This is a very classical question. Here is my two cents:

1. The formula you post is the general formula:

The problem is you need to re-write the minimum function as a piecewise function first and thus you can simplify the integral into the form you can integrate:

That is, the integral will be splited into two parts; Since now is the variable of the inner integral, it will be splited like this:

and you may proceed from here.

I assume you know the joint pdf here as each of them are just exponential random variables and they are independent.

2. However, the main spirit of this question is want you to figure out the following relation:

Try to understand and use this relationship to calculate the CDF of :

Once you find out the CDF you will know that actually is just another exponential random variables and of course the mean and variance are easy to find out.