LECTURE NOTES on PROBABILITY and STATISTICS Eusebius …

LECTURE NOTES

on

PROBABILITY and STATISTICS

Eusebius Doedel

TABLE OF CONTENTS

SAMPLE SPACES

Events

The Algebra of Events

Axioms of Probability

Further Properties

Counting Outcomes

Permutations

Combinations

1

5

6

9

10

13

14

21

CONDITIONAL PROBABILITY

Independent Events

45

63

DISCRETE RANDOM VARIABLES

Joint distributions

Independent random variables

Conditional distributions

Expectation

Variance and Standard Deviation

Covariance

71

82

91

97

101

108

110

SPECIAL DISCRETE RANDOM VARIABLES

The Bernoulli Random Variable

The Binomial Random Variable

The Poisson Random Variable

118

118

120

130

CONTINUOUS RANDOM VARIABLES

Joint distributions

Marginal density functions

Independent continuous random variables

Conditional distributions

Expectation

Variance

Covariance

Markov¡¯s inequality

Chebyshev¡¯s inequality

142

150

153

158

161

163

169

175

181

184

SPECIAL CONTINUOUS RANDOM VARIABLES

The Uniform Random Variable

The Exponential Random Variable

The Standard Normal Random Variable

The General Normal Random Variable

The Chi-Square Random Variable

187

187

191

196

201

206

THE CENTRAL LIMIT THEOREM

211

SAMPLE STATISTICS

The Sample Mean

The Sample Variance

Estimating the Variance of a Normal Distribution

Samples from Finite Populations

The Sample Correlation Coefficient

Maximum Likelihood Estimators

Hypothesis Testing

246

252

257

266

274

282

288

305

LEAST SQUARES APPROXIMATION

Linear Least Squares

General Least Squares

335

335

343

RANDOM NUMBER GENERATION

The Logistic Equation

Generating Random Numbers

Generating Uniformly Distributed Random Numbers

Generating Random Numbers using the Inverse Method

362

363

378

379

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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