Download Document

8
PROBABILITY
DISTRIBUTIONS
AND STATISTICS
Copyright © Cengage Learning. All rights reserved.
8.1
Distributions of Random
Variables
Copyright © Cengage Learning. All rights reserved.
Probability Distributions of
Random Variables
3
Probability Distributions of Random Variables
Since the random variable associated with an experiment is
related to the outcomes of the experiment, it is clear that
we should be able to construct a probability distribution
associated with the random variable rather than one
associated with the outcomes of the experiment.
Such a distribution is called the probability distribution of a
random variable and may be given in the form of a formula
or displayed in a table that gives the distinct (numerical)
values of the random variable X and the probabilities
associated with these values.
4
Probability Distributions of Random Variables
Thus, if x1, x2, . . . , xn are the values assumed by the
random variable X with associated probabilities P(X = x1),
P(X = x2), . . . ,P(X = xn), respectively, then the required
probability distribution of the random variable X may be
expressed in the form of the table shown in Table 3, where
pi = P(X = xi ), i = 1, 2, . . . , n.
Table 3
5
Probability Distributions of Random Variables
The probability distribution of a random variable X satisfies
1. 0  pi  1
i = 1, 2, . . . , n
2. p1 + p2 + · · · + pn = 1
In the next example, we illustrates the construction and
application of probability distributions.
6
Applied Example 6 – Waiting Lines
The following data give the number of cars observed
waiting in line at the beginning of 2-minute intervals
between 3 P.M. and 5 P.M. on a certain Friday at the drive-in
teller of Westwood Savings Bank and the corresponding
frequency of occurrence.
7
Applied Example 6 – Waiting Lines
cont’d
a. Find the probability distribution of the random variable X,
where X denotes the number of cars observed waiting in
line.
b. What is the probability that the number of cars observed
waiting in line in any 2-minute interval between 3 P.M.
and 5 P.M. on a Friday is less than or equal to 3?
Between 2 and 4, inclusive? Greater than 6?
8
Applied Example 6(a) – Solution
Dividing each number in the second row of the given table
by 60 (the sum of these numbers) gives the respective
probabilities associated with the random variable X when X
assumes the values 0, 1, 2, . . . , 8. (Here, we use the
relative frequency interpretation of probability).
For example,
P(X = 0) =
 .03
9
Applied Example 6(a) – Solution
P(X = 1) =
cont’d
 .15
and so on. The resulting probability distribution is shown
in Table 6.
Table 6
10
Applied Example 6(b) – Solution
cont’d
The probability that the number of cars observed waiting in
line is less than or equal to 3 is given by
P(X  3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)
= .03 + .15 + .27 + .20 = .65
The probability that the number of cars observed waiting in
line is between 2 and 4, inclusive, is given by
P(2  X  4) = P(2) + P(3) + P(4)
= .27 + .20 + .13 = .60
11
Applied Example 6(b) – Solution
cont’d
The probability that the number of cars observed waiting in
line is greater than 6 is given by
P(X > 6) = P(7) + P(8)
= .03 + .02 = .05
12
Histograms
13
Histograms
A probability distribution of a random variable may be
exhibited graphically by means of a histogram.
To construct a histogram of a particular probability
distribution, first 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.
14
Histograms
For example, the histogram of the probability distribution
appearing in Table 4 is shown in Figure 1.
Table 4
Histogram showing the probability distribution for the
number of heads occurring in three coin tosses
Figure 1
15
Histograms
Observe that in histogram, the area of a rectangle associated
with a value of a random variable X gives precisely the
probability associated with the value of X.
This follows because each such 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.
16
Histograms
For example, in the coin-tossing experiment of Example 1,
the event of obtaining at least two heads, which
corresponds to the event (X = 2) or (X = 3), is given by
P(X = 2) + P(X = 3)
and may be obtained from
the histogram depicted in
Figure 1 by adding the
areas associated with the
values 2 and 3 of the
random variable X.
Histogram showing the probability distribution for the
number of heads occurring in three coin tosses
Figure 1
17
Histograms
We obtain
P(X = 2) + P(X = 3) =
This result provides us with a method of computing the
probabilities of events directly from the knowledge of a
histogram of the probability distribution of the random
variable associated with the experiment.
18
Example 7
Suppose the probability distribution of a random variable X
is represented by the histogram shown in Figure 4.
Figure 4
Identify the part of the histogram whose area gives the
probability P(10  X  20).
19
Example 7 – Solution
The event (10  X  20) is the event consisting of outcomes
related to the values 10, 11, 12, . . . , 20 of the random
variable X.
The probability of this event P(10  X  20) is therefore
given by the shaded area of the histogram in Figure 5.
P(10  X  20)
Figure 5
20
Practice
p. 435 Exercises #24
p. 433 Self-Check Exercises #2
21
Assignment
p. 434 Exercises #13-19, 25-29
22