Page 67 - Fundamentals of Probability and Statistics for Engineers

50 Fundamentals of Probability and Statistics for Engineers

Using again the mass distribution analogy, let one unit of mass be distributed
over the (x, y) plane in such a way that the mass in any given region R is equal
to the probability that X and Y take values in R. Then JPDF F XY (x, y)
represents the total mass in the quadrant to the left and below the point
(x, y), inclusive of the boundaries. In the case where both X and Y are discrete,
all the mass is concentrated at a finite or countably infinite number of points in
the (x, y) plane as point masses. When both are continuous, the mass is
distributed continuously over the (x, y) plane.
It is clear from the definition that F XY (x, y) is nonnegative, nondecreasing in
x and y, and continuous to the left with respect to x and y. The following
properties are also a direct consequence of the definition:

9
F XY 1; 1† ˆ F XY 1; y†ˆ F XY x; 1† ˆ 0; >
>
>
>
F XY ‡1; ‡1† ˆ 1; =
3:17†
F XY x; ‡1† ˆ F X x†; >
>
>
F XY ‡1; y†ˆ F Y y†: >
;
For example, the third relation above follows from the fact that the joint event
X x \ Y ‡1 is the same as the event X x, since Y ‡1 is a sure event.
Hence,
F XY x; ‡1† ˆ P X x \ Y ‡1† ˆ P X x†ˆ F X x†:

Similarly, we can show that, for any x 1 , x 2 , y 1 , and y 2 such that x 1 < x 2 and
y 1 < y 2 , the probability P9x 1 < X x 2 \ y 1 < Y y 2 ) is given in terms of
F XY (x, y) by

P x 1 < X x 2 \ y 1 < Y y 2 †ˆ F XY x 2 ; y 2 † F XY x 1 ; y 2 †
3:18†
F XY x 2 ; y 1 †‡ F XY x 1 ; y 1 †;

which shows that all probability calculations involving random variables X and
Y can be made with the knowledge of their JPDF.
Finally, we note that the last two equations in Equations (3.17) show that
distribution functions of individual random variables are directly derivable
from their joint distribution function. The converse, of course, is not true. In
the context of several random variables, these individual distribution functions
are called marginal distribution functions. For example, F X (x) is the marginal
distribution function of X.
The general shape of F XY (x, y) can be visualized from the properties given in
Equations (3.17). In the case where X and Y are discrete, it has the appearance of
a corner of an irregular staircase, something like that shown in Figure 3.10. It rises
from zero to the height of one in the direction moving from the third quadrant to the

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