Question

Find expected value and variance of X, where X is number obtained on uppermost face when a fair die is thrown.

Answer 1

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.