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Question
Find expected value and variance of X, where X is number obtained on uppermost face when a fair die is thrown.
Solution
If a die is tossed, then the sample space for the random variable X is
S = {1, 2, 3, 4, 5, 6}
∴ P(X) = `1/6` ; X = 1, 2, 3, 4, 5, 6.
∴ E(X) = `Sigma_(X ∈ S) X . P(X)`
= `1(1/6)+2(1/6)+3(1/6)+4(1/6)+5(1/6)+6(1/6)`
= `1/6(1+2+3+4+5+6)`
=`21/ 6 = 7/ 2 = 3.5`
V (X) = E(X2) -[E(X)]2
`Sigma_(X ∈ S)X^2·P(X)-(7/ 2)^2`
= `[(1)^2(1/6)+(2)^2(1/6)+(3)^2(1/6)+(4)^2(1/6)+(5)^2(1/6)+(6)^2(1/6)] - 49/4`
= `1/6 (1+4+9+16+25+36)-49/4`
= `91/6 - 49/4=(182-147)/12=35/12=2.9167`
Hence, E (X) = 3.5 and V (X) = 2.9167.
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