 2 ) depend on . A common research interest of several authors (see [5-12] ) is to find the value of for which (1) = ( ) for every fixed ≠ 1 ( ∈ ) . This is the so-called normality condition. Furthermore, It is expected that at this point 2 (1) = 2 ( ) for every fixed ≠ 1 ( ∈ ) . Observe that ( ; , , ) and ( ; , ) are strictly monotone and have one turning point, furthermore ( ; , , ) > 0 and ( ; , ) > 0 for every , ∈ + , ∈ ℱ . . Which implies that the values of , at these turning points maximizes ( ; , ), ( ; , , ) respectively. Consequently by classical calculus, it is well known that these values of , at this turning point coincide with the mode of ( ; , ), ( ; , , ) respectively. We shall determine this values of , using the Rolle' s theorem. Now we state the following theorem which is equivalent to the (so-called) normality condition. On the Generalized Power Transformation of Left Truncated Normal Distribution 1 Global Journal of Science Frontier Research Volume XXI Issue IV Year 2021 2 ( F ) © 2021 Global Journals Version I 5. Iwueze Iheanyi S. (2007). Some Implications of Truncating the N (1, σ 2) Distribution to the left at Zero. Journal of Applied Sciences. 7(2) (2007) pp 189-195. R ef