Abstract
Actuarial models rely on probability distributions to represent uncertainty in insurance risk. In practice, however, real insurance datasets may violate classical distributional assumptions, which can affect model fit, risk estimation, pricing, and reserve adequacy. This project studies two major components of actuarial modeling: claim frequency and claim severity. Claim frequency is analyzed using the beMTPL97 Belgian motor third-party liability dataset, while claim severity is analyzed using the Danish fire insurance loss dataset. The main research question is whether classical actuarial assumptions, especially the Poisson assumption for claim frequency and light-tailed assumptions for claim severity, adequately describe real insurance data. The analysis begins with descriptive statistics and graphical diagnostics. For claim frequency, the sample mean is approximately 0.1239 and the variance is approximately 0.1350, producing a dispersion index of about 1.09. Since the variance exceeds the mean, the data show mild overdispersion relative to the Poisson model. A Negative Binomial model is then considered because it allows the variance to exceed the mean. For claim severity, the Danish fire loss data have mean approximately 3.3851 and variance approximately 72.3767, indicating strong right-skewness and heavy-tailed behavior. Q-Q plots and a log-log tail plot show that the Exponential model under-estimates extreme losses, while the Lognormal model improves fit in the center of the distribution but may still miss the most extreme tail behavior. Overall, the results support the hypothesis that real insurance data often violate classical actuarial distributional assumptions. The Poisson model is too restrictive for claim counts when overdispersion is present, and light-tailed severity models are inadequate when large losses occur more often than expected. These findings matter in actuarial practice because misspecified models can underestimate variability, distort premiums, weaken reserve estimates, and reduce the reliability of model-based risk decisions.