Let Xy, . ~ , XD are rid mof of x M x ( 6 ) = Erect ) = Seat fulda = eat e = [eater ] = (BEJ [ - 0+1 ] M x ( ( ) = (H-Bt ) - Cumulant generating function Kx (t ) = lng MX ( ( ) = loy CHBEN Kxit ) = - lug (1-Bt)Suppose that X1, . .., Xn are i.i.d with density f (x) = - 1 B ex/P, x >0, B20. a) Determine the cumulant generating function for a single observation of X. Kx(t) = - log(1 - Bt) O Kx(t) = exp(1 - Bt) O Kx (t) = B t - B O Kx (t) = 1 1 - Bt O Kx(t) = exp(1 - Bt)b) Compute the first and second derivative of the cumulant generating function. O K'x (t ) = 1 1 1 - Bt and Kx (t ) = (1 - Bt)2 O B 82 Kx (t ) = : 1 - Bt and Kx (t) = (1 - Bt)2 O B K'x(t) = B 1 - Bt and Kx (t ) = (1 - Bt)2 O K'x(t) = 1 - 3t and (1 - Bt) B Ky(t) = 82 O 1 - 3t K'y (t) = and B K* (t) = (1 - Bt)2 B