Applied Calculus - 7e - c 10.pdf
(
999 KB
)
Pobierz
10-W3979 11/2/06 1:27 PM Page 653
10
Probability and Calculus
W
here do we go from here?
Some of the top 10% of this
senior class will further their
education at one of the campuses
of the state university system.
In Example 6, page 688, we
determine the minimum grade
point average a senior needs to
be eligible for admission to one
of the state universities.
T
HE SYSTEMATIC STUDY of probability began in the seventeenth century,
when certain aristocrats wanted to discover superior strategies to use
in the gaming rooms of Europe. Some of the best mathematicians of the
period were engaged in this pursuit. Since then, applications of probability
have evolved in virtually every sphere of human endeavor that contains an
element of uncertainty.
In this chapter we take a look at the role of calculus in the study of
probability
involving a
continuous random variable.
We see how probability
can be used to find the average life span of a certain brand of color televi-
sion tube, the average waiting time for patients in a health clinic, and the
percentage of a current Mediterranean population who have serum choles-
terol levels between 160 and 180 mg/dL—to name but a few applications.
653
10-W3979 11/2/06 1:27 PM Page 654
654
10
PROBABILITY AND CALCULUS
10.1
Probability Distributions of Random Variables
Probability
We begin by mentioning some elementary notations important in the study of prob-
ability. For our purpose, an
experiment
is an activity with observable results called
outcomes
, or
sample points
. The totality of all outcomes is the
sample space
of the
experiment. A subset of the sample space is called an
event
of the experiment.
Now, given an event associated with an experiment, our primary objective is to
determine the likelihood that this event will occur. This likelihood, or
probability of
the event
, is a number between 0 and 1 and may be viewed as the proportionate
number of times that the event will occur if the experiment associated with the event
is repeated indefinitely under independent and similar conditions. By way of exam-
ple, let’s consider the simple experiment of tossing an unbiased coin and observing
whether it lands “heads” (H) or “tails” (T). Since the coin is
unbiased
, we see that
the probability of each outcome is
1
2
,
abbreviated
1
2
1
2
P
1
H
2
and
P
1
T
2
Discrete Random Variables
In many situations it is desirable to assign numerical values to the outcomes of an
experiment. For example, suppose an experiment consists of casting a die and ob-
serving the face that lands up. If we let
X
denote the outcome of the experiment, then
X
assumes one of the values 1, 2, 3, 4, 5, or 6. Because the values assumed by
X
depend on the outcomes of a chance experiment, the outcome
X
is referred to as a
random variable
. In this case the random variable
X
is also said to be
finite discrete
since it can assume only a finite number of integer values.
The function
P
that associates with each value of a random variable its proba-
bility of occurrence is called a
probability function
. It is
discrete
since its domain
consists of a finite set. In general, a discrete probability function is defined as
follows.
Discrete Probability Function
A
discrete probability function
P
with domain {
x
1
,
x
2
, . . . ,
x
n
} satisfies these
conditions:
1.
0
P
(
x
i
)
1
p
2.
P
(
x
1
)
P
(
x
2
)
P
(
x
n
)
1
Note
These conditions imply that the probability assigned to an outcome must
be nonnegative and less than or equal to 1 and that the sum of all probabilities must
be 1.
Histograms
A discrete probability function or distribution may be exhibited graphically by
means of a
histogram
. To construct a histogram of a probability distribution, first
10-W3979 11/2/06 1:27 PM Page 655
10.1
655
PROBABILITY DISTRIBUTIONS OF RANDOM VARIABLES
locate the values of the random variable on a number line. Then, above each such
number, erect a rectangle with width 1 and height equal to the probability associated
with that value of the random variable.
For example, consider the data in Table 1, the number of cars observed waiting
in line at 2-minute intervals between 3 and 5
P
.
M
. on a certain Friday at the drive-in
teller of the Westwood Savings Bank and the corresponding frequency of occur-
rence. If we divide each number on the right of the table by 60 (the sum of these
numbers), then we obtain the respective probabilities associated with the random
variable
X
, when
X
assumes the values 0, 1, 2, . . . , 8. For example,
TABLE 1
Number
Frequency
of
of
Cars
Occurrence
0
2
1
9
2
16
3
12
4
8
5
6
2
60
1
2
P
X
0
.03
6
4
7
2
9
60
8
1
1
2
p
P
X
1
.15,
The resulting probability distribution is shown in Table 2.
The histogram associated with this probability distribution is shown in Figure 1.
Observe that the area of a rectangle in a histogram is associated with a value of
a random variable
X
and that this area gives precisely the probability associated with
that value of
X
. This follows since each rectangle, by construction, has width 1 and
height corresponding to the probability associated with the value of the random
variable.
Another consequence arising from the method of construction of a histogram is
that the probability associated with more than one value of the random variable
X
is
given by the sum of the areas of the rectangles associated with those values of
X
. For
example, the probability that three or four cars are in line is given by
TABLE
2
Probability Distribution for the
Random Variable
X
x
P
(
X
x
)
0
.03
1
.15
2
.27
3
.20
4
.13
5
.10
6
.07
P
(
X
3)
P
(
X
4)
7
.03
which may be obtained from the histogram by adding the areas of the rectangles as-
sociated with the values 3 and 4 of the random variable
X
. Thus, the required proba-
bility is
8
.02
P
(
X
3)
P
(
X
4)
.20
.13
.33
0.3
0.2
0.1
FIGURE
1
Probability distribution of the number of
cars waiting in line.
x
012345678
10-W3979 11/2/06 1:27 PM Page 656
656
10
PROBABILITY AND CALCULUS
Continuous Random Variables
A random variable
x
that can assume any value in an interval is called a
continuous
random variable
. Examples of continuous random variables are the life span of a
light bulb, the length of a telephone call, the length of an infant at birth, the daily
amount of rainfall in Boston, and the life span of a certain plant species. For the
remainder of this chapter, we will be interested primarily in continuous random
variables.
Consider an experiment in which the associated random variable
x
has the inter-
val [
a
,
b
] as its sample space. Then an event of the experiment is any subset of [
a
,
b
].
For example, if
x
denotes the life span of a light bulb, then the sample space asso-
ciated with the experiment is [0,
), and the event that a light bulb selected at ran-
dom has a life span between 500 and 600 hours inclusive is described by the interval
[500, 600] or, equivalently, by the inequality 500
600. The probability that
the light bulb will have a life span of between 500 and 600 hours is denoted by
P
(500
x
600).
In general, we will be interested in computing
P
(
a
x
x
b
), the probability that
a random variable
x
assumes a value in the interval
a
b
. This computation is
based on the notion of a probability density function, which we now introduce.
x
Probability Density Function
A
probability density function
of a continuous random variable
x
in an inter-
val
I
, where
I
may be bounded or unbounded, is a nonnegative function
f
hav-
ing the following properties.
1.
The total area of the region under the graph of
f
is equal to 1 (Figure 2a).
2.
The probability that an observed value of the random variable
x
lies in the
interval [
a
,
b
] is given by
b
P
1
a
x
b
2
f
1
x
2
dx
a
(Figure 2b).
y
y
P
(
a
≤
x
≤
b
)
R
x
x
a
b
(a)
Area of
R
1
(b)
P
(
a
x
b
) is the probability that an
outcome of an experiment will lie between
a
and
b.
FIGURE
2
Notes
1.
A probability density function of a random variable
x
may be constructed using
methods that range from theoretical considerations of the problem on the one ex-
treme to an interpretation of data associated with the experiment on the other.
10-W3979 11/2/06 1:27 PM Page 657
10.1
657
PROBABILITY DISTRIBUTIONS OF RANDOM VARIABLES
2.
Property 1 states that the probability that a continuous random variable takes on a
value lying in its range is 1, a certainty, which is expected.
3.
Property 2 states that the probability that the random variable
x
assumes a value
in an interval
a
x
b
is given by the area of the region between the graph of
f
and the
x
-axis from
x
b
. Because the area under one point of the graph
of
f
is equal to zero, we see immediately that
P
(
a
a
to
x
x
b
)
P
(
a
x
b
)
P
(
a
x
b
)
P
(
a
x
b
).
EXAMPLE 1
Show that each of the following functions satisfies the nonnega-
tivity condition and Property 1 of probability density functions.
2
27
x
1
2
1
2
1
2
a.
f
x
x
1
1
x
4
1
3
e
1
2
1
1/3
2
x
1
2
b.
f
x
0
x
Solution
a.
Since the factors
x
and (
x
1) are both nonnegative, we see that
f
(
x
)
0
on [1, 4]. Next, we compute
4
4
2
27
x
2
27
x
2
1
x
1
2
dx
1
x
2
dx
1
1
4
2
27
a
1
1
2
x
2
3
x
3
b`
1
2
27
ca
64
3
a
1
1
2
8
b
3
bd
2
27
a
27
2
b
1
showing that Property 1 of probability density functions holds as well.
b.
First,
1
3
e
1
1/3
2
x
f
1
x
2
0
for all values of
x
in [0,
). Next,
b
1
1
1
1/3
2
x
dx
1
1/3
2
x
dx
3
e
lim
b
S
3
e
0
0
b
1
1/3
2
x
lim
b
S
e
`
0
1
1/3
2
b
lim
b
S
1
e
1
2
1
1
3
e
1
1/3
2
x
so the area under the graph of
f
1
x
2
is equal to 1, as we set out
to show.
EXAMPLE 2
a.
Determine the value of the constant
k
such that the function
f
(
x
)
kx
2
is a
probability density function on the interval [0, 5].
Plik z chomika:
Januszek66
Inne pliki z tego folderu:
Back Seat_ A Mumbai Tale - Aditya Kripalani.mobi
(755 KB)
Brief Wondrous Life of Oscar Wao, The - Junot Diaz.opf
(3 KB)
Don't Make Me Think, Revisited_ - Steve Krug.mobi
(9256 KB)
M. T. Anderson - Norumbegan 03 - The Empire of Gut and Bone # (v5.0).epub
(2209 KB)
M. T. Anderson - Norumbegan 02 - The Suburb Beyond the Stars # (v5.0).epub
(2105 KB)
Inne foldery tego chomika:
Dokumenty
Galeria
LUDLUM ROBERT
Midi - Kar
Mszał Rzymski PL
Zgłoś jeśli
naruszono regulamin