The elementary specific risk associated with the probability of the magnitude of the event between x and x C&V is: where (er) are the elements at risk and, the whole distribution, the specific risk is: 2 = J”” 6 er o fx) Fs (x) PH (3~) dx =Joo o( 0 (3~) Ps fx) 0H (x) dx In both expressions, the first two terms under the sign of integration depend only upon the structures, and the third one only upon the natural phenomenon. The first two terms thus define the vulnerability when taking into account the randomness of the properties of structures. It may be seen that this definition changes depending upon whether the hazard is introduced through its probability of exceedance or through its density of probability. 36 3. Simplifications and practical applications As the available data are not sufficient for treating the problem is the sophisticated way sketched above, and as such complexity is not desirable for practical purposes, simplifications appear to be necessary. The first simplification is to consider that the randomness of the strength of structures and, as a consequence, of their vulnerability, is negligible with respect to the variability of the hazard. In this case the functionti (x) has a profile rather similar to the one represented in Figure 3. Figure 3 From Figure 3 it may be seen that the elementary specific risk associated with the prob- ability of the magnitude x lying between x and x +dx is: y$=d (x) PH (x) cfx and for the whole range of magnitudes, the specific risk is: -“=J”” d (x) pH (x) dx er 0 Taking into account the particular values of o( for x < 3c, or x > xi! yields: ‘=Ix” o( (x) PH (x) dx i- OH (x,) er x D 
**Analysis**
The text appears to be discussing the probability of natural disasters and the risk associated with them. The language used is technical and focused on mathematical concepts, suggesting that the topic is related to the assessment and management of natural hazards. The presence of terms such as "hazard", "vulnerability", and "specific risk" further supports this interpretation.

**Category**
Natural resources and environment, Science and technology