Each kind of phenomenon defined, with respect to its effects on the site, by its magnitude (x,,) which is a variable or a set of variables. The distribution of (x) is generally known through the function 0 (x), which defines the probability of the magnitude x being exceeded within the period of reference. Alternatively, the functions F(x) = 1 - 0 (x) which defines the prob- ability of x not being exceeded, or p(x) = @?, which is the probability density function, can be used. dx The vulnerability may be expressed as the degree of damage inflicted on a structure or on a population of structures by a natural phenomenon of given magnitude. Let o( be this degree of damage which is expressed as a random function of x. It is a function of x 2. Theoretical background A complete solution of the problem should involve the randomness of the mechanical properties of the structures (especially their strength) and of their vulnerability as defined above. In this case, the evaluation of the risk should be performed in the following way: The distribution of the hazard (H) is known through its probability density function pH(1c) or the probability of exceedance 8 Ho x . The distribution of the strength (6’) is known through its probability density function pS (h) or the probability of non-exceedance Fs (x). 1 1 4x1 0 0 x xi-dx Figure 1 Figure 2 c From Figure 5, it may be seen that the probability of failure associated with the probability of the hazard x being comprised between x and x + dx is: df=pH FS dx From Figure 2, it may be seen that the probability of failure associated with the probability of the strength x lying between x and x $ dx is: so that the probability of failure for the whole distribution of x is: 
**Analysis**
The text appears to be discussing the probabilistic analysis of natural hazards, such as earthquakes or floods, and their impact on structures and populations. The language used is technical and focused on the mathematical modeling of the phenomena, suggesting that the text is related to scientific or engineering fields.

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