PROBABILITY
DISTRIBUTION AND
SAMPLING THEORY
III SEMESTER
2019 Admission Onwards
Complementary Course (STA3 C03)
B Sc MATHEMATICS
UNIVERSITY OF CALICUT
School of Distance Education
Calicut University- P.O,
Malappuram - 673635, Kerala.
19559
,UNIVERSITY OF CALICUT
School of Distance Education
Study Material
III SEMESTER
(2019 Admission Onwards)
Complementary Course (STA3 C03)
B Sc MATHEMATICS
PROBABILITY DISTRIBUTION
AND SAMPLING THEORY
Prepared by:
Dr. Aparna Aravindakshan M.
Assistant Professor,
Department of Statistics,
St. Joseph’s College,
Devagiri, Kozhikode.
Scrutinized by:
Dr. Rajasekharan K.E.,
Assistant Professor,
EMEA College of Arts & Science,
Kondotty.
,Contents
1 Standard Probability Distributions 1
1.1 Discrete Probability Distributions . . . . . . . . . . . . . . 2
1.1.1 Bernoulli Distribution . . . . . . . . . . . . . . . . 2
1.1.2 Binomial Distribution . . . . . . . . . . . . . . . . 3
1.1.3 Poisson Distribution . . . . . . . . . . . . . . . . . 17
1.1.4 Negative Binomial Distribution . . . . . . . . . . . 31
1.1.5 Geometric Distribution . . . . . . . . . . . . . . . 34
1.1.6 Discrete Uniform Distribution . . . . . . . . . . . . 40
1.2 Continuous Distributions . . . . . . . . . . . . . . . . . . 43
1.2.1 Continuous Uniform or Rectangular Distribution . 43
1.2.2 Gamma Distribution . . . . . . . . . . . . . . . . . 46
1.2.3 Exponential Distribution . . . . . . . . . . . . . . 49
1.2.4 Normal Distribution . . . . . . . . . . . . . . . . . 54
1.2.5 Log-normal Distribution . . . . . . . . . . . . . . . 89
i
, ii CONTENTS
1.2.6 Beta Distribution . . . . . . . . . . . . . . . . . . . 92
1.2.7 Pareto Distribution . . . . . . . . . . . . . . . . . . 95
1.2.8 Cauchy Distribution . . . . . . . . . . . . . . . . . 96
2 Limit Theorems 98
2.1 Chebyshev’s Inequality . . . . . . . . . . . . . . . . . . . . 98
2.2 Modes of Convergence . . . . . . . . . . . . . . . . . . . . 105
2.2.1 Convergence in Distribution . . . . . . . . . . . . . 105
2.2.2 Convergence in Probability . . . . . . . . . . . . . 105
2.3 Laws of Large Numbers . . . . . . . . . . . . . . . . . . . 106
2.3.1 Weak Law of Large Numbers . . . . . . . . . . . . 106
2.3.2 Bernoulli’s Law of Large Numbers . . . . . . . . . 109
2.4 Central Limit Theorem . . . . . . . . . . . . . . . . . . . 113
2.4.1 Lévy Central Limit Theorem . . . . . . . . . . . . 114
2.4.2 De-Moivre’s-Laplace Central Limit Theorem . . . 117
3 Sampling Methods 122
3.1 Non-probability Sampling . . . . . . . . . . . . . . . . . . 132
3.2 Probability Sampling . . . . . . . . . . . . . . . . . . . . . 133
3.2.1 Simple Random Sampling . . . . . . . . . . . . . . 133
3.2.2 Stratified Random Sampling . . . . . . . . . . . . 138
3.2.3 Systematic Random Sampling . . . . . . . . . . . . 141
3.2.4 Cluster Sampling . . . . . . . . . . . . . . . . . . . 144
4 Sampling Distributions 146
4.1 Sampling Distribution of Sample Mean . . . . . . . . . . . 149
4.2 Chi-square Distribution . . . . . . . . . . . . . . . . . . . 151
DISTRIBUTION AND
SAMPLING THEORY
III SEMESTER
2019 Admission Onwards
Complementary Course (STA3 C03)
B Sc MATHEMATICS
UNIVERSITY OF CALICUT
School of Distance Education
Calicut University- P.O,
Malappuram - 673635, Kerala.
19559
,UNIVERSITY OF CALICUT
School of Distance Education
Study Material
III SEMESTER
(2019 Admission Onwards)
Complementary Course (STA3 C03)
B Sc MATHEMATICS
PROBABILITY DISTRIBUTION
AND SAMPLING THEORY
Prepared by:
Dr. Aparna Aravindakshan M.
Assistant Professor,
Department of Statistics,
St. Joseph’s College,
Devagiri, Kozhikode.
Scrutinized by:
Dr. Rajasekharan K.E.,
Assistant Professor,
EMEA College of Arts & Science,
Kondotty.
,Contents
1 Standard Probability Distributions 1
1.1 Discrete Probability Distributions . . . . . . . . . . . . . . 2
1.1.1 Bernoulli Distribution . . . . . . . . . . . . . . . . 2
1.1.2 Binomial Distribution . . . . . . . . . . . . . . . . 3
1.1.3 Poisson Distribution . . . . . . . . . . . . . . . . . 17
1.1.4 Negative Binomial Distribution . . . . . . . . . . . 31
1.1.5 Geometric Distribution . . . . . . . . . . . . . . . 34
1.1.6 Discrete Uniform Distribution . . . . . . . . . . . . 40
1.2 Continuous Distributions . . . . . . . . . . . . . . . . . . 43
1.2.1 Continuous Uniform or Rectangular Distribution . 43
1.2.2 Gamma Distribution . . . . . . . . . . . . . . . . . 46
1.2.3 Exponential Distribution . . . . . . . . . . . . . . 49
1.2.4 Normal Distribution . . . . . . . . . . . . . . . . . 54
1.2.5 Log-normal Distribution . . . . . . . . . . . . . . . 89
i
, ii CONTENTS
1.2.6 Beta Distribution . . . . . . . . . . . . . . . . . . . 92
1.2.7 Pareto Distribution . . . . . . . . . . . . . . . . . . 95
1.2.8 Cauchy Distribution . . . . . . . . . . . . . . . . . 96
2 Limit Theorems 98
2.1 Chebyshev’s Inequality . . . . . . . . . . . . . . . . . . . . 98
2.2 Modes of Convergence . . . . . . . . . . . . . . . . . . . . 105
2.2.1 Convergence in Distribution . . . . . . . . . . . . . 105
2.2.2 Convergence in Probability . . . . . . . . . . . . . 105
2.3 Laws of Large Numbers . . . . . . . . . . . . . . . . . . . 106
2.3.1 Weak Law of Large Numbers . . . . . . . . . . . . 106
2.3.2 Bernoulli’s Law of Large Numbers . . . . . . . . . 109
2.4 Central Limit Theorem . . . . . . . . . . . . . . . . . . . 113
2.4.1 Lévy Central Limit Theorem . . . . . . . . . . . . 114
2.4.2 De-Moivre’s-Laplace Central Limit Theorem . . . 117
3 Sampling Methods 122
3.1 Non-probability Sampling . . . . . . . . . . . . . . . . . . 132
3.2 Probability Sampling . . . . . . . . . . . . . . . . . . . . . 133
3.2.1 Simple Random Sampling . . . . . . . . . . . . . . 133
3.2.2 Stratified Random Sampling . . . . . . . . . . . . 138
3.2.3 Systematic Random Sampling . . . . . . . . . . . . 141
3.2.4 Cluster Sampling . . . . . . . . . . . . . . . . . . . 144
4 Sampling Distributions 146
4.1 Sampling Distribution of Sample Mean . . . . . . . . . . . 149
4.2 Chi-square Distribution . . . . . . . . . . . . . . . . . . . 151
