Question

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

Answer 1

S={(1,1),(2,1),(3,1),(4,1),(5,1),(6,1)},
(1,2),(2,2),(3,2),(4,2),(5,2),(6,2),
(1,3),(2,3),(3,3),(4,3),(5,3),(6,3),
(1,4),(2,4),(3,4),(4,4),(5,4),(6,4),
(1,5),(2,5),(3,5),(4,5),(5,5),(6,5),
(1,6),(2,6),(3,6),(4,6),(5,6),(6,6)}
n(S)=36
Let A denotes the sum of the numbers=2,B denotes the sum of the numbers=3
C denotes the sum of the numbers=4,D denotes the sum of the numbers=5
E denotes the sum of the numbers=6,F denotes the sum of the numbers=7
G denotes the sum of the numbers=8,H denotes the sum of the numbers=9
I denotes the sum of the numbers=10,J denotes the sum of the numbers=11
K denotes the sum of the numbers=12
A={1,1},n(A)=1,P(A)=$\frac{1}{36}$
B={(1,2),(2,1)},n(B)=2,P(A)=$\frac{2}{36}$
C={(1,3),(2,2),(3,1)},n(C)=3,P(A)=$\frac{3}{36}$
D={(1,4),(2,3),(3,2),(4,1)},n(D)=4,P(A)=$\frac{4}{36}$
E={(1,5),(2,4),(3,3),(4,2),(5,1)},
n(E)=5,P(A)=$\frac{5}{36}$
F={(1,6),(2,5),(3,4),(4,3),(5,2),(6,1)},
n(F)=6,P(A)=$\frac{6}{36}$
G={(2,6),(3,5),(4,4),(5,3),(6,2)},
n(G)=5,P(A)=$\frac{5}{36}$
H={(3,6),(4,5),(5,4),(6,3)},
n(H)=4,P(A)=$\frac{4}{36}$
I={(4,6),(5,5),(6,4)},
n(I)=3,P(A)=$\frac{3}{36}$
J={(5,6),(6,5)},
n(J)=2,P(A)=$\frac{2}{36}$
K={6,6},n(K)=1,P(A)=$\frac{1}{36}$
Mean μ=∑ptxt
=$\frac{1}{36}$(2+6+12+20+30+42+40+36+30+22+12)
=$\frac{252}{36}$=7
Now ∑ptxt2
=$\frac{1}{36}$(4+1848+100+180+294+320+300+242+144)
=$\frac{1}{36}$×1974=$\frac{329}{6}$
Varience ∑ptxt2-(∑ptxt)2=$\frac{329}{6}$-(7)2=54.83=5.83
Standard deviation=$\sqrt{5.83}$=2.4(nearly)