Variance of a Cumulative Distribution Function of Normal Distribution

Suppose, XN(μ,σ2).

Can anyone help in finding the following : Var (Φ(X+cd)) ? Here, c and d are positive.

Here, Φ(x) is the “Cumulative Distribution Function” of the above-mentioned normal distribution.

Thanking you in advance.

Answer

Expanding on the comment by Dilip Sarwate, using this list of integrals of Gaussian functions gives

E[Φ(X+cd)]=1σϕ(xμσ)Φ(x+cd)dx=ϕ(x)Φ(σx+μ+cd)dx=Φ(μ+cσ2+d2) and

E[Φ(X+cd)2]=1σϕ(xμσ)Φ(x+cd)2dx=ϕ(x)Φ(σx+μ+cd)2dx=Φ(μ+cσ2+d2)2T(μ+cσ2+d2,d2σ2+d2) where T is Owen’s T function.

From these expressions, Var(Φ(X+cd)) follows.

Attribution
Source : Link , Question Author : Dwaipayan Gupta , Answer Author : gung – Reinstate Monica

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