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

= −1 dx; ⟹ | � � | = 1 | | −1 By substituting appropriately into equation (3.1) and simplifying, we have ( ; , , ) = ⎩⎪⎨ ⎪⎧ 1 −1 | | √2 Φ� − � −1 2 � 1 − � 2 , ∈ + , ∈ ℱ . . 0 ℎ (3.2) It now remain to show that ( ; , , ) given in equation (3.2) is a well-defined . It suffices to show that ∫ ( ; , , ) ∞ 0 = 1 . To see this we proceed as follows: � ( ; , , ) ∞ 0 = � 1 −1 | | √2 Φ � − � −1 2 � 1 − � 2 ∞ 0 = � 1 −1 −1 2 � 1 − � 2 ∞ 0 ; = 1 | | √2 Φ � − � Let = 1 ; ⟹ = 1− 1 du , substituting into the integral above gives � 1 −1 −1 2 � − � 2 1− 1 du ∞ 0 = � −1 2 � − � 2 du ∞ 0 Let = − ;⟹ = , substituting into the integral above gives � −1 2 2 dz ∞ − = � 1 √2 Φ � − � −1 2 2 dz ∞ − = � 1 Φ � − � �� 1 √2 � −1 2 2 dz ∞ − � = Φ � − � Φ � − � = 1 This is as required. IV. T he ℎ M oment about the M ean and the O rigin In this section, for all fixed ∈ , we solved for the ℎ moment of the random variable about the mean , which is also called the ℎ central moment is defined as ( , , ) = �( − ) ; , , �( ( ) for short). This implies 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 16 ( F ) Version I N otes