Climate risk financing programs in agriculture have caught the attention of researchers and policy makers over the last decade. Weather index insurance has emerged as a promising market-based risk financing mechanism. However, to develop a suitable weather index insurance mechanism it is essential to incorporate the distribution of underlying weather and climate risks to a specific event model that can minimize intra-seasonal basis risk. In this paper we investigate the erratic nature of rainfall patterns in Kenya using CHIRPS rainfall data from 1983-2017. We find that the patterns of rainfall are fractional, both erratic and persistent which is consistent with the Noah and Joseph effects well known in mathematics. The erratic nature of rainfall emerges from the breakdown of the convergence to a normal distribution. Instead we find that the distribution about the average is approximately lognormal, with an almost 50% higher chance of deficit rainfall below the mean versus adequate rainfall above the mean. We find that the rainfall patterns obey Hurst law and the measured Hurst coefficients for seasonal rainfall pattern across all years range from a low of 0.137 to a high above 0.685. To incorporate the erratic and persistent nature of seasonal rainfall, we develop a new approach to weather index insurance based upon the accumulated rainfall in any 21-day period falling below 60% of the long-term average for that same 21-day period. We argue that this approach is more satisfactory to matching drought conditions within and between various phenological stages of growth.