If a continuous random variable X has probability density function f (x) given by f(x)= array cc a x & , if 0 x < 1 a & , if 1 x < 2 3 a-a x & if 2 x 3 0 & , otherwise array , . then a has the value

Step-by-step solution

Since f (x) is the p.d.f. of X , aligned & _ - ^ f (x) d x=1 & ₀^1 a x ~d x+ ₁^2 ad x+ ₂^3(3 a - ax ) d x=1 & a [ x^2 2 ]₀^1+ a [x]₁^2+ [3 a x- a x^2 2 ]₂^3=1 aligned aligned & a ( 1 2 )+ a (1)+ ( 9 a 2 -4 a )=1 & 2 a =1 & a = 1 2 aligned