Fundamentals of Probability and Statistics for Engineers

194 Fundamentals of Probability and Statistics for EngineersIt 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 uniformdistribution in Example 3.7 (page 57). Indeed, Example 3.7 gives a typicalsituation in which the distribution given by Equation (7.5) is convenientlyapplied. Let us give one more example.Example 7.2. Problem: a warehouse receives merchandise and fills a specificorder for the same merchandise in any given day. Suppose that it receivesmerchandise with equal likelihood during equal intervals of time over theeight-hour working day and likewise for the order to be filled. (a) What is theprobability that the order will arrive after the merchandise is received and (b)what is the probability that the order will arrive within two hours after thereceipt of merchandise?Answer: let X be the time of receipt of merchandise expressed as a fraction ofan eight-hour working day, and let Y be the time of receipt of the ordersimilarly expressed. Then 1; for 0 x 1;f X …x† ˆ …7:6†0; elsewhere;and similarly for f Y (y). The joint probability density function (jpdf) of X and Yis, assuming independence, 1; for 0 x 1; and 0 y 1;f XY …x; y† ˆ0; elsewhere;and is shown in Figure 7.3.f XY (x,y)y111xFigure 7. 3 Joint probability density function, f XY (x,y), of X and Y in Example 7.2TLFeBOOK