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3. (a) (b) If X and Y are independent gamma random variables with parameters (a, A) and (,3, A), respec- tively. Compute the joint density of U = X + Y and V = . Are U and V independent random variables? What are their respective marginal distributions? Suppose there are n+m jobs to be performed, each (independently) taking an exponential amount of time with rate A to be completed and suppose that we have two workers to perform these jobs. Worker I will do jobs 1, 2, ' u ,n, and worker II will do the remaining m jobs. If we let X and Y denote the total working times of workers I and II, respectively, then what are the respective distributions of X and Y? What is the distribution of the proportion of the work that will be performed by worker