Overlapping Normal Distributions

A B
μ1
σ1 		μ2
σ2


Overlap area=

c2 Intersection point's X:
c1 Intersection point's X:

Two normal populations mixed equally (50/50 weight, equal N, equal σ):
The overall PDF is the average (mixture) of the two normals:f(x) = ½ φ(x; μ1, σ2) + ½ φ(x; μ2, σ2)
Using Standard normal as one pop, the other at μ=2:
PDF(x) = \frac{1}{2}\frac{1}{\sqrt{2\pi}}e^{-\frac{\left(x-2\right)^{2}}{2}}+\frac{1}{2}\frac{1}{\sqrt{2\pi}}e^{-\frac{x^{2}}{2}}