Question

For the random variable X with the given probability mass function as below, find the mean and variance.

`f(x) = {{:(2(x - 1), 1 < x ≤ 2),(0, "otherwise"):}`

Answer 1

Mean: `mu = "E"("X")`

= `int_(1/2)^2 x f(x) "d"x`

= `int_1^(1/2) x xx 2(x - 1)  "d"x`

= `2 int_1^2 (x^2 - x)  "d"x`

= `2[x^3/3 - x^2/2]_1^2`

= `2[8/3 - 4/2 - 1/3 + 1/2]`

= `2(7/3 - 3/2)`

= `2 xx 5/6`

= `5/3`

Variance: `"E"("X"^2)`

= `int_1^2 x^2 f(x)  "d"x`

=  `2int_1^2 (x^3 - x^2)  "d"x`

= `2[x^4/4 - x^3/3]_1^2`

= `2[16/4 - 8/3 - 1/4 + 1/3]`

= `2[15/4 - 7/3]`

= `2 xx 17/12`

= `17/6`

Var(X) = `"E"("X"^2) - ["E"("X")]^2`

= `17/6 - (5/3)^2`

= `17/6 - 25/9`

= `(51 - 50)/18`

= `1/18`