Constructing multivariate distributions for soil parameters - Risk and Reliability in Geotechnical Engineering - page 20

1. Simulate two independent standard uniform random variables (U 1 , U 2 ).

2. Simulate two independent standard normal random variables (X 1 , X 2 )

(1.30)

X =− ⋅ 2 ln()cos( U 2 π UX ) = − 2 ln( U ) in( ⋅ 2 π U )

1 1 2 2 1 2

The Marsaglia-Bray method circumvents the evaluation of trigonometric functions in
Equation 1.30 (Marsaglia and Bray 1964):

1. Simulate two independent standard uniform random variables (U 1 , U 2 ). Compute

VU = 2 − 1 VU = 2 − 1 (1.31)

1 1 2 2

2. Compute

1 2 2 2

(1.32)

RVV = +

If |R| ≥ 1 or R = 0, repeat step 1.

3. Simulate two independent standard normal random variables (X 1 , X 2 )

− 2 ln() R

2

− 2 ln() R

2

(1.33)

XV = XV =

1 1 2 2

R

2

R

2

1.2.4.3 Simulating normal random variable Y
The normal random variable Y with mean, μ, and standard deviation, σ, can be simulated
using

Y =+⋅ µσ X (1.34)

One can also simulate a table of iid Y samples containing n rows and d columns using
the MATLAB function normrnd (μ, σ, n , d ), without the need of going through the above
details.

1.3 bIVarIate norMal VeCtor

1.3.1 bivariate data
What are bivariate data? It is instructive to distinguish between univariate and bivari-
ate data. One may be tempted to say that if univariate data refer to one column of data
(e.g., undrained shear strength), then bivariate data simply refer to two columns of data
(e.g., undrained shear strength and undrained modulus). However, there is more than
one method of collecting two columns of data. One method is to extract the undrained
shear strength and undrained modulus from the same stress-strain curve. The undrained
shear strength ( s u ) is the ultimate stress. The undrained modulus (E u ) can be defined as the
secant Young's modulus at a stress level equal to one-third of the ultimate stress. The crux
here is that s u and E u are properties of a single soil sample. It is meaningful to take the