Modeling and simulation of bivariate distribution of shear strength parameters using copulas - Risk and Reliability in Geotechnical Engineering - page 81

(a) 3

(b) 3

2

2

1

1

0 0

-1 -1

-2

-2

-3 -3

-3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3

x 1

x 1

(c) 3

2

1

0

-1

-2

-3

-3 -2 -1 0 1 2 3

x 1

Figure 2.1 Contour plots for the bivariate distributions of X 1 and X 2 using a Gaussian copula. (a) θ = − 0.5,
τ = − 1/3, (b) θ = 0, τ = 0, and (c) θ = 0.5, τ = 1/3.

product-moment correlation coefficient, measures the degree of linear dependence between
X 1 and X 2 as (e.g., Mari and Kozt 2001)

= Cov

(, XX

1

)

ρ

2

(2.8)

σσ

12

where Cov( X 1 , X 2 ) is the covariance between X 1 and X 2 ; σ 1 and σ 2 are the standard devia-
tions of X 1 and X 2 , respectively. The effect of Pearson's rho on the shape of the bivariate
standard normal distributions can be found in Figure 1.12 of Chapter 1 . By definition of the
covariance, Pearson's rho can be further expressed as

+∞ +∞

x − µ x − µ

∫ ∫

ρ =

1 1 2 2

fx xxx (, ) d d (2.9)

12 12

σ σ

1 2

−∞ −∞

where μ 1 and μ 2 are the means of X 1 and X 2 , respectively. Substituting Equation 2.6 into
Equation 2.9, the integral relation between ρ and θ can be obtained as