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Anderson localization lasers exploit resonant cavities formed due to structural disorder. The inherent randomness in the structure of these cavities realizes a probability distribution in all cavity parameters such as quality factors, mode volumes, mode structures etc, implying resultant statistical fluctuations in the temporal behavior. Here, we provide the first, direct experimental measurements of temporal width distributions of Anderson localization lasing pulses in intrinsically and extrinsically disordered coupled-microresonator arrays. We first illustrate signature exponential decays in the spatial intensity distributions of the lasing modes that quantify their localized character, and then measure the temporal width distributions of the pulsed emission over several configurations. We observe a hitherto-unreported dependence of temporal widths on the disorder strength, wherein the widths show a single-peaked, left-skewed distribution in extrinsic disorder and a dual-peaked distribution in intrinsic disorder. We propose a model based on coupled rate equations for an emitter and an Anderson cavity with a random mode structure, which gives excellent quantitative and qualitative agreement with the experimental observations. The experimental and theoretical analyses bring to the fore the temporal complexity in Anderson localization based lasing systems., Comment: 9 pages, 9 figures