Fundamentals of Probability and Statistics for Engineers

Random Variables and Probability Distributions 57We also note the following important properties:Z 11Z 11f XY …x; y†dxdy ˆ 1;…3:27†Z 11f XY …x;y†dy ˆf X …x†;…3:28†Z 1f XY …x;y†dx ˆf Y …y†:1…3:29†Equation (3.27) follows from Equation (3.25) by letting x, y !‡1, ‡1, andthis shows that the total volume under the f XY (x,y) surface is unity. To givea derivation of Equation (3.28), we know thatZ 1 Z xF X …x† ˆF XY …x; ‡1† ˆ f XY …u; y†dudy:1 1Differentiating the above with respect to x gives the desired result immediately.The density functions f X (x) and f Y (y) in Equations (3.28) and (3.29) are nowcalled 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 between9 a.m. and 10 a.m., each not waiting more than 10 minutes for the other. If alltimes of arrival within the hour are equally likely for each person, and if theirtimes of arrival are independent, find the probability that they will meet.Answer: for a single continuous random variable X that takes all values overan interval a to b with equal likelihood, the distribution is called a uniformdistribution and its density function f X (x) has the form8< 1; for a x b;f X …x† ˆ b a …3:30†:0; elsewhere:The height of f X (x) over the interval (a, b) must be 1/(b a) in order that thearea is 1 below the curve (see Figure 3.14). For a two-dimensional case asdescribed in this example, the joint density function of two independent uniformlydistributed 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 past9 a.m. The jpdf f XY (x, y) thus takes the form shown in Figure 3.15 and is given by8< 1; for 0 x 60; and 0 y 60;f XY …x; y† ˆ 3600:0; elsewhere:TLFeBOOK