Marginal Normality Does Not Imply Bivariate Normality

ρ (X, Y)

-0.7

mix of normal distributions

This Demonstration shows\b an example of a bivariate distribution that has standard normal margins but is not bivariate normal. So as the title says, marginal normality does not imply bivariate normality. In contrast, it is simple to show that bivariate normality implies marginal normality.

The relevant densities are:

f

X

(x)=

1

2π

exp-

2

x

2

,

f

Y

(y)=

1

2π

exp-

2

y

2

,

f

X,Y

(x,y)=

1

2π

1-

2

ρ

exp-

2

x

+

2

y

-2xyρ

21-

2

ρ



+exp-

2

x

+

2

y

+2xyρ

21-

2

ρ



.