some useful formula for normal integration

The normal distribution function Φ[x], the quantile function ,  and the normal density function φ[x].

First, we will calculate the following intxfp[a,b]

Mathematica does not calculate intxfp, so we consider change of variables. First rewrite intxfp as follows.

Then, {x1,x2} is transformed to {y1,y2}.

The range of the integration is now expressed as

Finally we consider some asymptotic expansions.

Thus, F[x+oy] = F[x] + f[x] ()=F[x] + f[x] * expandF[x+oy,x], where x=Coefficient[x+oy,o,0].

Next, note that

Thus,  Q[F[x]+f[x] o y] = =expandQ[x,y].

Combining these asymptotic expansions, we obtain the following integration. Let func[z]=(az+b) + rem[z], where rem[z]=is of order We would like to calculate intfz=Q[.

The integrand F[func[z]]f[z] is  expandF[func[z],az+b]f[z] =   . We denote . The coefficients d's are obtained from a,b, and c's.

Now the integration is intfz=Q[. Since , and , we obtain
. Using expandQ, this becomes intfz=.

Now, the integration function may be written as intfz=dd2int[a,b,func2dd[a,b,rem,z],z];