(1)
(2)F_Z(z)=P{Z≤z}=P{X+Y≤z}=P{X=-1,Y≤z+1}+P{X=0,Y≤z}+P{X=1,Y≤z-1}
并且X与Y相互独立,所以
F_Z(z)=P{X=-1}P{Y≤z+1}+P{X=0}P{Y≤z}+P{X=1}P{Y≤z-1}=[P{Y≤z+1}+P{Y≤z}+P{Y≤z-1}]/3
①当z+1<0(z-1<-2)即z<-1时,f_Y(y)=0,从而F_Z(z)=0。
②当0≤z+1<1(-2≤z-1<-1)即-1≤z<0时,
P{Y≤z}=P{Y≤z-1}=0(f_Y(y)=0);F_Z(z)=(z+1)/3。
③当-1≤z-1<0(1≤z+1<2)即0≤z<1时,
P{Y≤z-1}=0;F_Z(z)=(z+1)/3。
④当0≤z-1<1(2≤z+1<3)即1≤z<2时,
F_Z(z)=(1+1+z-1)/3=(z+1)/3。
⑤当1≤z-1(3≤z+1)即z≥2时,
P{Y≤z+1}=1;P{Y≤z}=1;
F_Z(z)=(1+1+1)/3=1。
所以
即