See this page. Looks like you did it right.
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.