Page 222 - Fundamentals of Probability and Statistics for Engineers

Some Important Continuous Distributions 205
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

" 2
1 1 x m X
f XY x; y†ˆ exp 2
2 1=2
2 X Y 1 † 2 1 † X
x m X † y m Y † y m Y 2 #) 7:27†
2 ‡ ;
X Y Y
1; 1† < x; y† < 1; 1†:

Equation (7.27) describes the bivariate normal distribution. There are five param-
eters associated with it: m X , m Y , X (greater than 0), Y (greater than 0), and

"j j 1). A typical plot of this joint density function is given in Figure 7.8.

f XY (x,y)

x y

Figure 7.8 Bivariate normal distribution with m X ˆ m Y ˆ 0 and X ˆ Y

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