  For caching with nonuniform file popularity, we aim to characterize the
memory-rate tradeoff under uncoded cache placement. We consider the recently
proposed Modified Coded Caching Scheme (MCCS) with the optimized cache
placement based on the popularity-first approach to minimize the average
delivery rate. We introduce two information-theoretic lower bounds on the
average rate for caching under uncoded placement. For $K = 2$ users, we show
that the optimized MCCS attains the lower bound and is optimal for caching with
uncoded placement. For general $K$ users with distinct file requests, the
optimized MCCS attains the popularity-first-based lower bound. When there are
redundant file requests among $K$ users, we show a possible gap between the
optimized MCCS and the lower bounds, which is attributed to zero-padding
commonly used for coded delivery. We analyze the impact of zero-padding and its
limitation. Simulation study shows that the loss is very small in general and
only exists in some limited cases.
