Question

Find expected value and variance of X, the number on the uppermost face of a fair die.

Answer 1

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)`.