and FY = Fln *X*. To obtain the parameters of the normal

assumed probability density function. It is defined in

terms of the mean X and standard deviation FX as:

random variable *Y*, first the coefficient of variation of

2

FX

1

exp &

2 F2

F 2B

The standard deviation of *Y *is then calculated as:

When fitting the normal distribution, the mean of the

distribution is taken as the expected value of the

random variable. The cumulative distribution function

FY ' Fln *X *'

2

ln 1 % *V*X

for the normal distribution is not conveniently ex-

pressed in closed form but is widely tabulated and can

The standard deviation FY is in turn used to calculate

be readily computed by numerical approximation. It is

the expected value of *Y*:

a built-in function in most spreadsheet programs.

Although the normal distribution has limits of plus and

F2

minus infinity, values more than three or four standard

deviations from the mean have very low probability.

2

Hence, one empirical fitting method is to take

minimum and maximum reasonable values to be at

The density function of the lognormal variate *X *is:

plus and minus three or so standard deviations. The

normal distribution is commonly assumed to

ln *X *& *E Y*

characterize many random variables where the

2

1

1

exp &

coefficient of variation is less than about 30 percent.

FY

2B

2

For levees, these include soil density and drained

friction angle. Where the mean and standard deviation

are the only information known, it can be shown that

The shape of the distribution can be plotted from the

the normal distribution is the most unbiased choice.

above equation. Values on the cumulative distribution

function for *X *can be determined from the cumulative

distribution function of *Y (E[Y]*, FY) by substituting the

(1) When a random variable *X *is lognormally

distributed, its natural logarithm, ln *X*, is normally

distributed. The lognormal distribution has several

properties which often favor its selection to model

certain random variables in engineering analysis:

of the reliability index is based on the assumption that

As *X *is positive for any value of ln *X*,

capacity and demand are normally distributed and the

lognormally distributed random variables

limit state is the event that their difference, the safety

cannot assume values below zero.

margin *S*, is zero. The random variable *S *is then also

normally distributed and the reliability index is the

It often provides a reasonable shape in cases

distance by which *E[S] *exceeds zero in units of FS:

where the coefficient of variation is large

(>30 percent) or the random variable may

$'

'

assume values over one or more orders of

FS

F2 % F2

The central limit theorem implies that the

An alternative formulation (also shown in Figure B-2)

distribution of products or ratios of random

implies that capacity *C *and demand *D *are lognormally

variables approaches the lognormal distri-

distributed random variables. In this case ln *C *and ln

bution as the number of random variables

increases.

safety *FS *as the ratio *C/D*, then ln *FS *= (ln *C*) - (ln

(2) If the random variable *X *is lognormally

reliability index as the distance by which ln *FS*

distributed, then the random variable *Y *= ln *X *is

normally distributed with parameters *E[Y] *= *E*[ln *X*]

B-6

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