normal density with |z|: MGF and CGF

This is sample question to which the study guide does not provide an answer. I'd appreciate if you could check and make comments on my mistakes/narratives.

Question.

Let Z be a random variable with density

, for .

(a) show that f_Z is a valid density

(b) find the moment generating function of Z and give an open interval around the origin in which the moment generating function is well-defined.

(c) by considering the cumulant generating function or otherwise, evaluate E(Z) and Var (Z).

Answer.

(a) since |z|=-z if z<0, and |z|=z when , I split the integration into two parts:

after some eliminations of negative powers of exp.

(b) I use the same 'split interval' when I need to calculate Mx(t)

bounds on t: for the function M to be integrable on the whole real line, the following must hold: z(t-1)<0 and z(t+1)>0; this leads to -1<t<1.

(c) cumulant generating function

E(Z) can then be found as the first derivative of Kx(t) around zero, and Var (Z) can be found as the second derivative of Kx(t) around zero:

. If t=0, E(Z)=0

. If t=0, this evaluates to 2.