Variance of a max function

Question

Say $x_1$ and $x_2$ are normal random variables with known means and standard deviations and $C$ is a constant. If $y = \max(x_1,x_2,C)$, what is $\mathrm{Var}(y)$?

Well, I forgot to tell that $x_1$ and $x_2$ are independent.

Answer 1

I do not know the actual expression for Var($y$), but to get started, $y$ will not follow a normal distribution and its cumulative probability distribution $F_Y(x)$ will be the product of the cumulative distributions $F_{X1}(x)$ and $F_{X2}(x)$, floored to zero for $x<C$.

For an estimation of Var($y$) you may find this other question helpful: Expectation of the maximum of gaussian random variables

Answer 2

The solution is far from trivial, using Maple I obtained this: