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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"):}`
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Solution
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`
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