LECTURE NOTES on PROBABILITY and STATISTICS Eusebius …

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

SAMPLE SPACES DEFINITION : The sample space is the set of all possible outcomes of an experiment.

EXAMPLE : When we flip a coin then sample space is

where and

S = {H, T }, H denotes that the coin lands "Heads up" T denotes that the coin lands "Tails up".

For a "fair coin " we expect H and T to have the same "chance " of occurring, i.e., if we flip the coin many times then about 50 % of the outcomes will be H.

We say that the probability of H to occur is 0.5 (or 50 %) .

The probability of T to occur is then also 0.5.

1

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