Fundamentals of Probability and Statistics for Engineers

It takes the shape of a flat surface bounded by (a 1 ,b 1 ) along the x axis and
(a 2 ,b 2 ) along the y axis. We have seen an application of this bivariate uniform
distribution in Example 3.7 (page 57).Indeed, Example 3.7 gives a typical
situation in which the distribution given by Equation (7.5) is conveniently
applied. Let us give one more example.

Example 7.2.Problem: a warehouse receives merchandise and fills a specific
order for the same merchandise in any given day. Suppose that it receives
merchandise with equal likelihood during equal intervals of time over the
eight-hour working day and likewise for the order to be filled. (a) What is the
probability that theorder will arrive after the merchandise is received and (b)
what is the probability that theorder will arrive within two hours after the
receipt of merchandise?
Answer: let X be the time of receipt of merc handise expressed as a fraction of
an eight-hour working day, and let Y be the time of receipt of the order
similarly expressed. Then

and similarly for fY (y). The joint probability density function (jpdf) of X and Y
is, assuming independence,

and is shown in Figure 7.3.

fXY(x,y)

y

x

1

1

1

Figure 7. 3 Joint probability density function, fX Y (x,y), of X and Y in Example 7.2

194 Fundamentals of Probability and Statistics for Engineers

fX…x†ˆ

1 ; for 0x 1 ;

0 ; elsewhere;



… 7 : 6 †

fXY…x;y†ˆ

1 ; for 0x 1 ;and 0y 1 ;

0 ; elsewhere;