Question

Q. 5. Let X be any random variable, with moment generating function M(S) = E[es], and assume M(s) < o for all s E R. The cumulant generating function of X is defined as A(s) = log Ele**] = log M(s), SER Show the following identities: (1) A'(0) = E[X]. (2) A”(0) = Var(X). (3) A"(0) = E[(X - E[X]))). Using the inversion theorem for MGFs, argue the following: (4) If A'(s) = 0 for all s ER, then P(X= 0) = 1. (5) If A"(s) = 0 for all s ER, then P(X = c) = 1 for some constant CER. (6) If A"(s) = 0 for all s ER and A"(0) + 0, then X is normally distributed; that is, X ~ N(H1, 02) for some ER and a > 0.

Answer 1

Answer : Given; let X be any Sendom rariable, with moment genetliling function M (5) = E LesXJ, and assume. M(s) < oo for all gER. The Cumuleent generating function of x is destined ast. ^(5.) = log Elesx) = log M(5) SER On!(0) = E [x] dd (s) I dolsce. Proses Clog Mars)) = E(X) and on Mas) 1s=o = 7(E) and Ma(s) Ional = E(x). ☺ (0) = Vous () tem co some comme les

. Ma (5) Me" (5) - [ Mae' (s) 921 - [Ma (3)J2 s=o . 3 d It 3 - E(XRP) - LE (2)]* = var bre) "Co: E [12-E (2)%] he a ncss Isto = Mze(s) Mix'(1) - [Ma(s).JP) | EMA (S). 92 . Isso smacsoma"(s) + Macs) Mal" (s)] [Macs? mars CCMecs) 2)-2M2'(1) [Mers) man (3) fm* (374 S=0 - Ex E(22) + Ex?-2 E(X)2-2 E(X) [6C*?) - - "E(2)322 = E [de- € (2) 3J

@ N'(s) =0 for all Ser, then plot =o) - T => q n(s) =0 A SER d (s) ds moto 0 7 SER de Ma (s) = 0 VSER, P (2-0)=/ Yox

Answer 2

Answer : Given; let X be any Sendom rariable, with moment genetliling function M (5) = E LesXJ, and assume. M(s) < oo for all gER. The Cumuleent generating function of x is destined ast. ^(5.) = log Elesx) = log M(5) SER On!(0) = E [x] dd (s) I dolsce. Proses Clog Mars)) = E(X) and on Mas) 1s=o = 7(E) and Ma(s) Ional = E(x). ☺ (0) = Vous () tem co some comme les

. Ma (5) Me" (5) - [ Mae' (s) 921 - [Ma (3)J2 s=o . 3 d It 3 - E(XRP) - LE (2)]* = var bre) "Co: E [12-E (2)%] he a ncss Isto = Mze(s) Mix'(1) - [Ma(s).JP) | EMA (S). 92 . Isso smacsoma"(s) + Macs) Mal" (s)] [Macs? mars CCMecs) 2)-2M2'(1) [Mers) man (3) fm* (374 S=0 - Ex E(22) + Ex?-2 E(X)2-2 E(X) [6C*?) - - "E(2)322 = E [de- € (2) 3J

@ N'(s) =0 for all Ser, then plot =o) - T => q n(s) =0 A SER d (s) ds moto 0 7 SER de Ma (s) = 0 VSER, P (2-0)=/ Yox

Answer 3

Answer : Given; let X be any Sendom rariable, with moment genetliling function M (5) = E LesXJ, and assume. M(s) < oo for all gER. The Cumuleent generating function of x is destined ast. ^(5.) = log Elesx) = log M(5) SER On!(0) = E [x] dd (s) I dolsce. Proses Clog Mars)) = E(X) and on Mas) 1s=o = 7(E) and Ma(s) Ional = E(x). ☺ (0) = Vous () tem co some comme les

. Ma (5) Me" (5) - [ Mae' (s) 921 - [Ma (3)J2 s=o . 3 d It 3 - E(XRP) - LE (2)]* = var bre) "Co: E [12-E (2)%] he a ncss Isto = Mze(s) Mix'(1) - [Ma(s).JP) | EMA (S). 92 . Isso smacsoma"(s) + Macs) Mal" (s)] [Macs? mars CCMecs) 2)-2M2'(1) [Mers) man (3) fm* (374 S=0 - Ex E(22) + Ex?-2 E(X)2-2 E(X) [6C*?) - - "E(2)322 = E [de- € (2) 3J

@ N'(s) =0 for all Ser, then plot =o) - T => q n(s) =0 A SER d (s) ds moto 0 7 SER de Ma (s) = 0 VSER, P (2-0)=/ Yox