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Question
Solve the following problem :
The p.d.f. of the r.v. X is given by
f(x) = `{("k"/sqrt(x), "for" 0 < x < 4.),(0, "otherwise".):}`
Determine k, the c.d.f. of X, and hence find P(X ≤ 2) and P(X ≥ 1).
Solution
Given that f(x) represents 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
∴ 2k (2 – 0) = 1
∴ k = `(1)/(4)`
By definition of c.d.f.,
F(x) = P(X ≤ x)
= `int_0^4 "k"/sqrt(x)*dx`
= `"k"[2sqrt(x)]_0^x`
= `(1)/(4)[2sqrt(x)]_0^x`
= `sqrt(x)/(2)`
P(X ≤ 2) = F (2) = `sqrt(2)/(2) = (1)/sqrt(2)`
P(X ≥ 1) = 1 – P(X < 1)
= 1 – F(1)
= `1 - (sqrt1)/(2)`
= `(1)/(2)`.
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