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Question
Find expected value and variance of X, the number on the uppermost face of a fair die.
Solution
Let X denote the number on uppermost face.
∴ Possible values of X are 1, 2, 3, 4, 5, 6.
Each outcome is equiprobable.
∴ P(X = 1) = P(X = 2) = P(X = 3) = P(X = 4) = P(X = 5) = P(X = 6) = `(1)/(6)`
∴ Expected value of X
= E(X)
= \[\sum\limits_{i=1}^{6} x_i.\text{P}(x_i)\]
= `1 xx (1)/(6) + 2 xx (1)/(6) + 3 xx (1)/(6) + 4 xx (1)/() + 5 xx (1)/(6) + 6 xx (1)/(6)`
= `(1)/(6)(1 + 2 + 3 + 4 + 5 + 6)`
= `(21)/(6)`
= `(7)/(2)`
E(X2) = \[\sum\limits_{i=1}^{6} x_i^2.\text{P}(x_i)\]
= `1^2 xx (1)/(6) + 2^2 xx (1)/(6) + 3^2 xx (1)/(6) + 4^2 xx (1)/(6) + 5^2 xx (1)/(6) + 6^2 xx (1)/(6)`
= `(1)/(6)(1^2 + 2^ + 3^2 + 4^2 + 5^2 + 6^2)`
= `((6 xx 7 xx 13))/(6 xx 6)`
= `(91)/(6)`
∴ Variance of X
= Var(X)
= E(X2) – [E(X)]2
= `(91)/(6) - (7/2)^2`
= `(35)/(12)`.
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