Constructing multivariate distributions for soil parameters - Risk and Reliability in Geotechnical Engineering - page 23

where COV(,) denotes the covariance; σ i is the standard deviation of Yi; i ; and δ ij is the Pearson
(product-moment) correlation coefficient between Yi i and Y j . The mean and standard devia-
tion have been defined in the preceding discussion on a single normal random variable. The
only new concept required to describe a bivariate normal random vector is the covariance
(or correlation coefficient).
The PDF for the bivariate normal distribution is

1 −− − ( y µ )

T

C

− 1

( y µ )

f () y = exp (1.36)

2

2

2 π ⋅ C

where | C | is the determinant of C matrix; the superscript '-1' symbolically denotes the
matrix inverse.

1.3.2.1 Bivariate standard normal
Consider the case where X is standard normal (μ = 0 and σ = 1). Hence, the two random
variables (X 1 , X 2 ) are both standard normal, and their bivariate normal distribution is sim-
plified into

1 − xx

T

C

− 1

() =

f x exp

(1.37)

2

2

2 π ⋅ C

where

1 δ

C =

12

(1.38)

δ 1

12

1.3.2.2 Correlation coefficient
The Pearson product-moment correlation coefficient is defined as

( )

COVX X ,

i j

(1.39)

δ =

ij

σσ ⋅

i j

δ ij quantifies the degree of linear correlation between Xi i and X j . It is equal to 1 for perfectly
positive linear correlation, −1 for perfectly negative linear correlation, and 0 for uncor-
related Xi i and X j . Figure 1.12 shows the contour plots for two bivariate standard normal
distributions with δ 12 equal to 0.9 and 0 (random sequence initiated using randn['state', 13]).
The random samples from the distributions are also shown. For δ 12 = 0.9, a strong positive
linear correlation exists between X 1 and X 2 . For δ 12 = 0, X 1 and X 2 are uncorrelated.

1.3.3 estimation of δ 12
1.3.3.1 Method of moments
Given samples of (X 1 , X 2 ) that are bivariate standard normal, it is possible to estimate δ 12 .
The most common method is the method of moments: