LECTURE NOTES
on
PROBABILITY and STATISTICS
Eusebius Doedel
, TABLE OF CONTENTS
SAMPLE SPACES 1
Events 5
The Algebra of Events 6
Axioms of Probability 9
Further Properties 10
Counting Outcomes 13
Permutations 14
Combinations 21
CONDITIONAL PROBABILITY 45
Independent Events 63
DISCRETE RANDOM VARIABLES 71
Joint distributions 82
Independent random variables 91
Conditional distributions 97
Expectation 101
Variance and Standard Deviation 108
Covariance 110
,SPECIAL DISCRETE RANDOM VARIABLES 118
The Bernoulli Random Variable 118
The Binomial Random Variable 120
The Poisson Random Variable 130
CONTINUOUS RANDOM VARIABLES 142
Joint distributions 150
Marginal density functions 153
Independent continuous random variables 158
Conditional distributions 161
Expectation 163
Variance 169
Covariance 175
Markov’s inequality 181
Chebyshev’s inequality 184
SPECIAL CONTINUOUS RANDOM VARIABLES 187
The Uniform Random Variable 187
The Exponential Random Variable 191
The Standard Normal Random Variable 196
The General Normal Random Variable 201
The Chi-Square Random Variable 206
, THE CENTRAL LIMIT THEOREM 211
SAMPLE STATISTICS 246
The Sample Mean 252
The Sample Variance 257
Estimating the Variance of a Normal Distribution 266
Samples from Finite Populations 274
The Sample Correlation Coefficient 282
Maximum Likelihood Estimators 288
Hypothesis Testing 305
LEAST SQUARES APPROXIMATION 335
Linear Least Squares 335
General Least Squares 343
RANDOM NUMBER GENERATION 362
The Logistic Equation 363
Generating Random Numbers 378
Generating Uniformly Distributed Random Numbers 379
Generating Random Numbers using the Inverse Method 392
SUMMARY TABLES AND FORMULAS 403
on
PROBABILITY and STATISTICS
Eusebius Doedel
, TABLE OF CONTENTS
SAMPLE SPACES 1
Events 5
The Algebra of Events 6
Axioms of Probability 9
Further Properties 10
Counting Outcomes 13
Permutations 14
Combinations 21
CONDITIONAL PROBABILITY 45
Independent Events 63
DISCRETE RANDOM VARIABLES 71
Joint distributions 82
Independent random variables 91
Conditional distributions 97
Expectation 101
Variance and Standard Deviation 108
Covariance 110
,SPECIAL DISCRETE RANDOM VARIABLES 118
The Bernoulli Random Variable 118
The Binomial Random Variable 120
The Poisson Random Variable 130
CONTINUOUS RANDOM VARIABLES 142
Joint distributions 150
Marginal density functions 153
Independent continuous random variables 158
Conditional distributions 161
Expectation 163
Variance 169
Covariance 175
Markov’s inequality 181
Chebyshev’s inequality 184
SPECIAL CONTINUOUS RANDOM VARIABLES 187
The Uniform Random Variable 187
The Exponential Random Variable 191
The Standard Normal Random Variable 196
The General Normal Random Variable 201
The Chi-Square Random Variable 206
, THE CENTRAL LIMIT THEOREM 211
SAMPLE STATISTICS 246
The Sample Mean 252
The Sample Variance 257
Estimating the Variance of a Normal Distribution 266
Samples from Finite Populations 274
The Sample Correlation Coefficient 282
Maximum Likelihood Estimators 288
Hypothesis Testing 305
LEAST SQUARES APPROXIMATION 335
Linear Least Squares 335
General Least Squares 343
RANDOM NUMBER GENERATION 362
The Logistic Equation 363
Generating Random Numbers 378
Generating Uniformly Distributed Random Numbers 379
Generating Random Numbers using the Inverse Method 392
SUMMARY TABLES AND FORMULAS 403
