Fundamentals of Probability and Statistics for Engineers

We also note the following important properties:

Equation (3.27) follows from Equation (3.25) by letting and
this shows that the total volume under the fX Y (x,y) surface is unity. To give
a derivation of Equation (3.28), we know that

Differentiating the above with respect to x gives the desired result immediately.

The density functions fX (x) and f (^) Y (y) in Equations (3.28) and (3.29) are now
called the marginal density functions of X and Y, respectively.
Example 3.7.Problem: a boy and a girl plan to meet at a certain place between
9 a.m. and 10 a.m., each not waiting more than 10 minutes for the other. If all
times of arrival within the hour are equally likely for each person, and if their
times of arrival are independent, find the probability that they will meet.
Answer: for a single co ntinuous random variable X that takes all values over
an interval a to b with equal likelihood, the distribution is called a uniform
distribution and its den sity fu nction fX(x) has the form
The height of fX (x) over the interval (a, b) must be 1/(b a) in order that the
area is 1 below the curve (see F igure 3.14). F or a two-dimensional case as
described in this example, the joint density function of two independent uni-
formly distributed random variables is a flat surface within prescribed bounds.
The volume under the surface is unity.
Let the boy arrive at X minutes past 9 a.m. and the girl arrive at Y minutes past
9 a.m. The jpdf fX Y (x, y) thus takes the form shown in Figure 3.15 and is given by
Random Variables and Probability D istributions 57

Z 1

1

Z 1

1

fXY…x;y†dxdyˆ 1 ; … 3 : 27 †

Z 1

1

fXY…x;y†dyˆfX…x†;… 3 : 28 †

Z 1

1

fXY…x;y†dxˆfY…y†:… 3 : 29 †

x,y!‡1,‡1,

FX…x†ˆFXY…x;‡1†ˆ

Z 1

1

Zx

1

fXY…u;y†dudy:

fX…x†ˆ

1

ba

; foraxb;

0 ; elsewhere:

8

<

:

… 3 : 30 †



fXY…x;y†ˆ

1

3600

; for 0x 60 ;and 0y 60 ;

0 ; elsewhere:

8

<

: