Question

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

`f(x)={(0.5x",",0≤x≤2,,),(0",","otherwise",,):}`

Calculate (a) P (X ≤ 1) (b) P (0.5 ≤ X ≤ 1.5)   

Answer 1

f(x) = 0.5 x ,      0 ≤ x ≤ 2

      = 0 ,            otherwise

(a) P (X ≤ 1) = `int_0^1 0.5  "x dx"`

`= 0.5 int_0^1 "x  dx"`

`= 0.5 ["x"^2/2]_0^1`

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

 

∴ P (X ≤ 1) = `1/4`

(b) 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`