Probability Distributions

16.3 Probability Distributions

We note in Fig. 16.5 that the type IV distribution and the symmetrical type VII are unbounded (infinite) below and above. From the point of view of civil engineering applications this represents an extremely unlikely distribution. For example, all parameters or properties (see Table 16.1 ) are positive numbers (including zero).

The type V (the lognormal distribution), type III (the gamma), and type VI distributions are unbounded above. Hence, their use would be confined to those variables with an extremely large range of possible values. Some examples are the coefficient of permeability, the state of stress at various points in a body, the distribution of annual rainfall, and traffic variations.

The normal (Gaussian) distribution [ b(1) = 0, b(2) = 3], even though it occupies only a single point in the universe of possible distributions, is the most frequently used of probability models. Some asso- ciated properties were given in Table 16.2 . The normal distribution is the well-known symmetrical bell- shaped curve (see Fig. 16.6 ). Some tabular values are given in Table 16.3 . The table is entered by forming the standardized variable z for the normal variate x as

z = ----------- x – x

sx []

Tabular values yield the probabilities associated with the shaded areas shown in the figure: area = y (z).

Example 16.8

Assuming the strength s of concrete to be a normal variate with an expected value of – s = 6000 psi and a coefficient of variation of 14%, find (a) P [6000 £ s £ 7000], (b) P [5000 £ s £ 6000], and (c) P [s ≥ 7000]. Solution. The standard deviation is s[s] = (0.14)(6000) = 840 psi. Hence (see Fig. 16.6 ), • z = |(7000 – 6000)/840| = 1.19, y (1.19) = 0.383.

• By symmetry, P[5000 £ s £ 6000] = 0.383. • P[s ≥ 7000] = 0.500 – 0.383 = 0.117.

As might be expected from its name, the lognormal distribution (type V) is related to the normal distribution. If x is a normal variate and x = ln y or y = exp (x) then y is said to have a lognormal distribution. It is seen that the distribution has a minimum value of zero and is unbounded above. The probabilities associated with lognormal variates can be obtained very easily from those of mathematically corresponding normal variates (see Table 16.3 ). If E(y) and V(y) are the expected value and coefficient of variation of a lognormal variate, the corresponding normal variate x will have the expected value and standard deviation [Benjamin and Cornell, 1970]:

( 2 sx [] ) ln { 1 [ Vy () ] } (16.18a)

Ex [] =

Ey

ln 2 () – ( sx [] ) § 2 (16.18b)

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TABLE 16.3

Standard Normal Probability

– 2 (2p) exp – z [ ] p ( z )

for z > 2.2, y( z )艐 1 1 –1/2

2 2 2 z = x - x s[ x ]

Area = y ( z )

© 2003 by CRC Press LLC

Example 16.9

A live load of 20 kips is assumed to act on a footing. If the loading is assumed to be lognormally distributed, estimate the probability that a loading of 40 kips will be exceeded.

Solution. From Table 16.1 we have that the coefficient of variation for a live load, L, can be estimated as 25%; hence, from Eqs. (16.18) we have for the corresponding normal variate, x,

sx 2 [] 1 = ln 0.25 [ + ( 0.25 ) ] = (16.19a) and

Ex 2 [ ] 20 0.25 = ln 2.96 – ( ) § 2 = (16.19b) As x = ln L, the value of the normal variate x equivalent to 40 K is ln 40 = 3.69. We seek the equivalent

normal probability P[3.69 £ x]. The standardized normal variate is z = (3.69 – 2.96)/0.25 = 2.92. Hence, using Table 16.3 ,

P [ 40 £ L ] = 0.50 – y 2.92 ( ) = 0.500 0.498 – = 0.002 (16.20) As was noted with respect to Fig. 16.5 many and diverse distributions (as well as the normal, lognormal,

uniform, and exponential) can be obtained from the very versatile beta distribution. The beta distribution is treated in great detail by Harr [1977, 1987]. The latter reference also contains FORTRAN programs for beta probability distributions. Additional discussion is given below following Example 16.11.