Set ϕ(t)=M_{X}(t)=Ee^{tX} and ψ(t)=M_{Z}(t)=Ee^{tZ}=Ee^{t((X/4)-(3/4))}=Ee^{(t/4)X}e^{-((3t)/4)}=e^{-((3t)/4)}Ee^{(t/4)X}=e^{-((3t)/4)}ϕ((t/4))=e^{-((3t)/4)}e^{3(t/4)+8((t²)/(16))}=e^{((t²)/2)}.

We have ψ′(t)=te^{((t²)/2)} and ψ′′(t)=e^{((t²)/2)}+t²e^{((t²)/2)}, and so EZ=ψ′(0)=0, EZ²=ψ′′(0)=1 and VarZ=EZ²-(EZ)²=1

Best regards, Aurel Spataru