Following equations taken from List of integrals of Gaussian functions.
Function M(μ, σ)
returns expectation ∫ N(x|μ, σ) Φ(x) dx where Φ(x) is the standard cumulative normal.
Function V(μ, σ)
returns variance ∫ N(x| μ, σ) ⋅ (Φ(x) - M(μ, σ))² dx.
Function B(μ, σ)
returns expectation of square ∫ N(x|μ, σ) [Φ(x)]² dx.
using CDFIntegrals
using PyPlot # must be indepedently installed
μ, σ = 1, 2
x = -10:0.01:10
plot(x, M.(x,σ),"blue",label="mean of noisy cdfs")
plot(x, M.(x,σ) .+ V.(x,σ),"--r",label="variance")
plot(x, M.(x,σ) .- V.(x,σ),"--r")
plot(x, CDFIntegrals.Φ.(x),"g",label="cdf")
legend()