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a.
N=6 So value of Z = k , k=2,3,....,12
X and Y are independent.
pz(2) = P(X+Y=2) = P(X=1,Y=1) = P(X=1)P(Y=1) = (1/6)(1/6) = 1/36 = (2-1)/36
pz(3) = P(X+Y=3) = P(X=1,Y=2) + P(X=2,Y=1) = P(X=1)P(Y=2) + P(X=2)P(Y=1) = (1/6)(1/6) + (1/6)(1/6) = 2/36 = (3-1)/36
pz(4) = P(X+Y=4) = P(X=1,Y=3) + P(X=2,Y=2) + P(X=3,Y=1) = P(X=1)P(Y=3) + P(X=2)P(Y=2) + P(X=3)P(Y=1) = (1/6)(1/6) + (1/6)(1/6) + (1/6)(1/6) = 3/36 = (4-1)/36
pz(5) = P(X+Y=5) = P(X=1,Y=4) + P(X=2,Y=3) + P(X=3,Y=2) + P(X=4,Y=1) = P(X=1)P(Y=4) + P(X=2)P(Y=3) + P(X=3)P(Y=2) + P(X=4)P(Y=1) = (1/6)(1/6) + (1/6)(1/6) + (1/6)(1/6) + (1/6)(1/6) = 4/36 = (5-1)/36
pz(6) = P(X+Y=6) = P(X=1,Y=5) + P(X=2,Y=4) + P(X=3,Y=3) + P(X=4,Y=2) + P(X=5,Y=1) = P(X=1)P(Y=4) + P(X=2)P(Y=4) + P(X=3)P(Y=3) + P(X=4)P(Y=2) + P(X=5)P(Y=1) = (1/6)(1/6) + (1/6)(1/6) + (1/6)(1/6) + (1/6)(1/6) + (1/6)(1/6) = 5/36 = (6-1)/36
pz(7) = P(X+Y=7) = P(X=1,Y=6) + P(X=2,Y=5) + P(X=3,Y=4) + P(X=4,Y=3) + P(X=5,Y=2) + P(X=6,Y=1) = P(X=1)P(Y=6) + P(X=2)P(Y=5) + P(X=3)P(Y=4) + P(X=4)P(Y=3) + P(X=5)P(Y=2) + P(X=6)P(Y=1) = (1/6)(1/6) + (1/6)(1/6) + (1/6)(1/6) + (1/6)(1/6) + (1/6)(1/6) + (1/6)(1/6) = 6/36 = (13-7)/36
pz(8) = P(X+Y=8) = P(X=2,Y=6) + P(X=3,Y=5) + P(X=4,Y=4) + P(X=5,Y=3) + P(X=6,Y=2) = P(X=2)P(Y=6) + P(X=3)P(Y=5) + P(X=4)P(Y=4) + P(X=5)P(Y=3) + P(X=6)P(Y=2) = (1/6)(1/6) + (1/6)(1/6) + (1/6)(1/6) + (1/6)(1/6) + (1/6)(1/6) = 5/36 = (13-8)/36
pz(9) = P(X+Y=9) = P(X=3,Y=6) + P(X=4,Y=5) + P(X=5,Y=4) + P(X=6,Y=3) = P(X=3)P(Y=6) + P(X=4)P(Y=5) + P(X=5)P(Y=4) + P(X=6)P(Y=3) = (1/6)(1/6) + (1/6)(1/6) + (1/6)(1/6) + (1/6)(1/6) = 4/36 = (13-9)/36
pz(10) = P(X+Y=10) = P(X=4,Y=6) + P(X=5,Y=5) + P(X=6,Y=4) = P(X=4)P(Y=6) + P(X=5)P(Y=5) + P(X=6)P(Y=4) = (1/6)(1/6) + (1/6)(1/6) + (1/6)(1/6) = 3/36 = (13-10)/36
pz(11) = P(X+Y=11) = P(X=5,Y=6) + P(X=6,Y=5) = P(X=5)P(Y=6) + P(X=6)P(Y=5) = (1/6)(1/6) + (1/6)(1/6) = 2/36 = (13-11)/36
pz(12) = P(X+Y=12) = P(X=6,Y=6) = P(X=6)P(Y=6) = (1/6)(1/6) = 1/36 = (13-12)/36
.
b.
For general N ; value of Z = k , k = 2,3,....,2N
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