Quantile function

Hi everyone,
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?


Active Member
I think the confusion is that the F() on the LHS is not the same F(.) on the RHS, the one on the RHS is apparently the CDF of X, ie x for the uniform RV?

so having the CDF of Y now just invert ie set F(y) = gamma_0.5 (or whatever percentile) and solve.
My intuition says that the next stage is to compute the CDF of X, that is, the CDF(e^-(y/2))
This CDF is equal to 0 for e^(-y/2)<=0 ,
is equal to e^(-y/2) for 0< e^(-y/2) <1
Then it is one for e^(-y/2)>1
Just as I did for CDF of X in my first message

So, the result is 1 - F( e^-(y/2) ) = 1-e^(-y/2)
Then, I can write down the quantile function.
Can it work for you?


Active Member
yes, you have it in that last equation, so just need to take inverse function of that. real question is how will you check if ans is correct?