Probabilistic Methods in Geotechnical Engineering
Soils and rocks are among the most variable of all engineering materials, and as such are highly amenable to a probabilistic treatment. The application of statistical and probabilistic concepts to geotechnical analysis is a rapidly growing area of interes
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Series Editors: The Rectors Giulio Maier - Milan Jean Salençon - Palaiseau Wilhelm Schneider - Wien
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(',7('%< '9*5,)),7+6 &2/25$'26&+22/2)0,1(6*2/'(186$ *25'21$)(1721 '$/+286,(81,9(56,7 0 then means that an apple was selected. Now mathematics can be used on X, ie. if the fruit picking experiment is repeated n times and xi — A'i(s) is the outcome of the first experiment, X2 — ^2(s) the outcome of the second, etc., then the total number of apples picked is Yl7=i ^i- ^o^e that mathematics could not be used on the actual outcomes themselves, e.g. picking an apple is a real event which knows nothing about mathematics nor can it be used in a mathematical expression without first mapping the event to a number. For each outcome s, there is exactly one value oi x — X{s). but different values of s may lead to the same x. The above discussion illustrates in a rather simple way one of the primary motivation for the use of random variables - simply so that mathematics can be used. One other thing might be noticed in the previous paragraph. After the 'experiment' has taken place and the outcome is known, it is referred to using the lower case, Xi. That is Xi has a known fixed value while X is unknown. In other words x is a realization of the random variable X. This is a rather subtle distinction, but it is important to remember that X is unknown. The most that we can say about X is to specify what its likelihoods of taking on certain values are - we cannot say exactly what the value of X is. 3.1 Discrete Random Variables Discrete random variables are those that take on only discrete values {x\ .x^-- • •}• ie. have a countable number of outcomes. Note that countable just means that the outcomes can be numbered 1,2,..., however there could still be an infinite number of them. For example, our experiment might be to count the number of soil tests performed before one yields a cohesion of 200 MPa. This is a discrete random variable since we outcome is one of 0 , 1 , . . . , but the number may be very large or even (in concept) infinite (implying that a soil sample with cohesion 200 MPa was never found). Discrete Probability Distributions As mentioned previously, we can never know for certain what the value of a random variable is (if we do measure it, it becomes a realization - presumably the next measurement is again uncertain until it is measured, and so on). The most that we can say about a random variable is what its probability is of assum
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