Solution
Best of Three Normals
Show the problem again
X₁, X₂, X₃ are independent standard normal random variables. What is E[max(X₁, X₂, X₃)]? Give an exact closed form, or a decimal rounded to 4 places.
Worked solution
The answer is 3/(2√π) ≈ 0.8463. Start with two variables: max(X, Y) = (X + Y)/2 + |X − Y|/2. Since X − Y is normal with variance 2, E|X − Y| = √2·√(2/π) = 2/√π, so E[max(X, Y)] = 1/√π. Now a counting identity: if three numbers are sorted as p ≤ q ≤ r, the maxima of the three pairs are r, r and q, so max(X₁,X₂) + max(X₁,X₃) + max(X₂,X₃) = 2·max + median. Taking expectations, 3/√π = 2E[max] + E[median]. By the symmetry X ↦ −X, the median of three standard normals has mean 0. Therefore E[max] = 3/(2√π). The trick fails for four or more variables; the expected maximum of four normals has no comparably simple form.
Source: Classic Gaussian exercise in wide circulation; no traceable original source. Statement written for AxiomIQ.