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

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