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MTH/STA 561
RANDOM VARIABLES
1.10 Random Variables
Up to now our discussion of probability theory has been concerned with situations in
which the sample points of a sample space are arbitrary objects that may not be numbers.
In the experiment of tossing a coin, for example, the outcome is either head or tail which is
not a number. For many purposes, it is convenient to convert the outcome of an experiment
into a numerical value, in particular to be able to use the familiar structure of the real
numbers. Often we are not interested in the details associated with each sample point but
only in some numerical description of the outcome. For instance, the sample space with a
detailed description of each possible outcome of tossing a coin three times may be written
S = fHHH; HHT; HT H; HT T; T HH; T HT; T T H; T T T; g :
If we are concerned only with the number of heads that appear, then a numerical value of
0, 1, 2, or 3 will be assigned to each sample point.
Thus, we are concerned only with experiments in which the outcomes either are numerical themselves or have a numerical value assigned to them. The sample space or the induced
sample space (in the latter case) is then a space of numbers or a pace of vectors of numbers,
and the structure of such spaces allows analyses and descriptions that may not be possible
in the general case.
The number of heads in the three tosses of a coin is referred to as a random variable,
which is a random quantity determined, at least in part, by some chance mechanism on the
outcome of the experiment. The most frequent use of probability theory is the description
of random variables. We will study how random variables are de…ned and how to describe
their behavior, and, subsequently, how to specify their probability laws to various ways.
Suppose that we have an experiment associated with a sample space. Any random variable de…ned on the sample space can be referred to as a rule that associates a real number
with each element in the sample space. In other words, a random variable is a function
whose domain is the sample space and whose range is a set of real numbers. The formal
de…nition may be stated as follows.
De…nition 1. A random variable is a numerically valued function whose values correspond to the various outcomes of an experiment; that is, whose domain is the sample space.
We shall use a capital letter, say Y , to denote a random variable and its corresponding
lower case y to denote one of its values.
Example 1. Consider an experiment of tossing a pair of fair coins. The sample space is
S = fHH; HT; T H; T T g :
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Let Y be the number of heads observed. Then we see that
Sample Point
HH
HT
TH
TT
Y
2
1
1
0
Thus,
fY = 0g = fT T g ;
fY = 1g = fHT; T Hg ;
fY = 2g = fHHg :
Hence,
1
4
2
1
P fY = 1g =
=
4
2
1
P fY = 2g =
:
4
P fY = 0g =
Example 2. Consider an experiment of tossing three fair coins. The sample space is
S = fHHH; HHT; HT H; HT T; T HH; T HT; T T H; T T T g :
Let Y be the number of heads observed. We note that the random variable Y is a function
from S into a set of nonnegative integers de…ned by
Y (HHH) = 3
Y (T HH) = 2
Y (HHT ) = 2
Y (T HT ) = 1
Y (HT H) = 2
Y (T T H) = 1
Y (HT T ) = 1
Y (T T T ) = 0:
Thus, each possible value of Y represents an event of the sample space S. Thus,
fY
fY
fY
fY
= 0g
= 1g
= 2g
= 3g
=
=
=
=
fT T T g ;
fHT T; T HT; T T Hg ;
fHHT; HT H; T HHg ;
fHHHg :
Hence,
1
8
3
P fY = 1g =
8
3
P fY = 2g =
8
1
P fY = 3g =
:
8
P fY = 0g =
2
Example 3. A urn contains 3 white balls and 4 red balls. We draw two balls in
succession without replacement from the urn and let Y be the number of white balls drawn.
Then the sample space is
S = fW W; W R; RW; RRg :
Since Y is the number of white balls drawn, we see that
Sample Point
WW
WR
RW
RR
Y
2
1
1
0
Thus,
fY = 0g = fRRg
fY = 1g = fW R; RW g
fY = 2g = fW W g :
Hence,
P fY = 0g = P fRRg =
4 3
2
=
7 6
7
P fY = 1g = P fW Rg + P fRW g =
P fY = 2g = P fW W g =
4
3 4 4 3
+
=
7 6 7 6
7
3 2
1
= :
7 6
7
Example 4. A spinner can land in any of four positions, A, B, C, and D, with equal
probability. The spinner is used twice and the position noted each time. Let the random
variable Y denote the number of positions that the spinner did not land on. Compute the
probabilities for each value of Y .
Solution. The sample space is
S = fAA; AB; AC; AD; BA; BB; BC; BD; CA; CB; CC; CD; DA; DB; DC; DDg :
Since Y is the number of positions that the spinner did not land on, we see that
Sample Point
AA
AB
AC
AD
BA
BB
BC
BD
Sample Point
CA
CB
CC
CD
DA
DB
DC
DD
Y
3
2
2
2
2
3
2
2
3
Y
2
2
3
2
2
2
2
3
Thus,
fY = 2g = fAB; AC; AD; BA; BC; BD; CA; CB; CD; DA; DB; DCg
fY = 3g = fAA; BB; CC; DDg :
Hence,
12
3
=
16
4
4
1
P fY = 3g =
= :
16
4
P fY = 2g =
Example 5. Toss a pair of balanced dice and let Y be the sum of the the two numbers
that appear. The sample space is
S = f(y1 ; y2 ) j y1 = 1; 2;
; 6 and y2 = 1; 2;
; 6g :
Then
for (y1 ; y2 ) 2 S:
Y (y1 ; y2 ) = y1 + y2
In particular,
fY = 7g = f(1; 6) ; (2; 5) ; (3; 4) ; (4; 3) ; (5; 2) ; (6; 1)g
and
P fY = 7g =
6
1
= :
36
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Example 6. A single dart is tossed at a bull’s-eye target by an experienced player. We
are sure that the dart will hit somewhere within the outer ring, whose diameter is 8 inches.
For the convenience of illustration, we superimpose an (y1 ; y2 ) rectangular coordinate system
over the target, with its origin located at the center of the bull’s-eye. Then the sample space
of the experiment de…ned by observing the impact point of one dart thrown by the player is
given by
S = (y1 ; y2 ) j y12 + y22 64 ;
where the units used are inches and the impact point of the dart is recorded by its (y1 ; y2 )
coordinate (see Figure 1.6 ).
If we de…ne Y to be the distance between the dart’s impact point and the center of the
target (that is, the origin of p
the rectangular coordinate system), then Y is a random variable
that associates the number y12 + y22 with every element of S, that is,
q
Y (y1 ; y2 ) = y12 + y22
for (y1 ; y2 ) 2 S;
p
since the radial distance of any point (y1 ; y2 ) from the origin is y12 + y22 . For example, the
radial distance of point (3; 4) from the origin is 5 as shown in Figure 1.6.
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The range of random variables is the collection of real numbers associated with elements
of the sample space S. For instance, the ranges of the random variables of the preceding …ve
examples are, respectively, given by
f0; 1; 2g
f0; 1; 2; 3g
f0; 1; 2g
f2; 3g
f2; 3; ; 4;
; 12g
fyj 0 y 8g
for
for
for
for
for
for
Example
Example
Example
Example
Example
Example
1
2
3
4
5
6
The random variables in Examples 1 through 5 are discrete since their ranges are …nite sets,
while the random variable in Example 6 is continuous since its range is an uncountable set.
De…nition 2. A random variable Y is said to be discrete if its range is a …nite or countable set.
Likewise, when a random variable assumes its values on a continuous scale (or on an
uncountable set), it is called a continuous random variable. In most practical problems,
discrete random variables represent count data, such as the number of defectives in a sample
of n items or the number of phone calls per day in a certain type of switchboard, whereas
continuous random variables represent measured data, such as all possible heights, weights,
temperatures, distances, water levels of a river bank, or life periods.
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