Distribution functions

The probability of failure functions (cumulative distribution functiuons) F(x) and density functions f(x) are profiles that contain all the relevant information about the statistical properties of a random variable. The most useful information about distribution is its location (the mean) and its dispertion (the variance). There are four wide-spread distribution functions to describe reliability:

A.Exponential
B.Normal(Gaussian)
C.Weibull
D.Uniform

For structures consisting of a large number of structural elements with independent failures the reliability function is exponential.
The Weibull law is used if failures are connected with damage accumulation (wear-out, fatigue).
The uniform law is used if the only information about the distribution is lower and upper boundaries.
The normal distribution is the most frequently used. It is symmetrical about the mean value. At this mean the probability of failure and reliability are equal:
F(t) = R(t) = 0.5

The summed area under the probability density function is constant and always equal to 1. If the summation is made at segment [0,t] the sum is equal to the probability of failure F(t) to time t. The (failure) probability function F(t) is equal to zero at the left end or 1 at the right end of the axis (infinity).

The reliability function R(t) characterizes the probability of structural integrity. All distributions can be characterized by the mean and the dispersion or the standard deviation. There are special mathematical formulas to calculate the parameters.
For a small number of structural elements (10 for example) the reliability function of the whole structure corresponds to the distribution law of its elements. If elements are described by normal law, the structure has also normal distribution.