Probability - Joint Distribution: Poisson variables, Range of n random variables

A First Course in Probability (9th Edition) - Sheldon Ross
Problems and Theoretical Exercises in Chapter 6: Jointly Distributed Random Variables
P6.59: If X, Y, and Z are independent random variables having identical density functions f(x) = exp(-x), x in (0, infinity), derive the joint distribution of U = X + Y, V = X + Z, W = Y + Z.
E6.2: Suppose that the number of events occurring in a given time period is a Poisson random variable with parameter lambda. If each event is classified as a type i event with probability p_i, i = 1,...,n, sum p_i = 1, independently of other events, show that the numbers of type i events that occur, i = 1,...,n, are independent Poisson random variables with respective parameters (lambda)p_i, i = 1,...,n.
E6.30: Compute the density of the range of a sample of size n from a continuous distribution having density function f.