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

Section 8.5 Probability 579
SOLUTION
(a) Since IQ scores are normally distributed, we use the probability density function
given by Equation 3 with − 100 and − 15:
Ps85 < X < 115d − y 115
85
1
15s2
e 2sx2100d 2 ys2?15 2d dx
Recall from Section 7.5 that the function y − e 2x 2 doesn’t have an elementary antiderivative,
so we can’t evaluate the integral exactly. But we can use the numerical
integration capability of a calculator or computer (or the Midpoint Rule or Simpson’s
Rule) to estimate the integral. Doing so, we find that
Ps85 < X < 115d < 0.68
So about 68% of the population has an IQ score between 85 and 115, that is, within one
standard deviation of the mean.
(b) The probability that the IQ score of a person chosen at random is more than 140 is
PsX . 140d − y`
140
1
15s2
e 2sx2100d2 y450
dx
To avoid the improper integral we could approximate it by the integral from 140 to 200.
(It’s quite safe to say that people with an IQ over 200 are extremely rare.) Then
PsX . 140d < y 200
140
1
15s2
e 2sx2100d2 y450
dx < 0.0038
Therefore about 0.4% of the population has an IQ score over 140.
n
1. Let f sxd be the probability density function for the lifetime of a
manufacturer’s highest quality car tire, where x is measured in
miles. Explain the meaning of each integral.
(a) y 40,000
f sxd dx
(b) y`
f sxd dx
30,000
25,000
2. Let f std be the probability density function for the time it takes
you to drive to school in the morning, where t is measured in
minutes. Express the following probabilities as integrals.
(a) The probability that you drive to school in less than
15 minutes
(b) The probability that it takes you more than half an hour to
get to school
3. Let f sxd − 30x 2 s1 2 xd 2 for 0 < x < 1 and f sxd − 0 for all
other values of x.
(a) Verify that f is a probability density function.
(b) Find P(X < 1 3).
;
4. The density function
f sxd −
e 32x
s1 1 e 32x d 2
is an example of a logistic distribution.
(a) Verify that f is a probability density function.
(b) Find P(3 < X < 4d.
(c) Graph f. What does the mean appear to be? What about
the median?
5. Let f sxd − cys1 1 x 2 d.
(a) For what value of c is f a probability density function?
(b) For that value of c, find Ps21 , X , 1d.
6. Let f sxd − ks3x 2 x 2 d if 0 < x < 3 and f sxd − 0 if x , 0
or x . 3.
(a) For what value of k is f a probability density function?
(b) For that value of k, find PsX . 1d.
(c) Find the mean.
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