GATE 2004 question with modification

Question

Let X and Y be normal distribution with mean $\alpha$ and $\beta$ respectively. If $Z=max(X,Y)$

then the mean of Z is given by_______________

Comments

One query here, consider below figure for X and Y,,from the below figure max(X,Y) would mean the points of max of both the normal distributions right? one more query if we're considering the maximum points like i mentioned the resulting Z need not be a Normal Distribution right ?