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Question

Let X denote the sum of the numbers obtained when two fair dice are rolled. Find the variance and standard deviation of X.

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Solution

Let X denote the sum of the numbers obtained when two fair dice are rolled. So, X may have values 2, 3, 4, 5, 6, 7, 8, 9, 10, 11or 12.(as 1 can't be the sum of two numbers on fair dice)

P(X=2)=P[1,1]=136,P(X=3)=P[(1,2),(2,1)]=236

P(X=4)=P[(1,3),(2,2),(3,1)]=336

P(X=5)=P[(1,4),(2,3),(3,2),(4,1)]=436

P(X=6)=P[(1,5),(2,4),(3,3),(4,2),(5,1)]=536

P(X=7)=P[(1,6),(2,5),(3,4),(4,3),(5,2)(6,1)]=636

P(X=8)=P[(2,6),(3,5),(4,4),(5,3),(6,2)]=536

P(X=9)=P[(3,6),(4,5),(5,4),(6,3)]=436

P(X=10)=P[(4,6),(5,5),(6,4)]=336

P(X=11)=P[(5,6),(6,5)]=236,P(X=12)=P[6,6]=136

X 2 3 4 5 6 7 8 9 10 11 12P(X)136236336436536636536436336236136

Mean X =XP(X)=[2×1+3×2+4×3+5×4+6×5+7×6+8×5+9×4+10×3+11×2+12×1]36=25236=7

Variance X X2P(X)(Mean)2

[22×1+32×2+42×3+52×4+62×5+72×6+82×5+92×4+102×3+112×2+122×1]3672=19743649=1947176436=21036=356
Hence, SD=Variance=356=2.415


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