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

� � 1 ! ( ) =0 ( ; , , ) ∞ 0 = � ! =0 � ( ; , , ) ∞ 0 = � ! =0 � i+ 1 −1 | | √2 Φ � − � −1 2 � 1 − � 2 ∞ 0 = � ! =0 � ; , , � = � ! =0 −1 2 2 ∑ 2 r+n +1 2 r ! =⌊− ⌋ Γ � r + + 1 2 � 2 n +2 √2 Φ � − � For the moment generating function of , recall that at = 1, = , it follows that ( ; , , 1) = ( ; , ) . Hence ( ; , , 1) = � ( ; , , 1) ∞ 0 = � ( ; , ) ∞ 0 = ( ; , ) = ( ; , ) = � ! =0 −1 2 2 ∑ 2 r+ +1 2 r ! =⌊− ⌋ Γ � r + + 1 2 � 2 +2 √2 Φ � − � VI. E xistence of the B ell -S hape C urve A ssociated with ( ; , , ) and ( ; , ) Recall that ( ; , ) , the left truncated normal distribution of , which is given by ( ; , ) = 1 √2 Φ � − � −1 2 � − � 2 , ∈ + Is normal distribution in the region > 0 with mean 1 ( , , 1) and variance 2 ( , , 1) , where 1 ( , , 1) = −1 2 2 ∑ 2 r+22 r ! =⌊− ⌋ Γ � r + 2 2 � 2 3 √2 Φ � − � 2 ( , , 1) = �(−1) 2− � 2 2 − � 2− 2 =0 −1 2 2 ∑ 2 r+ +1 2 r ! =⌊− ⌋ Γ � r + + 1 2 � 2 +2 √2 Φ � − � 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 20 ( F ) Version I N otes