Random Variables and Probability Distributions | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET PDF Download

 Page 1


Random Variables and Probability 
Distributions 
Random Variables 
Suppose that to each point of a sample space we assign a number. We then have a 
function defined on the sample space. This function is called a random variable (or 
stochastic variable) or more precisely a random function (stochastic function). It is 
usually denoted by a capital letter such as ?? or ?? . In general, a random variable has some 
specified physical, geometrical, or other significance. 
EXAMPLE 2.1 Suppose that a coin is tossed twice so that the sample space is ?? =
{???? ,???? ,???? ,???? }. Let ?? represent the number of heads that can come up. With each 
sample point we can associate a number for ?? as shown in Table 2-1. Thus, for example, 
in the case of ???? (i.e., 2 heads), ?? =2 while for ???? ( 1 head), ?? =1. It follows that ?? is a 
random variable. 
Table 2-1 
Sample Point ???? ???? ???? ???? 
?? 2 1 1 0 
 
It should be noted that many other random variables could also be defined on this 
sample space, for example, the square of the number of heads or the number of heads 
minus the number of tails. 
A random variable that takes on a finite or countably infinite number of values (see page 
4) is called a discrete random variable while one which takes on a no countably infinite 
number of values is called a no discrete random variable. 
Discrete Probability Distributions 
Let X be a discrete random variable, and suppose that the possible values that it can 
assume are given by ?? 1
,?? 2
, ?? 3
,…, arranged in some order. Suppose also that these values 
are assumed with probabilities given by 
?? (?? =?? ?? )=?? (?? ?? ) ?? =1,2,… (1) 
It is convenient to introduce the probability function, also referred to as probability 
distribution, given by 
?? (?? =?? )=?? (?? ) (2) 
For ?? =?? ?? , this reduces to (1) while for other values of ?? ,?? (?? )=0. 
Page 2


Random Variables and Probability 
Distributions 
Random Variables 
Suppose that to each point of a sample space we assign a number. We then have a 
function defined on the sample space. This function is called a random variable (or 
stochastic variable) or more precisely a random function (stochastic function). It is 
usually denoted by a capital letter such as ?? or ?? . In general, a random variable has some 
specified physical, geometrical, or other significance. 
EXAMPLE 2.1 Suppose that a coin is tossed twice so that the sample space is ?? =
{???? ,???? ,???? ,???? }. Let ?? represent the number of heads that can come up. With each 
sample point we can associate a number for ?? as shown in Table 2-1. Thus, for example, 
in the case of ???? (i.e., 2 heads), ?? =2 while for ???? ( 1 head), ?? =1. It follows that ?? is a 
random variable. 
Table 2-1 
Sample Point ???? ???? ???? ???? 
?? 2 1 1 0 
 
It should be noted that many other random variables could also be defined on this 
sample space, for example, the square of the number of heads or the number of heads 
minus the number of tails. 
A random variable that takes on a finite or countably infinite number of values (see page 
4) is called a discrete random variable while one which takes on a no countably infinite 
number of values is called a no discrete random variable. 
Discrete Probability Distributions 
Let X be a discrete random variable, and suppose that the possible values that it can 
assume are given by ?? 1
,?? 2
, ?? 3
,…, arranged in some order. Suppose also that these values 
are assumed with probabilities given by 
?? (?? =?? ?? )=?? (?? ?? ) ?? =1,2,… (1) 
It is convenient to introduce the probability function, also referred to as probability 
distribution, given by 
?? (?? =?? )=?? (?? ) (2) 
For ?? =?? ?? , this reduces to (1) while for other values of ?? ,?? (?? )=0. 
In general, ?? (?? ) is a probability function if 
1. ?? (?? )=0 
2. ?
?? ??? (?? )=1 
where the sum in 2 is taken over all possible values of ?? . 
EXAMPLE 2.2 Find the probability function corresponding to the random variable ?? of 
Example 2.1. Assuming that the coin is fair, we have 
?? (???? )=
1
4
 ?? (???? )=
1
4
 ?? (???? )=
1
4
 ?? (???? )=
1
4
 
Then 
?? (?? =0)=?? (???? )=
1
4
?? (?? =1)=?? (???? ????? )=?? (???? )+?? (???? )=
1
4
+
1
4
=
1
2
?? (?? =2)=?? (???? )=
1
4
 
The probability function is thus given by Table 2-2. 
Table 2-2 
?? 0 1 2 
?? (?? ) 1/4 1/2 1/4 
 
Distribution Functions for Random 
Variables 
The cumulative distribution function, or briefly the distribution function, for a random 
variable ?? is defined by 
?? (?? )=?? (?? =?? ) (3) 
where ?? is any real number, i.e., -8<?? <8. 
The distribution function ?? (?? ) has the following properties: 
1. ?? (?? ) is nondecreasing [i.e., ?? (?? )=?? (?? ) if ?? =?? ]. 
2. lim
?? ?-8
??? (?? )=0;lim
?? ?8
??? (?? )=1. 
3. ?? (?? ) is continuous from the right [i.e., lim
h?0
+??? (?? +h)=?? (?? ) for all ?? ]. 
Page 3


Random Variables and Probability 
Distributions 
Random Variables 
Suppose that to each point of a sample space we assign a number. We then have a 
function defined on the sample space. This function is called a random variable (or 
stochastic variable) or more precisely a random function (stochastic function). It is 
usually denoted by a capital letter such as ?? or ?? . In general, a random variable has some 
specified physical, geometrical, or other significance. 
EXAMPLE 2.1 Suppose that a coin is tossed twice so that the sample space is ?? =
{???? ,???? ,???? ,???? }. Let ?? represent the number of heads that can come up. With each 
sample point we can associate a number for ?? as shown in Table 2-1. Thus, for example, 
in the case of ???? (i.e., 2 heads), ?? =2 while for ???? ( 1 head), ?? =1. It follows that ?? is a 
random variable. 
Table 2-1 
Sample Point ???? ???? ???? ???? 
?? 2 1 1 0 
 
It should be noted that many other random variables could also be defined on this 
sample space, for example, the square of the number of heads or the number of heads 
minus the number of tails. 
A random variable that takes on a finite or countably infinite number of values (see page 
4) is called a discrete random variable while one which takes on a no countably infinite 
number of values is called a no discrete random variable. 
Discrete Probability Distributions 
Let X be a discrete random variable, and suppose that the possible values that it can 
assume are given by ?? 1
,?? 2
, ?? 3
,…, arranged in some order. Suppose also that these values 
are assumed with probabilities given by 
?? (?? =?? ?? )=?? (?? ?? ) ?? =1,2,… (1) 
It is convenient to introduce the probability function, also referred to as probability 
distribution, given by 
?? (?? =?? )=?? (?? ) (2) 
For ?? =?? ?? , this reduces to (1) while for other values of ?? ,?? (?? )=0. 
In general, ?? (?? ) is a probability function if 
1. ?? (?? )=0 
2. ?
?? ??? (?? )=1 
where the sum in 2 is taken over all possible values of ?? . 
EXAMPLE 2.2 Find the probability function corresponding to the random variable ?? of 
Example 2.1. Assuming that the coin is fair, we have 
?? (???? )=
1
4
 ?? (???? )=
1
4
 ?? (???? )=
1
4
 ?? (???? )=
1
4
 
Then 
?? (?? =0)=?? (???? )=
1
4
?? (?? =1)=?? (???? ????? )=?? (???? )+?? (???? )=
1
4
+
1
4
=
1
2
?? (?? =2)=?? (???? )=
1
4
 
The probability function is thus given by Table 2-2. 
Table 2-2 
?? 0 1 2 
?? (?? ) 1/4 1/2 1/4 
 
Distribution Functions for Random 
Variables 
The cumulative distribution function, or briefly the distribution function, for a random 
variable ?? is defined by 
?? (?? )=?? (?? =?? ) (3) 
where ?? is any real number, i.e., -8<?? <8. 
The distribution function ?? (?? ) has the following properties: 
1. ?? (?? ) is nondecreasing [i.e., ?? (?? )=?? (?? ) if ?? =?? ]. 
2. lim
?? ?-8
??? (?? )=0;lim
?? ?8
??? (?? )=1. 
3. ?? (?? ) is continuous from the right [i.e., lim
h?0
+??? (?? +h)=?? (?? ) for all ?? ]. 
Distribution Functions for Discrete 
Random Variables 
The distribution function for a discrete random variable ?? can be obtained from its 
probability function by noting that, for all ?? in (-8,8), 
?? (?? )=?? (?? =?? )=??
?? =?? ??? (?? ) (4)
 
where the sum is taken over all values ?? taken on by ?? for which ?? =?? . 
If ?? takes on only a finite number of values ?? 1
,?? 2
,…,?? ?? , then the distribution function is 
given by 
?? (?? )=
{
 
 
 
 
0 -8<?? <?? 1
?? (?? 1
) ?? 1
=?? <?? 2
?? (?? 1
)+?? (?? 2
) ?? 2
=?? <?? 3
? ?
?? (?? 1
)+?+?? (?? ?? ) ?? ?? =?? <8
(5) 
EXAMPLE 2.3 (a) Find the distribution function for the random variable ?? of Example 
2.2. (b) Obtain its graph. 
(a) The distribution function is 
?? (?? )=
{
 
 
 
 
0 -8<?? <0
1
4
0=?? <1
3
4
1=?? <2
1 2=?? <8
 
(b) The graph of ?? (?? ) is shown in Fig. 2-1. 
Page 4


Random Variables and Probability 
Distributions 
Random Variables 
Suppose that to each point of a sample space we assign a number. We then have a 
function defined on the sample space. This function is called a random variable (or 
stochastic variable) or more precisely a random function (stochastic function). It is 
usually denoted by a capital letter such as ?? or ?? . In general, a random variable has some 
specified physical, geometrical, or other significance. 
EXAMPLE 2.1 Suppose that a coin is tossed twice so that the sample space is ?? =
{???? ,???? ,???? ,???? }. Let ?? represent the number of heads that can come up. With each 
sample point we can associate a number for ?? as shown in Table 2-1. Thus, for example, 
in the case of ???? (i.e., 2 heads), ?? =2 while for ???? ( 1 head), ?? =1. It follows that ?? is a 
random variable. 
Table 2-1 
Sample Point ???? ???? ???? ???? 
?? 2 1 1 0 
 
It should be noted that many other random variables could also be defined on this 
sample space, for example, the square of the number of heads or the number of heads 
minus the number of tails. 
A random variable that takes on a finite or countably infinite number of values (see page 
4) is called a discrete random variable while one which takes on a no countably infinite 
number of values is called a no discrete random variable. 
Discrete Probability Distributions 
Let X be a discrete random variable, and suppose that the possible values that it can 
assume are given by ?? 1
,?? 2
, ?? 3
,…, arranged in some order. Suppose also that these values 
are assumed with probabilities given by 
?? (?? =?? ?? )=?? (?? ?? ) ?? =1,2,… (1) 
It is convenient to introduce the probability function, also referred to as probability 
distribution, given by 
?? (?? =?? )=?? (?? ) (2) 
For ?? =?? ?? , this reduces to (1) while for other values of ?? ,?? (?? )=0. 
In general, ?? (?? ) is a probability function if 
1. ?? (?? )=0 
2. ?
?? ??? (?? )=1 
where the sum in 2 is taken over all possible values of ?? . 
EXAMPLE 2.2 Find the probability function corresponding to the random variable ?? of 
Example 2.1. Assuming that the coin is fair, we have 
?? (???? )=
1
4
 ?? (???? )=
1
4
 ?? (???? )=
1
4
 ?? (???? )=
1
4
 
Then 
?? (?? =0)=?? (???? )=
1
4
?? (?? =1)=?? (???? ????? )=?? (???? )+?? (???? )=
1
4
+
1
4
=
1
2
?? (?? =2)=?? (???? )=
1
4
 
The probability function is thus given by Table 2-2. 
Table 2-2 
?? 0 1 2 
?? (?? ) 1/4 1/2 1/4 
 
Distribution Functions for Random 
Variables 
The cumulative distribution function, or briefly the distribution function, for a random 
variable ?? is defined by 
?? (?? )=?? (?? =?? ) (3) 
where ?? is any real number, i.e., -8<?? <8. 
The distribution function ?? (?? ) has the following properties: 
1. ?? (?? ) is nondecreasing [i.e., ?? (?? )=?? (?? ) if ?? =?? ]. 
2. lim
?? ?-8
??? (?? )=0;lim
?? ?8
??? (?? )=1. 
3. ?? (?? ) is continuous from the right [i.e., lim
h?0
+??? (?? +h)=?? (?? ) for all ?? ]. 
Distribution Functions for Discrete 
Random Variables 
The distribution function for a discrete random variable ?? can be obtained from its 
probability function by noting that, for all ?? in (-8,8), 
?? (?? )=?? (?? =?? )=??
?? =?? ??? (?? ) (4)
 
where the sum is taken over all values ?? taken on by ?? for which ?? =?? . 
If ?? takes on only a finite number of values ?? 1
,?? 2
,…,?? ?? , then the distribution function is 
given by 
?? (?? )=
{
 
 
 
 
0 -8<?? <?? 1
?? (?? 1
) ?? 1
=?? <?? 2
?? (?? 1
)+?? (?? 2
) ?? 2
=?? <?? 3
? ?
?? (?? 1
)+?+?? (?? ?? ) ?? ?? =?? <8
(5) 
EXAMPLE 2.3 (a) Find the distribution function for the random variable ?? of Example 
2.2. (b) Obtain its graph. 
(a) The distribution function is 
?? (?? )=
{
 
 
 
 
0 -8<?? <0
1
4
0=?? <1
3
4
1=?? <2
1 2=?? <8
 
(b) The graph of ?? (?? ) is shown in Fig. 2-1. 
 
Fig. 2-1 
The following things about the above distribution function, which are true in general, 
should be noted. 
1. The magnitudes of the jumps at 0,1,2 are 
1
4
,
1
2
,
1
4
 which are precisely the 
probabilities in Table 2-2. This fact enables one to obtain the probability function 
from the distribution function. 
2. Because of the appearance of the graph of Fig. 2-1, it is often called a staircase 
function or step function. The value of the function at an integer is obtained from 
the higher step; thus the value at 1 is 
3
4
 and not 
1
4
. This is expressed 
mathematically by stating that the distribution function is continuous from the 
right at 0,1,2. 
3. As we proceed from left to right (i.e. going upstairs), the distribution function 
either remains the same or increases, taking on values from 0 to 1. Because of 
this, it is said to be a monotonically increasing function. 
It is clear from the above remarks and the properties of distribution functions that the 
probability function of a discrete random variable can be obtained from the distribution 
function by noting that 
Page 5


Random Variables and Probability 
Distributions 
Random Variables 
Suppose that to each point of a sample space we assign a number. We then have a 
function defined on the sample space. This function is called a random variable (or 
stochastic variable) or more precisely a random function (stochastic function). It is 
usually denoted by a capital letter such as ?? or ?? . In general, a random variable has some 
specified physical, geometrical, or other significance. 
EXAMPLE 2.1 Suppose that a coin is tossed twice so that the sample space is ?? =
{???? ,???? ,???? ,???? }. Let ?? represent the number of heads that can come up. With each 
sample point we can associate a number for ?? as shown in Table 2-1. Thus, for example, 
in the case of ???? (i.e., 2 heads), ?? =2 while for ???? ( 1 head), ?? =1. It follows that ?? is a 
random variable. 
Table 2-1 
Sample Point ???? ???? ???? ???? 
?? 2 1 1 0 
 
It should be noted that many other random variables could also be defined on this 
sample space, for example, the square of the number of heads or the number of heads 
minus the number of tails. 
A random variable that takes on a finite or countably infinite number of values (see page 
4) is called a discrete random variable while one which takes on a no countably infinite 
number of values is called a no discrete random variable. 
Discrete Probability Distributions 
Let X be a discrete random variable, and suppose that the possible values that it can 
assume are given by ?? 1
,?? 2
, ?? 3
,…, arranged in some order. Suppose also that these values 
are assumed with probabilities given by 
?? (?? =?? ?? )=?? (?? ?? ) ?? =1,2,… (1) 
It is convenient to introduce the probability function, also referred to as probability 
distribution, given by 
?? (?? =?? )=?? (?? ) (2) 
For ?? =?? ?? , this reduces to (1) while for other values of ?? ,?? (?? )=0. 
In general, ?? (?? ) is a probability function if 
1. ?? (?? )=0 
2. ?
?? ??? (?? )=1 
where the sum in 2 is taken over all possible values of ?? . 
EXAMPLE 2.2 Find the probability function corresponding to the random variable ?? of 
Example 2.1. Assuming that the coin is fair, we have 
?? (???? )=
1
4
 ?? (???? )=
1
4
 ?? (???? )=
1
4
 ?? (???? )=
1
4
 
Then 
?? (?? =0)=?? (???? )=
1
4
?? (?? =1)=?? (???? ????? )=?? (???? )+?? (???? )=
1
4
+
1
4
=
1
2
?? (?? =2)=?? (???? )=
1
4
 
The probability function is thus given by Table 2-2. 
Table 2-2 
?? 0 1 2 
?? (?? ) 1/4 1/2 1/4 
 
Distribution Functions for Random 
Variables 
The cumulative distribution function, or briefly the distribution function, for a random 
variable ?? is defined by 
?? (?? )=?? (?? =?? ) (3) 
where ?? is any real number, i.e., -8<?? <8. 
The distribution function ?? (?? ) has the following properties: 
1. ?? (?? ) is nondecreasing [i.e., ?? (?? )=?? (?? ) if ?? =?? ]. 
2. lim
?? ?-8
??? (?? )=0;lim
?? ?8
??? (?? )=1. 
3. ?? (?? ) is continuous from the right [i.e., lim
h?0
+??? (?? +h)=?? (?? ) for all ?? ]. 
Distribution Functions for Discrete 
Random Variables 
The distribution function for a discrete random variable ?? can be obtained from its 
probability function by noting that, for all ?? in (-8,8), 
?? (?? )=?? (?? =?? )=??
?? =?? ??? (?? ) (4)
 
where the sum is taken over all values ?? taken on by ?? for which ?? =?? . 
If ?? takes on only a finite number of values ?? 1
,?? 2
,…,?? ?? , then the distribution function is 
given by 
?? (?? )=
{
 
 
 
 
0 -8<?? <?? 1
?? (?? 1
) ?? 1
=?? <?? 2
?? (?? 1
)+?? (?? 2
) ?? 2
=?? <?? 3
? ?
?? (?? 1
)+?+?? (?? ?? ) ?? ?? =?? <8
(5) 
EXAMPLE 2.3 (a) Find the distribution function for the random variable ?? of Example 
2.2. (b) Obtain its graph. 
(a) The distribution function is 
?? (?? )=
{
 
 
 
 
0 -8<?? <0
1
4
0=?? <1
3
4
1=?? <2
1 2=?? <8
 
(b) The graph of ?? (?? ) is shown in Fig. 2-1. 
 
Fig. 2-1 
The following things about the above distribution function, which are true in general, 
should be noted. 
1. The magnitudes of the jumps at 0,1,2 are 
1
4
,
1
2
,
1
4
 which are precisely the 
probabilities in Table 2-2. This fact enables one to obtain the probability function 
from the distribution function. 
2. Because of the appearance of the graph of Fig. 2-1, it is often called a staircase 
function or step function. The value of the function at an integer is obtained from 
the higher step; thus the value at 1 is 
3
4
 and not 
1
4
. This is expressed 
mathematically by stating that the distribution function is continuous from the 
right at 0,1,2. 
3. As we proceed from left to right (i.e. going upstairs), the distribution function 
either remains the same or increases, taking on values from 0 to 1. Because of 
this, it is said to be a monotonically increasing function. 
It is clear from the above remarks and the properties of distribution functions that the 
probability function of a discrete random variable can be obtained from the distribution 
function by noting that 
?? (?? )=?? (?? )- lim
?? ??? -
??? (?? ) (6)
 
Continuous Random Variables 
A no discrete random variable ?? is said to be absolutely continuous, or simply 
continuous, if its distribution function may be represented as 
?? (?? )=?? (?? =?? )=? ?
?? -8
??? (?? )???? (-8<?? <8) (7) 
where the function ?? (?? ) has the properties 
1. ?? (?? )=0 
2. ?
-8
8
??? (?? )???? =1 
It follows from the above that if ?? is a continuous random variable, then the probability 
that ?? takes on any one particular value is zero, whereas the interval probability that ?? 
lies between two different values, say, ?? and ?? , is given by 
?? (?? <?? <?? )=? ?
?? ?? ??? (?? )???? (8) 
EXAMPLE 2.4 If an individual is selected at random from a large group of adult males, 
the probability that his height ?? is precisely 68 inches (i.e., 68.000… inches) would be 
zero. However, there is a probability greater than zero than ?? is between 67.000… inches 
and 68.500… inches, for example. 
A function ?? (?? ) that satisfies the above requirements is called a probability function or 
probability distribution for a continuous random variable, but it is more often called a 
probability density function or simply density function. Any function ?? (?? ) satisfying 
Properties 1 and 2 above will automatically be a density function, and required 
probabilities can then be obtained from (8). 
EXAMPLE 2.5 (a) Find the constant c such that the function 
?? (?? )={
?? ?? 2
0<?? <3
0 otherwise 
 
is a density function, and (b) compute ?? (1<?? <2). 
(a) Since ?? (?? ) satisfies Property 1 if ?? =0, it must satisfy Property 2 in order to be a 
density function. Now 
? ?
8
-8
?? (?? )???? =? ?
3
0
?? ?? 2
???? =
?? ?? 3
3
|
0
3
=9?? 
and since this must equal 1 , we have ?? =1/9.