A random variable X has following p.d.f. f ( x )= k x(1-x), 0 x 1 and P (x>a)= 20 27 , then a=

Step-by-step solution

The constant k is determined by the normalization condition ₀¹ kx(1-x) dx = 1 . k ₀¹ (x - x^2) dx = k [ x^2 2 - x^3 3 ]₀¹ = k ( 1 6 ) = 1 Thus k = 6 and f(x) = 6x(1-x) . Given P(x > a) = 20 27 , we require _ a ¹ 6x(1-x) dx = 20 27 . 6 _ a ¹ (x - x^2) dx = 6 [ x^2 2 - x^3 3 ]_ a ¹ = 1 - 3a^2 + 2a^3 = 20 27 Multiplying through by 27 yields 54a^3 - 81a^2 + 7 = 0 . Testing a = 1 3 gives 54 ( 1 27 ) - 81 ( 1 9 ) + 7 = 2 - 9 + 7 = 0 . The correct choice is A .