How to calculate the multiple integrals where the integral domain is based on the sum of normal distribution random variables?

The integral is shown below:

And how to use python to calculate pi (better if we don't need to code for each pi)?

• Hi: A sum of iid normals is still normal with mean equal to the sum of the individual means and variance equal to the sum of the variances so you probably don't need python. – mark leeds Aug 29 '19 at 3:57
• Thanks! I think you're right. Sadly I cannot upload the picture successfully, otherwise I can ask more detials. Actually I find that that the integral I asked can be transformed to the integral w.r.t a multivariate normal distribution with certain integral domain. I think I have to use Python to help me calculate the probability.. And as there are many p(i) need to be calculated, I also want to write some codes that can loop for each p(i) rather than I define a function for each one...Do you have any idea? – HenryLiu Aug 29 '19 at 6:27
• Hi: I'm not clear on why you need Python. Just figure out the mean of each $X_i$ and the variance of each $X_i$. Then add them up and you then have one normal random variable with a mean and a variance and you can look it up in a cumulative normal distribution table. – mark leeds Aug 29 '19 at 11:47