Section 1.5

# Section 1.5 - Contents I Probability 5 1 Sets and...

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Contents I Probability 5 1 Sets and Probability 7 1.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.1.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . 8 1.1.2 Venn Diagrams . . . . . . . . . . . . . . . . . . . . . . 14 1.1.3 Proportions . . . . . . . . . . . . . . . . . . . . . . . . 21 1.2 Counting Techniques . . . . . . . . . . . . . . . . . . . . . . . 34 1.3 Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 1.3.1 Random Experiments, Sample Spaces and Events . . . 59 1.3.2 Probability . . . . . . . . . . . . . . . . . . . . . . . . 65 1.4 Conditional Proportions and Probabilities . . . . . . . . . . . 84 1.4.1 Conditional Proportions . . . . . . . . . . . . . . . . . 84 1.4.2 Conditional Probabilities . . . . . . . . . . . . . . . . 87 1.4.3 Multiplicative Rule . . . . . . . . . . . . . . . . . . . . 91 1.4.4 Independence . . . . . . . . . . . . . . . . . . . . . . . 92 1.5 Compound Experiments . . . . . . . . . . . . . . . . . . . . . 107 1.5.1 Finding Probabilities and Conditional Probabilities in Compound Experiments . . . . . . . . . . . . . . . . . 108 1.5.2 Notation for Events in Compound Experiments . . . . 117 1.5.3 Using the Multiplicative Rule to Find Probabilities in Compound Experiments . . . . . . . . . . . . . . . . . 119 1.5.4 More Examples . . . . . . . . . . . . . . . . . . . . . . 124 1.6 Bayes’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 137 2 Variables 153 2.1 Variables and their Distributions . . . . . . . . . . . . . . . . 154 2.1.1 Basic Definitions of Variables and Random Variables . 154 2.1.2 Classifying Variables . . . . . . . . . . . . . . . . . . . 157 2.1.3 Distributions of Variables and Random Variables . . . 158 1

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2 CONTENTS 2.1.4 Distributions of Categorical Variables . . . . . . . . . 160 2.1.5 Distributions of Numerical Discrete Variables . . . . . 165 2.1.6 Distributions of Numerical Continuous Variables . . . 170 2.2 Percentiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 2.3 Mean and Standard Deviation . . . . . . . . . . . . . . . . . . 201 2.3.1 The Mean . . . . . . . . . . . . . . . . . . . . . . . . . 201 2.3.2 Variance and Standard Deviation . . . . . . . . . . . . 210 2.4 Joint Distributions . . . . . . . . . . . . . . . . . . . . . . . . 225 2.5 Association Between Variables . . . . . . . . . . . . . . . . . 242 2.5.1 Conditional Distributions . . . . . . . . . . . . . . . . 242 2.5.2 Independence . . . . . . . . . . . . . . . . . . . . . . . 246 2.5.3 Observing Association in Scatterplots . . . . . . . . . 249 2.5.4 Covariance . . . . . . . . . . . . . . . . . . . . . . . . 253 2.5.5 Correlation . . . . . . . . . . . . . . . . . . . . . . . . 257 2.5.6 Properties of Correlation . . . . . . . . . . . . . . . . 263 2.6 Combining Variables . . . . . . . . . . . . . . . . . . . . . . . 277 2.6.1 Y = f ( X ) . . . . . . . . . . . . . . . . . . . . . . . . . 277 2.6.2 Y = aX + b . . . . . . . . . . . . . . . . . . . . . . . . 279 2.6.3 W = f ( X, Y ) . . . . . . . . . . . . . . . . . . . . . . . 283 2.6.4 W = aX + bY + c . . . . . . . . . . . . . . . . . . . . 284 2.6.5 Portfolio Analysis . . . . . . . . . . . . . . . . . . . . 291 2.6.6 Sums of Many Variables . . . . . . . . . . . . . . . . . 294 3 Important Families of Distributions 303 3.1 Using Distributions to Find Probabilities . . . . . . . . . . . 304 3.2 Binomial and Hypergeometric Distributions . . . . . . . . . . 307 3.2.1 The Binomial Distribution . . . . . . . . . . . . . . . . 307 3.2.2 The Hypergeometric Distribution . . . . . . . . . . . . 315 3.2.3 The Binomial Distribution as an Approximation to the Hypergeometric Distribution . . . . . . . . . . . . 322 3.3 The Poisson and Exponential Distributions . . . . . . . . . . 331 3.3.1 Poisson Processes . . . . . . . . . . . . . . . . . . . . . 331 3.3.2 Poisson Distributions . . . . . . . . . . . . . . . . . . . 332 3.3.3 Exponential Distributions . . . . . . . . . . . . . . . . 338 3.4 The Uniform and Normal Distributions . . . . . . . . . . . . 348 3.4.1 Uniform Distributions . . . . . . . . . . . . . . . . . . 348 3.4.2 Normal Distributions . . . . . . . . . . . . . . . . . . . 351 3.5 Sampling Distributions . . . . . . . . . . . . . . . . . . . . . . 373 3.5.1 Mean and Standard Deviation of Σ X and ¯ X . . . . . 375 3.5.2 Complete Distribution of Σ X and ¯ X . . . . . . . . . . 377
CONTENTS 3 3.5.3 The Normal Approximation to the Binomial Distribu- tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387 3.5.4 Normal Approximation to the Poisson Distribution . . 391 II Statistics 397 4 Estimation 399 4.1 Point Estimation . . . . . . . . . . . . . . . . . . . . . . . . . 400 4.1.1 Parameters, Statistics and Point Estimators . . . . . . 400 4.1.2 Assessing the Quality of a Point Estimator . . . . . . 402 4.1.3 Point Estimators for Common Parameters . . . . . . . 406 4.2 Introduction to Interval Estimation . . . . . . . . . . . . . . . 415 4.2.1 Guiding Example and Definition . . . . . . . . . . . . 415 4.2.2 Confidence Interval for μ when σ is Known . . . . . . 420 4.2.3 Determining the Sample Size . . . . . . . . . . . . . . 423 4.2.4 A Closer Look at Confidence Intervals . . . . . . . . . 424 4.3 More Confidence Intervals . . . . . . . . . . . . . . . . . . . . 428 4.3.1 Estimating μ when σ is not known . . . . . . . . . . . . 428 4.3.2 Estimating a population proportion, p . . . . . . . . . . 436 5 Hypothesis Testing 445

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4 CONTENTS
Part I Probability 5

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Chapter 1 Sets and Probability 7

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1.5. COMPOUND EXPERIMENTS 107 1.5 Compound Experiments If an experiment involves doing more than one action we’ll call it a compound experiment . The outcomes of such experiments are
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