Specifically, suppose $X$ and $Y$ are normal random variables (independent but not necessarily identically distributed). Given any particular $a$, is there a nice formula for $P(\max(X,Y)\leq x)$ or similar concepts? Do we know that $\max(X,Y)$ is normally distributed, maybe a formula for mean and standard deviation in terms of those for $X$ and $Y$? I checked the usual places (wikipedia, google) but didn't find anything.
The max of two non-identical Normals can be expressed as an Azzalini skew-Normal distribution. See, for instance, a 2007 working paper/presentation by Balakrishnan
A Skewed Look at Bivariate and Multivariate Order Statistics
Prof. N. Balakrishnan
Working paper / presentation (2007)
A recent paper by (Nadarajah and Kotz - viewable here) gives some properties of max$(X,Y)$:
Nadarajah, S. and Kotz, S. (2008), "Exact Distribution of the Max/Min of Two Gaussian Random Variables", IEEE TRANSACTIONS ON VERY LARGE SCALE INTEGRATION (VLSI) SYSTEMS, VOL. 16, NO. 2, FEBRUARY 2008
For earlier work, see:
A. P. Basu and J. K. Ghosh, “Identifiability of the multinormal and other distributions under competing risks model,” J. Multivariate Anal., vol. 8, pp. 413–429, 1978
H. N. Nagaraja and N. R. Mohan, “On the independence of system life distribution and cause of failure,” Scandinavian Actuarial J., pp. 188–198, 1982.
Y. L. Tong, The Multivariate Normal Distribution. New York: Springer-Verlag, 1990.
One can also use a computer algebra system to automate the calculation. For example, given $X \sim N(\mu_1, \sigma_1^2)$ with pdf $f(x)$, and $Y \sim N(\mu_2, \sigma_2^2)$ with pdf $g(y)$:
... the pdf of $Z = max(X,Y)$ is:
where I am using the
Maximum function from the mathStatica package of Mathematica, and
Erf denotes the error function.