If the function f defined by f ( x )= cases K ( x - x ^2 ) & if 0 < x < 1 0 & , other wise cases is the p.d.f. of a r.v.x, then the value of P ( X < 1 2 ) is

Step-by-step solution

Given f is a the p.d.f. of a r.v.x. aligned & ₀^1 f ( x ) dx =1 ₀^1 ~K ( x - x ^2 ) dx =1 & ~K [ x ^2 2 - x ^3 3 ]₀^1=1 K ( 1 2 - 1 3 )=1 K 6 =1 & ~K =6 & . P ( X < 1 2 )= ₀^ 1 2 6 ( x - x ^2 ) dx = [ 6 x ^2 2 ]₀^1- [ 6 x ^3 3 ]₀^1 ] & = [3 x ^2-2 x ^3 ]₀^ 1 2 = 3 4 - 2 8 = 3 4 - 1 4 = 1 2 aligned