This is a problem I've come across in independent study...
Let $\xi_{1}, \xi_{2}, \xi_{3}$ be i.i.d. N(0,1). Can anything be said about the distribution of $a\max\{\xi_{1}, \xi_{2}\} + b \xi_{3}$ for $a,b>0$? If that is too difficult, it would also help me a lot to know the distribution of $a\max\{\xi_{1}, \xi_{2}\}$ alone. Thanks for all help...