I just wanted to make sure I was doing these problems correctly, as I don't have the solutions to any of them, or previous work to base my answers on.

given an exponential distribution compute:

a)

what I was thinking it's simply

b)

c)

d) compute with rates and

based on my book the probabilistic minimum of is just (no proof was given) so in this case it would be also I would like to show that is the case, with 2 variable. I thought it would be:

but that give me 1 and

another way I considered was

I finally considered but I don't know what to pick since they are both different.