A brother and sister. The brother eats a random fraction X of a pizza, where X is uniformly distributed between 0 and 1. The sister then eats a random fraction Y of the total pizza (not of the remaining pizza).

i) Find the joint density function for X and Y.

ii) If less than half the pizza is remaining after both brother and sister are done eating, what is the probability that neither of the boys individually ate more than half the pizza?