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Question
Suppose error involved in making a certain measurement is continuous r.v. X with p.d.f.
`"f(x)" = {("k"(4 - x^2) "for –2 ≤ x ≤ 2,"),(0 "otherwise".):}`
P(–1 < x < 1)
Solution
Given that f(x) represents a p.d.f. of r.v.X.
∴ `int_(-2)^2 "f(x) dx" = 1`
∴ `int_(-2)^2 "k" (4 − x^2) "dx" = 1`
∴ `"k"[4x − x^3/3]_(-2)^(2) = 1`
∴ `"k"[(8 − 8/3) − (−8 + 8/3)] = 1`
∴ `"k"(16/3 + 16/3) = 1`
∴ `"k"(32/3) = 1`
∴ k = `3/32`
`"F(x)" = int_(-2)^(x) "f(x) dx"`
`= int_(-2)^(x) "k" (4 − x^2) "dx"`
`= 3/32 [4x − x^3/3]_(-2)^(x)`
`= 3/32 [4x − x^3/3 + 8 − 8/3]`
∴ `"F(x)" = 3/32 [4x − x^3/3 + 16/3]`
P(–1 < x < 1)
= F(1) – F(– 1)
= `3/32 (4 – 1/3 + 16/3) – 3/32 (– 4 + 1/3 + 16/3)`
= `3/32 (9 – 5/3)`
= `3/32 (22/3)`
= `11/16`
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