Is it possible to get a closed-form solution for the expected value of the maximum of three normally distributed variables that are independent but NOT identically distributed?
In particular, I am interested in an application where:
$X_1\sim N(μ,σ^2ε_1 )$
$X_2\sim N(βμ,β^2σ^2ε_2 )$
$X_3 \sim N(β^2 μ,β^4σ^2ε_3 )$
Where $ε_i$'s are arbitrary constants between 0 and 1 and $\beta$ is a constant between 0 and 1.