I have a normal random variable that varies with time t as
where
Detection criterion is when the random variable
So what is the probability that detection occurs at time t=i?
Since
where
where
But when I did a monte-carlo run of the problem I am having the mismatch between my calculation and monte carlo as in the figure below.
Now my question is where did I go wrong in my calculation?
-Update: I found the bug. My assumption about the independence of X terms
is not correct
i.e.
instead the detection detection criterion should be written as
Which can be simplified as
The two terms on the right hand side are multivariate CDFs with marginal normal distributions and covariance given by
However, this multivariate CDF is too computationally intensive. I would like to find an easy way to compute this probability