Aedes aegypti is the main vector of multiple diseases, such as Dengue, Zika, and Chikungunya. Due to modifications in weather patterns, its geographical range is continuously evolving. Temperature is a key factor for its expansion into regions with cool winters, but rainfall can also have a strong impact on the colonization of these regions, since larvae emerging after a rainfall are likely to die at temperatures below $10^{\circ}$C. As climate change is expected to affect rainfall regimes, with a higher frequency of heavy storms and an increase in drought-affected areas, it is important to understand how different rainfall scenarios may shape Ae. aegypti's range. We develop a model for the population dynamics of Ae. aegypti, coupled with a rainfall model to study the effect of the temporal distribution of rainfall on mosquito abundance. Using a fracturing process, we then investigate the effect of a higher variability in the daily rainfall. As an example, we show that rainfall distribution is necessary to explain the geographic range of Ae. aegypti in Taiwan, an island characterized by rainy winters in the north and dry winters in the south. We also predict that a higher variability in the rainfall time distribution will decrease the maximum abundance of Ae. aegypti during the summer. An increase in daily rainfall variability will likewise enhance its extinction probability. Finally, we obtain a nonlinear relationship between dry season duration and extinction probability. These findings can have a significant impact on our ability to predict disease outbreaks.