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Question
X is r.v. with p.d.f. f(x) = `"k"/sqrt(x)`, 0 < x < 4 = 0 otherwise then x E(X) = _______
Options
`(1)/(3)`
`(4)/(3)`
`(2)/(3)`
1
Solution
X is r.v. with p.d.f. f(x) = `"k"/sqrt(x)`, 0 < x < 4 = 0 otherwise then x E(X) = `bbunderline((4)/(3))`
Explanation:
Since E(X) = `int_(-oo)^(oo) xf(x)*dx`
Since f(x) is a p.d.f. of r.v.X
∴ `int_0^4 "k"/sqrt(x)*dx` = 1
∴ `"k" [2sqrt(x)]_0^4` = 1
∴ `2"k"[sqrt(x)]_0^4` = 1
∴ `2"k"[sqrt(4) - sqrt(0)]` = 1
∴ 2k [2 – 0] = 1
∴ 4k = 1
∴ k = `(1)/(4)`
∴ E(X) = `int_0^4x((1/4)/sqrt(x))*dx`
= `(1)/(4) int_0^4 sqrt(x)*dx`
= `(1)/(4)[(x^(3/2))/(3/2)]_0^4`
= `(1)/(4) xx (2)/(3)[x^(3/2)]_0^4`
= `(1)/(6)[(4)^(3/2) - (0)^(3/2)]`
= `(1)/(6)[8 - 0]`
= `(8)/(6)`
∴ E(X) = `(4)/(3)`.
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