Let's assume I have two normal distributions, X and Y with known expected values and standard deviations. How to calculate the distribution of greater values from random pairs of values from these distributions? If it's not clear, step by step looks like that:
Based on E(X) and D(X) I sample one result, got 10. Based on E(Y) and D(Y) I sample one result, got 15. 15 is greater than 10, so we put 15 in our result set Z. If we'd receive 10 from X and 8 from Y, we put 10 in result set Z.
What are the parameters E(Z) and D(Z) (and is it still a normal distribution)?
If $X=a+Z_1$ and $Y=b+Z_2$ where $Z_1$ and $Z_2$ are $N(0,1)$ and independent, then $$M=\max(X,Y)=\frac{1}{2}(X+Y+|X-Y|)$$ is certainly not Gaussian since the density of $M$ is $\Phi(m-b)\phi(m-a)+\Phi(m-b)\phi(m-b)$ where $\phi$ is the density of $N(0,1)$ and $\Phi(y)=\int_{-\infty}^y\phi(x)dx.$ The expectation of $M$ is given by the following formula, since the density of $Z=\frac{1}{\sqrt{2}}(Z_1-Z_2)$ is $\phi$ : $$E(M)=\frac{a+b}{2}+\int_{-\infty}^{\infty}|a-b+\sqrt{2}z|\phi(z)dz.$$ This is computable only by numerical analysis if $a\neq b.$
I have not computed $E(M^2)$ but if $a$ and $b$ are large enough an approximation of $M^2$ is $$\max(X^2,Y^2)=\frac{1}{2}(X^2+Y^2+|X^2-Y^2|).$$ But I am afraid that both $E(M^2)$ and $E(\max(X^2,Y^2))$ end up with a double integral.