Fundamentals of Probability and Statistics for Engineers

7.2.3 Multivariate Normal Distribution

Consider two random variables X and Y. They are said to be jointly normal if
their joint density function takes the form

Equation (7.27) describes the bivariate normal distribution. There are five param-
eters associated with it: mX ,mY , X (greater than 0), Y (greater than 0), and
A typical plot of this joint density function is given in F igure 7.8.

x y

fXY(x,y)

Figure 7. 8 Bivariate normal distribution with mX mY 0and XY

Some Important Continuous Distributions 205

fXY…x;y†ˆ

1

2 XY… 1 ^2 †^1 =^2

exp 

1

2 … 1 ^2 †

xmX

X

 " 2

 2 

…xmX†…ymY†

XY

‡

ymY

Y

 2 #)

;

…1;1†<…x;y†<…1;1†:

… 7 : 27 †

 

"jj1).

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