Are there numerical integration methods we can use to approximate the integral of p(x) and its inverse, rather than solving it exactly?
What if some part of the CDF is flat, i.e. the inversion function is not injective?
To avoid the situation that the inverse function does not exist, should we only evaluate it in the domains where the PDF is not zero?
You mentioned that P is invertible because it is monotonically nondecreasing. Isn't this insufficient? For instance, a constant function is monotonically nondecreasing, but it is not invertible.
Why can't we draw samples from the range of x instead of the range of \psi?
What is if the inversion doesn't exist/p(x) isn't invertible?
Could we not execute a kind of binary search via sampling by starting in the center instead of inverting the function? We would stop once we reach a certain distance from the actual value.
is inversion method not viable if the inverse is not defined?
CDF of any distribution is Uniform([0,1]).
I wonder if Monte Carlo can still perform correctly if you do not know the expression of PDF?
What happens when the function is not invertible?
How do we ensure the inverse gets easier?
How do we get those quantities/what happens if we don't know/cannot invert it? Doesn't inverting it mean knowing which value of the random variable the CDF's value corresponds to?