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Drought indicators are used as triggers for action and so are the foundation of drought monitoring and early warning. The computation of drought indicators like the standardized precipitation index (SPI) and standardized streamflow index (SSI) require a statistical probability distribution to be fitted to the observed data. Both precipitation and streamflow have a lower bound at zero, and their empirical distributions tend to have positive skewness. For deriving the SPI, the Gamma distribution has therefore often been a natural choice. The concept of the SSI is newer and there is no consensus regarding distribution. In the present study, twelve different probability distributions are fitted to streamflow and catchment average precipitation for four durations (1, 3, 6, and 12 months), for 121 catchments throughout the UK. The more flexible three- and four-parameter distributions generally do not have a lower bound at zero, and hence may attach some probability to values below zero. As a result, there is a censoring of the possible values of the calculated SPIs and SSIs. This can be avoided by using one of the bounded distributions, such as the reasonably flexible three-parameter Tweedie distribution, which has a lower bound (and potentially mass) at zero. The Tweedie distribution has only recently been applied to precipitation data, and only for a few sites. We find it fits both precipitation and streamflow data nearly as well as the best of the traditionally used three-parameter distributions, and should improve the accuracy of drought indices used for monitoring and early warning.