We present a mathematical model for the spatial dynamics of DNA replication. Using this model we determine the probability distribution for the time at which each chromosomal position is replicated. From this we show, contrary to previous reports, that mean replication time curves cannot be used to directly determine origin parameters. We demonstrate that the stochastic nature of replication dynamics leaves a clear signature in experimentally measured population average data, and we show that the width of the activation time probability distribution can be inferred from this data. Our results compare favorably with experimental measurements in Saccharomyces cerevisae.