Multidimensional Random Variables - Random Signals and Processes Primer with MATLAB - page 163

Therefore, as opposed to one-dimensional random variable, where the probability
represents the area below the density function, for two-dimensional variable, the
probability is a volume below a joint density function.
The properties of one-dimensional PDF can be easily applied to the
two-dimensional PDF:
P.1

f X 1 X 2 ðx 1 ; x 2 Þ 0 : (3.36)

1 1

P.2

f X 1 X 2 ðx 1 ; x 2 Þ d x 1 d x 2 ¼ 1 :

(3.37)

1 1

x 1

ð

x 2

ð

P.3

F X 1 X 2 ðx 1 ; x 2 Þ¼ f X 1 X 2 ðx 1 ; x 2 Þ d x 1 d x 2 : (3.38)

1 1

ð 1

x 1

P.4

F X 1 ðx 1 Þ¼ f X 1 X 2 ðx 1 ; x 2 Þ d x 1 d x 2 : (3.39)

1 1

x 2

ð 1

F X 2 ðx 2 Þ¼ f X 1 X 2 ðx 1 ; x 2 Þ d x 1 d x 2 :

(3.40)

1 1

x 22

ð

x 12

ð

P.5

Pfx 11 < X 1 x 12 ; x 21 < X 2 x 22 g¼ f X 1 X 2 ðx 1 ; x 2 Þ d x 1 d x 2 : (3.41)

x 21 x 11

1

P.6

f X 1 ðx 1 Þ¼ f X 1 X 2 ðx 1 ; x 2 Þ d x 2 : (3.42a)

1

1

f X 2 ðx 2 Þ¼ f X 1 X 2 ðx 1 ; x 2 Þ d x 1 : (3.42b)

1

Example 3.2.6 A two-dimensional random variable has a constant joint density
function in the shaded area A , shown in Fig. 3.8 . Find the expression for the joint
density function.