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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.

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You need to know the joint distribution function of $X_1$ and $X_2$. Are they given to be independent but you neglected to tell us so? –  Dilip Sarwate Jul 30 '12 at 15:06
    
Presumably the $x_i$ are independent? –  copper.hat Jul 30 '12 at 15:59
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Answer: a complicated function of $C$ and of the means and standard deviations of $x_1$ and $x_2$. –  Did Jul 30 '12 at 17:35

2 Answers 2

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

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You re right. Actually, I already have the solution to expected value of max of two (or more) normal random variables. It is below. –  user36831 Aug 1 '12 at 11:56

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

enter image description here

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