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

578 Chapter 8 Further Applications of Integration
Notice the result of Example 4(b): Even though the mean waiting time is 5 minutes,
only 37% of callers wait more than 5 minutes. The reason is that some callers have to
wait much longer (maybe 10 or 15 minutes), and this brings up the average.
Another measure of centrality of a probability density function is the median. That is
a number m such that half the callers have a waiting time less than m and the other callers
have a waiting time longer than m. In general, the median of a probability density
function is the number m such that
y`
m f sxd dx − 1 2
This means that half the area under the graph of f lies to the right of m. In Exercise 9 you
are asked to show that the median waiting time for the company described in Example 4
is approximately 3.5 minutes.
Normal Distributions
Many important random phenomena—such as test scores on aptitude tests, heights and
weights of individuals from a homogeneous population, annual rainfall in a given location—are
modeled by a normal distribution. This means that the probability density
function of the random variable X is a member of the family of functions
3
f sxd − 1
s2
e 2sx2d2 ys2 2 d
The standard deviation is denoted by
the lowercase Greek letter (sigma).
You can verify that the mean for this function is . The positive constant is called
the stan dard deviation; it measures how spread out the values of X are. From the bellshaped
graphs of members of the family in Figure 5, we see that for small values of
the values of X are clustered about the mean, whereas for larger values of the values
of X are more spread out. Statisticians have methods for using sets of data to estimate
and .
y
1
s=
2
s=1
s=2
FIGURE 5
Normal distributions
0 m
x
The factor 1yss2 d is needed to make f a probability density function. In fact, it
can be verified using the methods of multivariable calculus that
y
0.02
y`
2`
1
s2 e2sx2d2 ys2 2 d
dx − 1
0.01
0 60
FIGURE 6
80 100 120 140
x
Example 5 Intelligence Quotient (IQ) scores are distributed normally with mean
100 and standard deviation 15. (Figure 6 shows the corresponding probability density
function.)
(a) What percentage of the population has an IQ score between 85 and 115?
(b) What percentage of the population has an IQ above 140?
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