ICAI Notes- Theoretical Distributions | Quantitative Aptitude for CA Foundation PDF Download

 Page 1


THEORETICAL DISTRIBUTIONS
16
CHAPTER
In chapter thirteen, it may be recalled, we discussed frequency distribution. In a similar manner,
we may think of a probability distribution where just like distributing the total frequency to
different class intervals, the total probability (i.e. one) is distributed to different mass points in
case of a discrete random variable or to different class intervals in case of a continuous random
variable. Such a probability distribution is known as Theoretical Probability Distribution, since
such a distribution exists in theory. We need to study theoretical probability distribution for the
following important factors:
The Students will be introduced in this chapter to the techniques of developing discrete and
continuous probability distributions and its applications.
Discrete Probability
Distributions
Theoretical Probability
Distributions
Continuous Probability
Distributions
Binomial
Distribution
Poisson
Distribution
Normal
Distribution
CHAPTER OVERVIEW
© The Institute of Chartered Accountants of India
Page 2


THEORETICAL DISTRIBUTIONS
16
CHAPTER
In chapter thirteen, it may be recalled, we discussed frequency distribution. In a similar manner,
we may think of a probability distribution where just like distributing the total frequency to
different class intervals, the total probability (i.e. one) is distributed to different mass points in
case of a discrete random variable or to different class intervals in case of a continuous random
variable. Such a probability distribution is known as Theoretical Probability Distribution, since
such a distribution exists in theory. We need to study theoretical probability distribution for the
following important factors:
The Students will be introduced in this chapter to the techniques of developing discrete and
continuous probability distributions and its applications.
Discrete Probability
Distributions
Theoretical Probability
Distributions
Continuous Probability
Distributions
Binomial
Distribution
Poisson
Distribution
Normal
Distribution
CHAPTER OVERVIEW
© The Institute of Chartered Accountants of India
16.2
STATISTICS
(a) An observed frequency distribution, in many a case, may be regarded as a sample i.e. a
representative part of a large, unknown, boundless universe or population and we may be
interested to know the form of such a distribution. By fitting a theoretical probability
distribution to an observed frequency distribution of, say, the lamps produced by a
manufacturer, it may be possible for the manufacturer to specify the length of life of the
lamps produced by him up to a reasonable degree of accuracy. By studying the effect of a
particular type of missiles, it may be possible for our scientist to suggest the number of such
missiles necessary to destroy an army position. By knowing the distribution of smokers, a
social activist may warn the people of a locality about the nuisance of active and passive
smoking and so on.
(b) Theoretical probability distribution may be profitably employed to make short term
projections for the future.
(c) Statistical analysis is possible only on the basis of theoretical probability distribution. Setting
confidence limits or testing statistical hypothesis about population parameter(s) is based on
the probability distribution of the population under consideration.
A probability distribution also possesses all the characteristics of an observed distribution. We
define mean ? ?? , median ?? ? ? , mode 
?
?? ? , standard deviation ? ?? etc. exactly same way we have
done earlier. Again a probability distribution may be either a discrete probability distribution or a
Continuous probability distribution depending on the random variable under study. Two important
discrete probability distributions are (a) Binomial Distribution and (b) Poisson distribution.
Some important continuous probability distributions are
Normal Distribution
One of the most important and frequently used discrete probability distribution is Binomial
Distribution. It is derived from a particular type of random experiment known as Bernoulli process
named after the famous mathematician Bernoulli. Noting that a 'trial' is an attempt to produce a
particular outcome which is neither certain nor impossible, the characteristics of Bernoulli trials
are stated below:
(i) Each trial is associated with two mutually exclusive and exhaustive outcomes, the occurrence
of one of which is known as a 'success' and as such its non occurrence as a 'failure'. As an
example, when a coin is tossed, usually occurrence of a head is known as a success and its
non–occurrence  i.e. occurrence of a tail is known as a failure.
© The Institute of Chartered Accountants of India
Page 3


THEORETICAL DISTRIBUTIONS
16
CHAPTER
In chapter thirteen, it may be recalled, we discussed frequency distribution. In a similar manner,
we may think of a probability distribution where just like distributing the total frequency to
different class intervals, the total probability (i.e. one) is distributed to different mass points in
case of a discrete random variable or to different class intervals in case of a continuous random
variable. Such a probability distribution is known as Theoretical Probability Distribution, since
such a distribution exists in theory. We need to study theoretical probability distribution for the
following important factors:
The Students will be introduced in this chapter to the techniques of developing discrete and
continuous probability distributions and its applications.
Discrete Probability
Distributions
Theoretical Probability
Distributions
Continuous Probability
Distributions
Binomial
Distribution
Poisson
Distribution
Normal
Distribution
CHAPTER OVERVIEW
© The Institute of Chartered Accountants of India
16.2
STATISTICS
(a) An observed frequency distribution, in many a case, may be regarded as a sample i.e. a
representative part of a large, unknown, boundless universe or population and we may be
interested to know the form of such a distribution. By fitting a theoretical probability
distribution to an observed frequency distribution of, say, the lamps produced by a
manufacturer, it may be possible for the manufacturer to specify the length of life of the
lamps produced by him up to a reasonable degree of accuracy. By studying the effect of a
particular type of missiles, it may be possible for our scientist to suggest the number of such
missiles necessary to destroy an army position. By knowing the distribution of smokers, a
social activist may warn the people of a locality about the nuisance of active and passive
smoking and so on.
(b) Theoretical probability distribution may be profitably employed to make short term
projections for the future.
(c) Statistical analysis is possible only on the basis of theoretical probability distribution. Setting
confidence limits or testing statistical hypothesis about population parameter(s) is based on
the probability distribution of the population under consideration.
A probability distribution also possesses all the characteristics of an observed distribution. We
define mean ? ?? , median ?? ? ? , mode 
?
?? ? , standard deviation ? ?? etc. exactly same way we have
done earlier. Again a probability distribution may be either a discrete probability distribution or a
Continuous probability distribution depending on the random variable under study. Two important
discrete probability distributions are (a) Binomial Distribution and (b) Poisson distribution.
Some important continuous probability distributions are
Normal Distribution
One of the most important and frequently used discrete probability distribution is Binomial
Distribution. It is derived from a particular type of random experiment known as Bernoulli process
named after the famous mathematician Bernoulli. Noting that a 'trial' is an attempt to produce a
particular outcome which is neither certain nor impossible, the characteristics of Bernoulli trials
are stated below:
(i) Each trial is associated with two mutually exclusive and exhaustive outcomes, the occurrence
of one of which is known as a 'success' and as such its non occurrence as a 'failure'. As an
example, when a coin is tossed, usually occurrence of a head is known as a success and its
non–occurrence  i.e. occurrence of a tail is known as a failure.
© The Institute of Chartered Accountants of India
16.3 THEORETICAL DISTRIBUTIONS
(ii) The trials are independent.
(iii) The probability of a success, usually denoted by p, and hence that of a failure, usually denoted
by q = 1–p, remain unchanged throughout the process.
(iv) The number of trials is a finite positive integer.
A discrete random variable x is defined to follow binomial distribution with parameters n and p,
to be denoted by x ~ B (n, p), if the probability mass function of x is given by
f (x)  = p (X = x)  =   
n x n-x
x
c p q for x = 0, 1, 2, …., n
                             = 0, otherwise   ……… (16.1)
We may note the following important points in connection with binomial distribution:
(a) As n >0, p, q ? 0, it follows that f(x) ? 0 for every x
Also 
?
x
f(x) = f(0)  + f(1)  + f(2)  + …..+ f(n) = 1………(16.2)
(b) Binomial distribution is known as biparametric distribution as it is characterised by two
parameters n and p. This means that if the values of n and p are known, then the
distribution is known completely.
(c) The mean of the binomial distribution is given by  ?  = np …. (16.3)
(d) Depending on the values of the two parameters, binomial distribution may be unimodal
or bi- modal. 
?
? , the mode of binomial distribution, is given by 
?
? = the largest integer
contained in (n+1)p if (n+1)p is a non-integer  (n+1)p and (n+1)p - 1
if (n+1)p is an integer ….(16.4)
(e) The variance of the binomial distribution is given by
        
2
?
 = npq                                                                       ………. (16.5)
       Since p and q are numerically less than or equal to 1, npq < np
       ? variance of a binomial variable is always less than its mean.
Also variance of X attains its maximum value  at p = q = 0.5 and this maximum value
is n/4.
(f) Additive property of binomial distribution.
If X and Y are two independent variables such that
X~B (n
1
, P)
and   Y~B (n
21
P)
Then (X+Y) ~B (n
1
 + n
2
 , P)    …………………………….. (16.6)
© The Institute of Chartered Accountants of India
Page 4


THEORETICAL DISTRIBUTIONS
16
CHAPTER
In chapter thirteen, it may be recalled, we discussed frequency distribution. In a similar manner,
we may think of a probability distribution where just like distributing the total frequency to
different class intervals, the total probability (i.e. one) is distributed to different mass points in
case of a discrete random variable or to different class intervals in case of a continuous random
variable. Such a probability distribution is known as Theoretical Probability Distribution, since
such a distribution exists in theory. We need to study theoretical probability distribution for the
following important factors:
The Students will be introduced in this chapter to the techniques of developing discrete and
continuous probability distributions and its applications.
Discrete Probability
Distributions
Theoretical Probability
Distributions
Continuous Probability
Distributions
Binomial
Distribution
Poisson
Distribution
Normal
Distribution
CHAPTER OVERVIEW
© The Institute of Chartered Accountants of India
16.2
STATISTICS
(a) An observed frequency distribution, in many a case, may be regarded as a sample i.e. a
representative part of a large, unknown, boundless universe or population and we may be
interested to know the form of such a distribution. By fitting a theoretical probability
distribution to an observed frequency distribution of, say, the lamps produced by a
manufacturer, it may be possible for the manufacturer to specify the length of life of the
lamps produced by him up to a reasonable degree of accuracy. By studying the effect of a
particular type of missiles, it may be possible for our scientist to suggest the number of such
missiles necessary to destroy an army position. By knowing the distribution of smokers, a
social activist may warn the people of a locality about the nuisance of active and passive
smoking and so on.
(b) Theoretical probability distribution may be profitably employed to make short term
projections for the future.
(c) Statistical analysis is possible only on the basis of theoretical probability distribution. Setting
confidence limits or testing statistical hypothesis about population parameter(s) is based on
the probability distribution of the population under consideration.
A probability distribution also possesses all the characteristics of an observed distribution. We
define mean ? ?? , median ?? ? ? , mode 
?
?? ? , standard deviation ? ?? etc. exactly same way we have
done earlier. Again a probability distribution may be either a discrete probability distribution or a
Continuous probability distribution depending on the random variable under study. Two important
discrete probability distributions are (a) Binomial Distribution and (b) Poisson distribution.
Some important continuous probability distributions are
Normal Distribution
One of the most important and frequently used discrete probability distribution is Binomial
Distribution. It is derived from a particular type of random experiment known as Bernoulli process
named after the famous mathematician Bernoulli. Noting that a 'trial' is an attempt to produce a
particular outcome which is neither certain nor impossible, the characteristics of Bernoulli trials
are stated below:
(i) Each trial is associated with two mutually exclusive and exhaustive outcomes, the occurrence
of one of which is known as a 'success' and as such its non occurrence as a 'failure'. As an
example, when a coin is tossed, usually occurrence of a head is known as a success and its
non–occurrence  i.e. occurrence of a tail is known as a failure.
© The Institute of Chartered Accountants of India
16.3 THEORETICAL DISTRIBUTIONS
(ii) The trials are independent.
(iii) The probability of a success, usually denoted by p, and hence that of a failure, usually denoted
by q = 1–p, remain unchanged throughout the process.
(iv) The number of trials is a finite positive integer.
A discrete random variable x is defined to follow binomial distribution with parameters n and p,
to be denoted by x ~ B (n, p), if the probability mass function of x is given by
f (x)  = p (X = x)  =   
n x n-x
x
c p q for x = 0, 1, 2, …., n
                             = 0, otherwise   ……… (16.1)
We may note the following important points in connection with binomial distribution:
(a) As n >0, p, q ? 0, it follows that f(x) ? 0 for every x
Also 
?
x
f(x) = f(0)  + f(1)  + f(2)  + …..+ f(n) = 1………(16.2)
(b) Binomial distribution is known as biparametric distribution as it is characterised by two
parameters n and p. This means that if the values of n and p are known, then the
distribution is known completely.
(c) The mean of the binomial distribution is given by  ?  = np …. (16.3)
(d) Depending on the values of the two parameters, binomial distribution may be unimodal
or bi- modal. 
?
? , the mode of binomial distribution, is given by 
?
? = the largest integer
contained in (n+1)p if (n+1)p is a non-integer  (n+1)p and (n+1)p - 1
if (n+1)p is an integer ….(16.4)
(e) The variance of the binomial distribution is given by
        
2
?
 = npq                                                                       ………. (16.5)
       Since p and q are numerically less than or equal to 1, npq < np
       ? variance of a binomial variable is always less than its mean.
Also variance of X attains its maximum value  at p = q = 0.5 and this maximum value
is n/4.
(f) Additive property of binomial distribution.
If X and Y are two independent variables such that
X~B (n
1
, P)
and   Y~B (n
21
P)
Then (X+Y) ~B (n
1
 + n
2
 , P)    …………………………….. (16.6)
© The Institute of Chartered Accountants of India
16.4
STATISTICS
Applications of Binomial Distribution
Binomial distribution is applicable when the trials are independent and each trial has just two
outcomes success and failure. It is applied in coin tossing experiments, sampling inspection plan,
genetic experiments and so on.
Example 16.1: A coin is tossed 10 times. Assuming the coin to be unbiased, what is the probability
of getting
(i) 4 heads?
(ii) at least 4 heads?
(iii) at most 3 heads?
Solution: We apply binomial distribution as the tossing are independent of each other. With
every tossing, there are just two outcomes either a head, which we call a success or a tail, which
we call a failure and the probability of a success (or failure) remains constant throughout.
Let X denotes the no. of heads. Then X follows binomial distribution with parameter n = 8 and
p = 1/2 (since the coin is unbiased). Hence q = 1 – p = 1/2
The probability mass function of X is given by
f(x)  = 
n
c
x
  p
x
 q
n-x
      = 
10
c
x
 . (1/2)
x
 . (1/2)
10-x
      =   
10
x
10
c
2
= 
10
c
x
 / 1024        for x = 0, 1, 2, ……….10
(i) probability of getting 4 heads
= f (4)
= 
10
c
4
 / 1024
= 210 / 1024
= 105 / 512
(ii) probability of getting at least 4 heads
= P (X ? 4)
= P (X = 4) + P (X = 5) + P (X = 6) + P(X = 7) +P (X = 8)
 = 
10
c
4
 / 1024 + 
10
c
5
 / 1024 + 
10
c
6
 / 1024 +
 10
c
7
 / 1024 +
 10
c
8
 /1024 +
 10
c
9
 /1024 +
 10
c
10
 /1024
© The Institute of Chartered Accountants of India
Page 5


THEORETICAL DISTRIBUTIONS
16
CHAPTER
In chapter thirteen, it may be recalled, we discussed frequency distribution. In a similar manner,
we may think of a probability distribution where just like distributing the total frequency to
different class intervals, the total probability (i.e. one) is distributed to different mass points in
case of a discrete random variable or to different class intervals in case of a continuous random
variable. Such a probability distribution is known as Theoretical Probability Distribution, since
such a distribution exists in theory. We need to study theoretical probability distribution for the
following important factors:
The Students will be introduced in this chapter to the techniques of developing discrete and
continuous probability distributions and its applications.
Discrete Probability
Distributions
Theoretical Probability
Distributions
Continuous Probability
Distributions
Binomial
Distribution
Poisson
Distribution
Normal
Distribution
CHAPTER OVERVIEW
© The Institute of Chartered Accountants of India
16.2
STATISTICS
(a) An observed frequency distribution, in many a case, may be regarded as a sample i.e. a
representative part of a large, unknown, boundless universe or population and we may be
interested to know the form of such a distribution. By fitting a theoretical probability
distribution to an observed frequency distribution of, say, the lamps produced by a
manufacturer, it may be possible for the manufacturer to specify the length of life of the
lamps produced by him up to a reasonable degree of accuracy. By studying the effect of a
particular type of missiles, it may be possible for our scientist to suggest the number of such
missiles necessary to destroy an army position. By knowing the distribution of smokers, a
social activist may warn the people of a locality about the nuisance of active and passive
smoking and so on.
(b) Theoretical probability distribution may be profitably employed to make short term
projections for the future.
(c) Statistical analysis is possible only on the basis of theoretical probability distribution. Setting
confidence limits or testing statistical hypothesis about population parameter(s) is based on
the probability distribution of the population under consideration.
A probability distribution also possesses all the characteristics of an observed distribution. We
define mean ? ?? , median ?? ? ? , mode 
?
?? ? , standard deviation ? ?? etc. exactly same way we have
done earlier. Again a probability distribution may be either a discrete probability distribution or a
Continuous probability distribution depending on the random variable under study. Two important
discrete probability distributions are (a) Binomial Distribution and (b) Poisson distribution.
Some important continuous probability distributions are
Normal Distribution
One of the most important and frequently used discrete probability distribution is Binomial
Distribution. It is derived from a particular type of random experiment known as Bernoulli process
named after the famous mathematician Bernoulli. Noting that a 'trial' is an attempt to produce a
particular outcome which is neither certain nor impossible, the characteristics of Bernoulli trials
are stated below:
(i) Each trial is associated with two mutually exclusive and exhaustive outcomes, the occurrence
of one of which is known as a 'success' and as such its non occurrence as a 'failure'. As an
example, when a coin is tossed, usually occurrence of a head is known as a success and its
non–occurrence  i.e. occurrence of a tail is known as a failure.
© The Institute of Chartered Accountants of India
16.3 THEORETICAL DISTRIBUTIONS
(ii) The trials are independent.
(iii) The probability of a success, usually denoted by p, and hence that of a failure, usually denoted
by q = 1–p, remain unchanged throughout the process.
(iv) The number of trials is a finite positive integer.
A discrete random variable x is defined to follow binomial distribution with parameters n and p,
to be denoted by x ~ B (n, p), if the probability mass function of x is given by
f (x)  = p (X = x)  =   
n x n-x
x
c p q for x = 0, 1, 2, …., n
                             = 0, otherwise   ……… (16.1)
We may note the following important points in connection with binomial distribution:
(a) As n >0, p, q ? 0, it follows that f(x) ? 0 for every x
Also 
?
x
f(x) = f(0)  + f(1)  + f(2)  + …..+ f(n) = 1………(16.2)
(b) Binomial distribution is known as biparametric distribution as it is characterised by two
parameters n and p. This means that if the values of n and p are known, then the
distribution is known completely.
(c) The mean of the binomial distribution is given by  ?  = np …. (16.3)
(d) Depending on the values of the two parameters, binomial distribution may be unimodal
or bi- modal. 
?
? , the mode of binomial distribution, is given by 
?
? = the largest integer
contained in (n+1)p if (n+1)p is a non-integer  (n+1)p and (n+1)p - 1
if (n+1)p is an integer ….(16.4)
(e) The variance of the binomial distribution is given by
        
2
?
 = npq                                                                       ………. (16.5)
       Since p and q are numerically less than or equal to 1, npq < np
       ? variance of a binomial variable is always less than its mean.
Also variance of X attains its maximum value  at p = q = 0.5 and this maximum value
is n/4.
(f) Additive property of binomial distribution.
If X and Y are two independent variables such that
X~B (n
1
, P)
and   Y~B (n
21
P)
Then (X+Y) ~B (n
1
 + n
2
 , P)    …………………………….. (16.6)
© The Institute of Chartered Accountants of India
16.4
STATISTICS
Applications of Binomial Distribution
Binomial distribution is applicable when the trials are independent and each trial has just two
outcomes success and failure. It is applied in coin tossing experiments, sampling inspection plan,
genetic experiments and so on.
Example 16.1: A coin is tossed 10 times. Assuming the coin to be unbiased, what is the probability
of getting
(i) 4 heads?
(ii) at least 4 heads?
(iii) at most 3 heads?
Solution: We apply binomial distribution as the tossing are independent of each other. With
every tossing, there are just two outcomes either a head, which we call a success or a tail, which
we call a failure and the probability of a success (or failure) remains constant throughout.
Let X denotes the no. of heads. Then X follows binomial distribution with parameter n = 8 and
p = 1/2 (since the coin is unbiased). Hence q = 1 – p = 1/2
The probability mass function of X is given by
f(x)  = 
n
c
x
  p
x
 q
n-x
      = 
10
c
x
 . (1/2)
x
 . (1/2)
10-x
      =   
10
x
10
c
2
= 
10
c
x
 / 1024        for x = 0, 1, 2, ……….10
(i) probability of getting 4 heads
= f (4)
= 
10
c
4
 / 1024
= 210 / 1024
= 105 / 512
(ii) probability of getting at least 4 heads
= P (X ? 4)
= P (X = 4) + P (X = 5) + P (X = 6) + P(X = 7) +P (X = 8)
 = 
10
c
4
 / 1024 + 
10
c
5
 / 1024 + 
10
c
6
 / 1024 +
 10
c
7
 / 1024 +
 10
c
8
 /1024 +
 10
c
9
 /1024 +
 10
c
10
 /1024
© The Institute of Chartered Accountants of India
16.5 THEORETICAL DISTRIBUTIONS
=
210 + 252 + 210 +120 + 45 + 10 + 1
1024
= 848 / 1024
(iii ) probability of getting at most 3 heads
= P (X ? 3)
= P (X = 0) + P (X = 1) + P (X = 2) + P (X = 3)
= f (0) + f (1) + f (2) + f (3)
= 
10
c
0
 / 1024 + 
10
c
1
 / 1024 + 
10
c
2
 / 1024 +
10
c
3
 / 1024
=
1+10 + 45 +120
1024
= 176 / 1024
= 11/64
Example 16.2: If 15 dates are selected at random, what is the probability of getting two Sundays?
Solution: If X denotes the number at Sundays, then it is obvious that X follows binomial
distribution with parameter n = 15 and p = probability of a Sunday in a week = 1/7 and
q = 1 – p = 6 / 7.
Then f(x) = 
15
c
x 
(1/7)
x
. (6/7)
15–x
.
for x = 0, 1, 2,……….. 15.
Hence the probability of getting two Sundays
= f(2)
=
 15
c
2
 (1/7)
2
 . (6/7)
15–2
= 
13
15
105 6
7
?
? 0.29
Example 16.3: The incidence of occupational disease in an industry is such that the workmen
have a 10% chance of suffering from it. What is the probability that out of 5 workmen, 3 or more
will contract the disease?
Solution: Let X denote the number of workmen in the sample. X follows binomial with parameters
n = 5 and p = probability that a workman suffers from the occupational
disease = 0.1
Hence q = 1 – 0.1 = 0.9.
© The Institute of Chartered Accountants of India