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 . . . . . . . . . . . . 22
1.1.4 Negative Binomial Distribution . . . . . . 40
1.1.5 Geometric Distribution . . . . . . . . . . 42
1.1.6 Discrete Uniform Distribution . . . . . . . 50
i
, ii
1.2 Continuous Distributions . . . . . . . . . . . . . 55
1.2.1 Continuous Uniform or Rectangular Dis-
tribution . . . . . . . . . . . . . . . . . . 55
1.2.2 Gamma Distribution . . . . . . . . . . . . 59
1.2.3 Exponential Distribution . . . . . . . . . 62
1.2.4 Normal Distribution . . . . . . . . . . . . 68
1.2.5 Log-normal Distribution . . . . . . . . . . 114
1.2.6 Beta Distribution . . . . . . . . . . . . . . 118
1.2.7 Pareto Distribution . . . . . . . . . . . . . 121
1.2.8 Cauchy Distribution . . . . . . . . . . . . 123
2 Limit Theorems 125
2.1 Chebyshev’s Inequality . . . . . . . . . . . . . . . 126
2.2 Modes of Convergence . . . . . . . . . . . . . . . 134
2.2.1 Convergence in Distribution . . . . . . . . 134
2.2.2 Convergence in Probability . . . . . . . . 134
2.3 Laws of Large Numbers . . . . . . . . . . . . . . 135
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 . . . . . . . . . . . . 22
1.1.4 Negative Binomial Distribution . . . . . . 40
1.1.5 Geometric Distribution . . . . . . . . . . 42
1.1.6 Discrete Uniform Distribution . . . . . . . 50
i
, ii
1.2 Continuous Distributions . . . . . . . . . . . . . 55
1.2.1 Continuous Uniform or Rectangular Dis-
tribution . . . . . . . . . . . . . . . . . . 55
1.2.2 Gamma Distribution . . . . . . . . . . . . 59
1.2.3 Exponential Distribution . . . . . . . . . 62
1.2.4 Normal Distribution . . . . . . . . . . . . 68
1.2.5 Log-normal Distribution . . . . . . . . . . 114
1.2.6 Beta Distribution . . . . . . . . . . . . . . 118
1.2.7 Pareto Distribution . . . . . . . . . . . . . 121
1.2.8 Cauchy Distribution . . . . . . . . . . . . 123
2 Limit Theorems 125
2.1 Chebyshev’s Inequality . . . . . . . . . . . . . . . 126
2.2 Modes of Convergence . . . . . . . . . . . . . . . 134
2.2.1 Convergence in Distribution . . . . . . . . 134
2.2.2 Convergence in Probability . . . . . . . . 134
2.3 Laws of Large Numbers . . . . . . . . . . . . . . 135
