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Topic: Integrals of gaussians / normal distributions
Replies: 0

 Maxime Posts: 1 Registered: 7/15/13
Integrals of gaussians / normal distributions
Posted: Jul 15, 2013 9:39 PM

Hi,
I'm searching for calculate some integrals with gaussians, or with normal distributions.
Let's define :
phi_A(x) the normal distribution with std=igma_A and mean=u_A.
phi_B(x) the normal distribution with std=igma_B and mean=u_B.
PHI_A(x) the cumulative of phi_A(x).
PHI_B(x) the cumulative of phi_B(x).

So I want to get the mean of phi_A*PHI_B (and variance, as well).
Therefore, I have to calculate this integral :

int(x*phi_A(x)*PHI_B(x),x=-Inf..+Inf)

If A=B, it's trivial, and given here :
http://en.wikipedia.org/wiki/List_of_integrals_of_Gaussian_functions

But as A and B are different (ie, mu_A and mu_B are different, sigma_A and sigma_B are different), it's not so easy. I tried to do it by parts, but x*Phi_B is not integrable...

Would you have any idea for getting this ?
Many many thanks in advance !

Max