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

Theorem 6.1 Let ( ; , . ) be a truncated normal distribution and ( ; ( ), . ( 0 ), 0 ) the generalized 0 -power transformation of ( ; , . ) induced by = 0 , then ( ; ( 0 ), . ( 0 ), 0 ) has a Bell-shape that coincide with ( ; , . ) if thereexists a sequence � � =1 ∞ ⊂ ( 1 , 2 ) ⊂ + and at least one point 0 ∈ ( 1 , 2 ) such that the � � =1 ∞ converges to 0 ∈ ( 1 , 2 ) (i.e. ⟶ 0 ⟶ ∞ ) and 0 is a zero solution to the problem : ( ; ( 0 ), . ( 0 ), 0 ) (6.1) ℎ : = 0 (6.2) provided ( ; , . ) ≤ ( 0 ; , . ) ∀ ∈ + . Proof. Observe that ( ; , . ) is bounded above and continuous, hence by boundedness above it follows there exist a positive constant such that ( ; , . ) ≤ ∀ ∈ + 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 22 ( F ) Version I and by continuity in + , it follows that there exists a constant 0 ∈ + such that = ( 0 ; , . ) , hence we must have 0 = 0 . This justisfy the existence of such 0 . Hence the problem is equivalent to : ( ; ( 0 ), . ( 0 ), 0 ) (6.3) ℎ : = 0 (6.4) Now, suppose for contradiction that there is no such ∈ + (recall that . , . . . ) that satisfies the maximization problem. This implies that for every ∈ + , the maximization problem becomes : ( ; ( 0 ), . ( 0 ), 0 ) ℎ : ≠ 0 If ≠ 0 , it implies that there is an ≠ 0 such that = 0 ± , hence the maximization problem becomes : ( ; ( 0 ), . ( 0 ), 0 ) ℎ : = 0 ± It then follows that Observe that this is a contradiction to the maximality of at 0 since ≠ 0 . And converly, if the maximality condition of holds, it ⟹ { ( 0 ± ; , . ): ∀ ≠ 0} < = { ( 0 ± ; , . ): ∀ ≠ 0} ⟹⟸ . N otes