Evaluating expected improvement in Mathematica

Question

There is a concept in optimization called expected improvement where you sequentially search for the value of x that optimizes the problem. Here is a screen shot of the general idea of the derivation:

Now, what I am trying to figure out is how to evaluate the expected value of the improvement function in Mathematica?
Here is code that I have tried, but for some reason it is not returning the answer I would have expected. Can anyone please point out why what I am doing is wrong, or give Mathematica code that does the correct evaluation?
Here is my code:
Expectation[Max[f-Y,0], Y [Distributed] NormalDistribution[mu,sigma]]

which returns the following incorrect expression:

Anton Antonov · Accepted Answer

[...] which returns the following incorrect expression:

No, the expression computed by Mathematica is correct.
Below that is verified.
1

"where ?, ? are the PDF and CDF of standard normal distribution, respectively."

Definitions as in OP's question:
[Phi][x_] := Evaluate[PDF[NormalDistribution[0, 1], x]];
[CapitalPhi][x_] := Evaluate[CDF[NormalDistribution[0, 1], x]];

2
Code from OP's question:
res1 = Expectation[
  Max[f - ([Mu] + [Sigma] [Epsilon]), 0], [Epsilon] [Distributed] 
   NormalDistribution[0, 1], Assumptions -> [Sigma] > 0]

(* (E^(-((f - [Mu])^2/(
  2 [Sigma]^2))) (E^((f - [Mu])^2/(2 [Sigma]^2)) f Sqrt[[Pi]] - 
   E^((f - [Mu])^2/(2 [Sigma]^2)) Sqrt[[Pi]] [Mu] + Sqrt[2] [Sigma] + 
   E^((f - [Mu])^2/(2 [Sigma]^2)) f Sqrt[[Pi]]
     Erf[(f - [Mu])/(Sqrt[2] [Sigma])] - 
   E^((f - [Mu])^2/(2 [Sigma]^2))
     Sqrt[[Pi]] [Mu] Erf[(f - [Mu])/(Sqrt[2] [Sigma])]))/(2 Sqrt[[Pi]]) *)

3
Simplified result:
res1 = FullSimplify[res1]

(* (E^(-((f - [Mu])^2/(2 [Sigma]^2))) [Sigma])/Sqrt[2 [Pi]] + 
 1/2 (f - [Mu]) (1 + Erf[(f - [Mu])/(Sqrt[2] [Sigma])]) *)

4
The right hand side of the last EI(x) formula in OP's  question:
res2 = (f - [Mu])*[CapitalPhi][(f - [Mu])/[Sigma]] + [Sigma]*[Phi][(
    f - [Mu])/[Sigma]]

(* (E^(-((f - [Mu])^2/(2 [Sigma]^2))) [Sigma])/Sqrt[2 [Pi]] + 
 1/2 (f - [Mu]) Erfc[-((f - [Mu])/(Sqrt[2] [Sigma]))] *)

5
Compare the results:
FullSimplify[res1 == res2]

(* True *)

Evaluating expected improvement in Mathematica

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