Question

Let X be amount of time for which a book is taken out of library by randomly selected student and suppose X has p.d.f

f (x) = 0.5x, for 0 ≤ x ≤ 2 and = 0 otherwise.

Calculate: P(0.5 ≤ x ≤ 1.5)

Answer 1

f(x) = 0.5 x ,      0 ≤ x ≤ 2

      = 0 ,            otherwise

P (0.5 ≤ X ≤ 1.5)

= `int_0.5^1.5 0.5  "x"`

`= 0.5  int_0.5^1.5 "x dx"`

`= 1/2 xx ["x"^2/2]_0.5^1.5`

`= 1/2 xx 1/2 [(1.5)^2 - (0.5)^2]`

`= 1/2 xx 1/2 [2.25 - 0.25]`

`= 1/4 xx 2 = 1/2`

P (0.5 ≤ X ≤ 1.5) = `1/2`