The thread below contains the ideas you need for the first part:
For the second part, you should know how to get the pdf of Y/2 from the pdf of Y ...... And you should know that the pdf of the sum of two random variables is the convolution of their seperate pdf's.
To get the pdf of the random variable max(X, Y), use the same idea that I used in my reply to your question about showing
E(U) = 1/3.
so, i got the density of X+1/2 Y = lambda /2 * e^(-lambda y)
and i got density for max : lambda square * e^(-lamda( x+y))
is it true??
The U and V take on values in the region defined by X > 0 and Y > 0, that is, the region defined by U > 0 and 2V - U > 0. Sketch this region. Then you'll see that the integral terminals are U = 0 (lower) and U = 2V (upper).
You should check your final answer by:
1. Finding the pdf g(x) for Y/2, and
2. Calculating the convolution f(x) * g(x) (you'll need to be careful with the integral terminals).