Multidimensional Random Variables - Random Signals and Processes Primer with MATLAB - page 217

0.1

0.05

0

-0.05

-0.1

8

6 10

4

5

2

0

0

-5

y2

y1

Fig. 3.37 Joint PDF in Exercise M.3.7

3.8 Questions

Q.3.1. Can the same marginal density functions possibly result in different joint
density functions?
Q.3.2. In general, is a knowledge of the marginal PDFs sufficient to specify the
joint PDF?
Q.3.3. The random variables X 1 and X 2 are independent. Does it mean that the
transformed variables Y 1 ¼ g 1 ( X 1 ) and Y 2 ¼ g 2 ( X 2 ) are also independent?
Q.3.4. In which condition is the conditional distribution equal to the marginal
distribution F X 1 ðx 1 jX 2 x 2 Þ¼F X 1 ðx 1 Þ ?
Q.3.5. Is the necessary condition f X 1 X 2 ðx 1 ; x 2 Þ¼f X 1 ðx 1 Þf X 2 ðx 2 Þ , for the indepen-
dence of two random variables X 1 and X 2 , also a sufficient condition?
Q.3.6. Is the following true?

Pfa < X 1 b; c < X 2 dg 6¼ F X 1 X 2 ðb; dÞF X 1 X 2 ða; cÞ: (3.336)

Q.3.7. Is that the following probability true?

Pfx 1 < X 1 x 1 þ d x 1 ; x 2 < X 2 x 2 þ d x 2 g¼f X 1 X 2 ðx 1 ; x 2 Þ d x 1 d x 2 : (3.337)

Q.3.8. For three random variables X 1 , X 2 , and X 3 we have:

f X 1 X 2 ðx 1 ; x 2 Þ¼f X 1 ðx 1 Þf X 2 ðx 2 Þ; (3.338)

f X 1 X 3 ðx 1 ; x 3 Þ¼f X 1 ðx 1 Þf X 3 ðx 3 Þ; (3.339)