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Question
Given the p.d.f. of a continuous r.v. X ,
f (x) = `x^2/ 3` , for –1 < x < 2 and = 0 otherwise
Determine c.d.f. of X hence find P( X > 0)
Solution
Then F(x) = ` int_(-∞)^x f (x) dx`
=` int_(-∞)^-1 f (x) dx + int_(-1)^x f (x) dx`
= 0 + `int_(-1)^x x^2/3 dx = 1/3int_(-1)^x x^2 dx`
= `1/3[x^3/3]_-1^x`
= `1/3[x^3/3-(-1/3)]`
∴ f(x) = `(x^3+1)/9`
P (X > 0) = 1 - P(X ≤ 0 )
= 1 - [F(0) - F(-1)]
= 1 - `[((0^3+1)/9)-(((-1)^3+1)/9)]`
= 1 - `(1/9-0) = 1 - 1/9=8/9`
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