# The inverse cumulative distribution function evaluated at Halton draws [duplicate]

Sandor and Train (2004, Quasi-random simulation of discrete choice models) mention that "A randomized Halton sequence is a set of draws from the uniform distribution. To obtain draws from density $$f$$ with cumulative distribution $$F$$, each element c is transformed as $$F^{-1}(c)$$." A same explanation is given in Train (2006, Discrete choice methods with simulation). Therefore the statement is correct. My question is as follows. I expect $$F^{-1}(c)$$ to give a "probability" value. But the statement says that $$F^{-1}(c)$$ is a draw from $$f$$. So I could not understand the statement. Which theory should I check to understand this? This is an econometric question for scientific research that regards estimation of the mixed logit model.

• Thanks. I need one more step to figure out if I understand it. Consider my original post. F^-1(c) = d where F is the standard normal cumulative distribution function and c has a uniform distribution and d has a standard normal distribution. The link confirms that F(d) = c has a uniform distribution. Is this it? The whole thing I am wondering about is that I thought it is always the case that F^-1(c) gives a probability value whereas in my original post F^-1(c) does not give a probability value but a draw from the density. Aug 29, 2021 at 13:40
• Check the two threads marked as duplicates. If they don’t answer your question, feel free to comment or edit the question specifying what’s still unclear.
– Tim
Aug 29, 2021 at 13:51
• The two threads together now answer my question. Thank you. If my question needs to be deleted, fine. Though the question could be instructive for those who are new and learning about pseudo-random sampling. Aug 29, 2021 at 14:00
• It doesn’t need to be deleted. The opposite: for some reason you didn’t find the threads yourself, so maybe someone else would also easier find your question and get the links from here.
– Tim
Aug 29, 2021 at 14:11
• OK, I do not delete the question. I did not find the mentioned threads because I did not know how to search for terms that would bring me those threads. Thank you once again. Aug 29, 2021 at 14:15