The p.d.f of a random variable x is given by `f(x)=(1)/(4a), 0 lt x lt 4a, (a gt 0)`
`=0,` otherwise
and `P(x lt (3a)/(2))=kP(xgt(5a)/(2)` then k= . . .

Options

A. 1
B. `(1)/(4)`
C. `(1)/(8)`
D. `(1)/(2)`

Answer

A

Explanation

We have p.d.f of a random variable x is given by
`f(x)=(1)/(4a), 0 lt x lt 4a. (a gt 0)`
and also, `p(x lt (3a)/(2))=kp(x gt (5a)/(2))`
`implies int_(0)^((3a)/(2))f(x)dx=k int_((5a)/(2))^(4a)f(x)dx`
`implies int_(0)^((3a)/(2))((1)/(4a))dx=k int_((5a)/(2))^(4a)((1)/(4a))dx`
`implies(1)/(4a)int_(0)^((3a)/(2))dx=(k)/(4a)int_((5a)/(2))^(4a)dx`
`implies(x)_(0)^((3a)/(2))=k(x)_((5a)/(2))^(4a)`
`implies((4a)/(2)-0)=k(4a-(5a)/(2))`
`implies(3a)/(2)=k((3a)/(2))impliesk=1`