  Given a statistical model for the request frequencies and sizes of data
objects in a caching system, we derive the probability density of the size of
the file that accounts for the largest amount of data traffic. This is
equivalent to finding the required size of the cache for a caching placement
that maximizes the expected byte hit ratio for given file size and popularity
distributions. The file that maximizes the expected byte hit ratio is the file
for which the product of its size and popularity is the highest -- thus, it is
the file that incurs the greatest load on the network. We generalize this
theoretical problem to cover factors and addends of arbitrary order statistics
for given parent distributions. Further, we study the asymptotic behavior of
these distributions. We give several factor and addend densities of widely-used
distributions, and verify our results by extensive computer simulations.
