The joint density is the product of the three marginals.
You want ...
Turtles A, B, and C are racing .Assume the times for each turtle to finsih the race are independent and randomly distributed exponential random variables with mean 1 hr, 2hr, and 3hrs, respectively. What is the probability of each turtle winning?
I'm drawing a blank...
Do you find the expected value of the Joint distribution of the 2 other turtles and set that equal to the probabilty that turtle A's time is less than that? and because they are independent, you can just multiply their cdf's to get the joint probability?