I'm trying to solve the following problem:

I have a uniform random variable X with:

1 if 0 < x < 1

0 otherwise

Now, I computed the CDF in the following way:

0 if x <= 0

x if 0 < x < 1

1 if x equal or higher than 1

Then, the problem says, let Y = -2log X; it asks to find the quantile function.

The quantile function is the inverse of CDF. So I wrote:

F(y) =P(Y<= y) = P (-2log X <= y) = P (log x >= -y/2) = P ( x >= e ^ (-y/2) ) = 1 - P ( x <= e ^ (-y/2) ) = 1 - F( e^-(y/2) )

Then, what should I do next?

Thanks