1. Probabilistic modeling of the histogram of a random load

The transition from statistical analysis of random loading to probabilistic modeling with mathematical analysis is always a delicate one. Indeed, the purpose of modeling random loading is to determine a theoretical probability of events occurring, based on the observation of a series of values ( x _i ) _N of a random variable ( X). Whereas descriptive statistics seek to make a judgement about the random variable in relation to pre-selected values, mathematical statistics seek to make a judgement about the probability of occurrence of different values. The histogram (figure 1 a) becomes a "density estimator" and the cumulative diagram is a "probability estimator" (distribution function). Regardless...