The plot below models a sample of departure delays of flights, the x-axis is minutes delayed. The mean is about 10 (red line) and std is 36. The distribution has a very long tail (some flights are 800 minute delayed). The blue line is a fitted normal (mean=10, std=36) and the purple distribution is a shifted Poison (such that it can handle negative values).
Both of them completely miss the data as the data has a long tail and is very skewed. How might I model the probability of flight delays best?