A continuous random variable X has CDF given by Fx 0 if x

A continuous random variable X has CDF given by

F(x) = 0 if x<1

= 2(x - 2 + 1/x) if 1 <= x <= 2

= 1 if 2 < x

Find the 100 by pth percentile of the distribution with p = 1/3

F(xp) =1/3

As from the cdf the xp lies between 1 & 2 because if xp<0 F(xp)=0 and if xp>2 F(xp) =1

so clearly 1<=xp<=2

writing xp as x for simplification

F(x) = 2(x-2+ 1/x) =1/3

solving the equation

6(x -2 + 1/x) =1

6x - 12 + 6/x =1

6x - 12 = 1 -6/x

6x - 12 = (x-6)/x

6x^2 - 12x = x - 6

6x^2 -13x +6 =0

solutions to equations

But as x lies in [1,2]

x= 1.5 is the required solution