Question

Let f(x) = `{{:(1/x^2",", 1 < x < oo","  "be the p.d.f of"),(0",",  "otherwise"):}`
a r.v. X. If C₁ = {x : 1 < x <2} and C₂ = {x : 4 < x < 5}, then P(C₁ ∪ C₂) =______.

Options

• `1/20`

• `7/20`

• `11/20`

• `13/20`

Answer 1

Let f(x) = `{{:(1/x^2",", 1 < x < oo","  "be the p.d.f of"),(0",",  "otherwise"):}`
a r.v. X. If C₁ = {x : 1 < x <2} and C₂ = {x : 4 < x < 5}, then P(C₁ ∪ C₂) = `11/20`.

Explanation:

P(C₁ ∪ C₂) = P(C₁) + P(C₂)

= `int_1^2 "f"(x) "d"x + int_4^5 "f"(x)"d"x`

= `int _1^2 1/x^2 "d"x + int_4^5 1/x^2 "d"x`

= `[(-1)/x]_1^2 + [(-1)/x]_4^5`

= `- 1/2 + 1 - 1/5 + 1/4`

= `11/20`