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प्रश्न
Find expected value and variance of X, where X is number obtained on uppermost face when a fair die is thrown.
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उत्तर
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) = `sum_(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
`sum_(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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