I have the following stochastic optimisation problem that I would like to implement:

# min(2x_1 + 3x_2)
# s.t. P(X + Y <= x_1 and Y <= x_2)>=0.95
#      0<=x_1<=2, 0<=x_2<=4
#where X is Gaussian with mean 1 and standard deviation 0.1,
#      Y is Gaussian with mean 2 and standard deviation 0.5,
#      and the correlation between X and Y is -0.4

I have tried using Pyomo, PuLP and Gurobi, but I could not implement the probabilistic constraint to function properly (either because of errors that I could not debug after hours of work or because the program returned x_1 = 0, x_2 = 0 as a result which is clearly wrong).

I’m convinced that there should be a fairly easy way to implement this problem. Does anybody know a compact and nice solution to this?