Question

If the probability density function continuous random variable X is

f(x) = k(1 - x₂); 0 < x < 1
      = 0  ;    otherwise

Then P`(0 < X < 1/2)` = ?

Options

• `9/11`

• `11/16`

• `2/3`

• `15/17`

Answer 1

`11/16`

Explanation:

Since, f(x) is the p.d.f. of X.

`therefore int_(- ∞)^∞ "f"(x) "dx"` = 1

`=> int_(- ∞)^0 "f"(x) "dx" + int_0^1 "f"(x) "dx" + int_0^∞ "f"(x) "dx"` = 1

`=> 0 + int_0^1 "k" (1 - x^2) "dx" + 0 = 1`

`=> "k" [x - x^3/3]_0^1` = 1

`2/3 "k" = 1`

`=> "k" = 3/2`

`"P"(0 < "X" < 1/2) = int_0^(1/2) "f"(x) "dx" = int_0^(1/2) 3/2 (1 - x^2) "dx"`

`= 3/2[x - x^3/3]_0^(1/2)`

`= 3/2(1/2 - 1/24) = 11/16`