James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Section 15.4 Applications of Double Integrals 1023
find the probability that a moviegoer waits a total of less than 20 minutes before taking
his or her seat.
SOLUTION Assuming that both the waiting time X for the ticket purchase and the
waiting time Y in the refreshment line are modeled by exponential probability density
functions, we can write the individual density functions as
f 1 sxd −H 0 if x , 0
1
10 e2xy10 if x > 0
f 2 syd −H 0 if y , 0
1
5 e2yy5 if y > 0
Since X and Y are independent, the joint density function is the product:
1
50
f sx, yd − f 1 sxd f 2 syd −H
e2xy10 e 2yy5
0
if x > 0, y > 0
otherwise
We are asked for the probability that X 1 Y , 20:
PsX 1 Y , 20d − PssX, Yd [ Dd
y
20
D
x+y=20
where D is the triangular region shown in Figure 8. Thus
PsX 1 Y , 20d − y f sx, yd dA − y 20
y 202x 1
50
0 0
e2xy10 e 2yy5 dy dx
D
− 1
50 y 20
fe 2xy10 s25de 2yy5 y−202x
g y−0 dx
0
0
20
x
− 1
10 y 20
e 2xy10 s1 2 e sx220dy5 d dx
0
FIGURE 8
− 1
10 y 20
se 2xy10 2 e 24 e xy10 d dx
0
− 1 1 e 24 2 2e 22 < 0.7476
This means that about 75% of the moviegoers wait less than 20 minutes before taking
their seats.
■
Expected Values
Recall from Section 8.5 that if X is a random variable with probability density function
f, then its mean is
− y`
x f sxd dx
2`
Now if X and Y are random variables with joint density function f, we define the X-mean
and Y-mean, also called the expected values of X and Y, to be
11 1 − y x f sx, yd dA
R 2
2 − y y f sx, yd dA
R 2
Notice how closely the expressions for 1 and 2 in (11) resemble the moments M x and
M y of a lamina with density function in Equations 3 and 4. In fact, we can think of
probability as being like continuously distributed mass. We calculate probability the way
we calculate mass—by integrating a density function. And because the total “probability
mass” is 1, the expressions for x and y in (5) show that we can think of the expected values
of X and Y, 1 and 2 , as the coordinates of the “center of mass” of the probability
distribution.
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