Multidimensional Random Variables - Random Signals and Processes Primer with MATLAB - page 220

This result shows us that if the random variables are independent, we
can obtain the joint density from the marginal densities.
A.3.3. Let us suppose that

Pfx 1 < X 1 x 1 þ d x 1 ; x 2 < X 2 x 2 þ d x 2 g¼Pfy 1 < Y 1
y 1 þ d y 1 ; y 2 < Y 2 y 2 þ d y 2 g:

(3.350)

From ( 3.350 ), we have:

f X 1 X 2 ðx 1 ; x 2 Þ d x 1 d x 2 ¼ f Y 1 Y 2 ðy 1 ; y 2 Þ d y 1 d y 2 : (3.351)

Knowing that the random variables X 1 and X 2 are independent, we get:

f X 1 ðx 1 Þf X 2 ðx 2 Þ d x 1 d x 2 ¼ f Y 1 Y 2 ðy 1 ; y 2 Þ d y 1 d y 2 : (3.352)

The joint distribution of Y 1 and Y 2 is:

y 1

ð

y 2

ð

gðx 1 Þ

ð

gðx 2 Þ

ð

F Y 1 Y 2 ðy 1 ;y 2 Þ¼ f Y 1 Y 2 ðy 1 ;y 2 Þ d y 1 d y 2 ¼ f X 1 X 2 ðx 1 ;x 2 Þ d x 1 d x 2

1 1 1 1

gðx 1 Þ

ð

gðx 2 Þ

ð

y 1

ð

y 2

ð

¼ f X 1 ðx 1 Þf X 2 ðx 2 Þ d x 1 d x 2 ¼ f X 1 ðx 1 Þ d x 1 f X 2 ðx 2 Þ d x 2 ¼F Y 1 ðy 1 ÞF Y 2 ðy 2 Þ:

1 1 1 1

(3.353)

The joint distribution function is equal to the product of the marginal
distributions. As a consequence, the variables Y 1 and Y 2 are independent.
A.3.4. The conditional distribution can be rewritten as:

P f X 1 x 1 ; X 2 x 2 g
PfX 2 x 2 g

F X 1 X 2 ð x 1 ; x 2 Þ
F X 2 ðx 2 Þ

F X 1 ðx 1 jX 2 x 2 Þ¼ ¼ : (3.354)

If the random variables X 1 and X 2 are independent, then

F X 1 X 2 ðx 1 ; x 2 Þ¼F X 1 ðx 1 ÞF X 2 ðx 2 Þ; (3.355)

resulting in:

F X 1 ð x 1 Þ F X 2 ð x 2 Þ
F X 2 ðx 2 Þ

F X 1 ðx 1 jX 2 x 2 Þ¼ ¼ F X 1 ðx 1 Þ: (3.356)

If the random variables X 1 and X 2 are independent, then the conditional
distribution of variable X 1 , given the other variable X 2 , is equal to the
marginal distribution of X 1 .