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.
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).
(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.