Mathematica cannot take such integrals, even for zero-mean normal distributions. So, it is highly unlikely that they can expressed in closed form.
Here is the image of the corresponding Mathematica notebook:
An exceptional trivial case is when the two normal distributions are the same:
That is, here we get $F(y)^2/2=\Phi(y)^2/2$, by the substitution $t=\Phi(x)$.
Mathematica cannot take such integrals, even for zero-mean normal distributions. So, it is highly unlikely that they can expressed in closed form.
Here is the image of the corresponding Mathematica notebook:
Mathematica cannot take such integrals, even for zero-mean normal distributions. So, it is highly unlikely that they can expressed in closed form.
Here is the image of the corresponding Mathematica notebook:
An exceptional trivial case is when the two normal distributions are the same:
That is, here we get $F(y)^2/2=\Phi(y)^2/2$, by the substitution $t=\Phi(x)$.
Mathematica cannot take such integrals, even for zero-mean normal distributions. So, it is highly unlikely that they can expressed in closed form.
Here is the image of the corresponding Mathematica notebook:
