1.1.1. | Introduction to Random Variables | |
1.1.2. | Probability Mass Functions of Discrete Random Variables | |
1.1.3. | Cumulative Distribution Functions for Discrete Random Variables |
1.2.1. | Probability Density Functions of Continuous Random Variables | |
1.2.2. | Calculating Probabilities With Continuous Random Variables | |
1.2.3. | Continuous Random Variables Over Infinite Domains | |
1.2.4. | Cumulative Distribution Functions for Continuous Random Variables | |
1.2.5. | Median, Quartiles and Percentiles of Continuous Random Variables | |
1.2.6. | Finding the Mode of a Continuous Random Variable |
1.3.1. | One-to-One Transformations of Discrete Random Variables | |
1.3.2. | Many-to-One Transformations of Discrete Random Variables | |
1.3.3. | The Distribution Function Method | |
1.3.4. | The Change-of-Variables Method for Continuous Random Variables | |
1.3.5. | The Distribution Function Method With Many-to-One Transformations |
1.4.1. | Bayes' Theorem | |
1.4.2. | Extending Bayes' Theorem | |
1.4.3. | The Law of Total Probability (Extended) |
2.5.1. | Expected Values of Discrete Random Variables | |
2.5.2. | Properties of Expectation for Discrete Random Variables | |
2.5.3. | Moments of Discrete Random Variables |
2.6.1. | Variance of Discrete Random Variables | |
2.6.2. | Properties of Variance for Discrete Random Variables | |
2.6.3. | Skewness of Discrete Random Variables |
2.7.1. | Moments of Continuous Random Variables | |
2.7.2. | Expected Values of Continuous Random Variables | |
2.7.3. | Variance of Continuous Random Variables | |
2.7.4. | Skewness of Continuous Random Variables | |
2.7.5. | The Rule of the Lazy Statistician |
2.8.1. | Moment-Generating Functions | |
2.8.2. | Calculating Raw Moments Using Moment-Generating Functions | |
2.8.3. | Calculating Variance and Standard Deviation Using Moment-Generating Functions | |
2.8.4. | Identifying Discrete Distributions From Moment Generating Functions | |
2.8.5. | Identifying Continuous Distributions From Moment-Generating Functions | |
2.8.6. | Properties of Moment-Generating Functions | |
2.8.7. | Further Properties of Moment-Generating Functions |
3.9.1. | The Discrete Uniform Distribution | |
3.9.2. | Mean and Variance of Discrete Uniform Distributions | |
3.9.3. | Modeling With Discrete Uniform Distributions |
3.10.1. | The Bernoulli Distribution | |
3.10.2. | Modeling With the Bernoulli Distribution | |
3.10.3. | Mean and Variance of the Bernoulli Distribution |
3.11.1. | The Binomial Distribution | |
3.11.2. | Modeling With the Binomial Distribution | |
3.11.3. | Mean and Variance of the Binomial Distribution | |
3.11.4. | The Cumulative Distribution Function of the Binomial Distribution |
3.12.1. | The Poisson Distribution | |
3.12.2. | Modeling With the Poisson Distribution | |
3.12.3. | Mean and Variance of the Poisson Distribution | |
3.12.4. | The Cumulative Distribution Function of the Poisson Distribution | |
3.12.5. | The Poisson Approximation of the Binomial Distribution |
3.13.1. | The Geometric Distribution | |
3.13.2. | Modeling With the Geometric Distribution | |
3.13.3. | Mean and Variance of the Geometric Distribution |
3.14.1. | The Negative Binomial Distribution | |
3.14.2. | Modeling With the Negative Binomial Distribution | |
3.14.3. | Mean and Variance of the Negative Binomial Distribution |
3.15.1. | The Hypergeometric Distribution | |
3.15.2. | Modeling With the Hypergeometric Distribution |
4.16.1. | The Continuous Uniform Distribution | |
4.16.2. | Mean and Variance of Continuous Uniform Distributions | |
4.16.3. | Modeling With Continuous Uniform Distributions |
4.17.1. | The Standard Normal Distribution | |
4.17.2. | The Z-Score | |
4.17.3. | The Normal Distribution | |
4.17.4. | Mean and Variance of the Normal Distribution | |
4.17.5. | Percentage Points of the Standard Normal Distribution | |
4.17.6. | Modeling With the Normal Distribution |
4.18.1. | Approximating Discrete Random Variables as Continuous | |
4.18.2. | Normal Approximations of Binomial Distributions | |
4.18.3. | The Normal Approximation of the Poisson Distribution |
4.19.1. | The Exponential Distribution | |
4.19.2. | Modeling With the Exponential Distribution | |
4.19.3. | Mean and Variance of the Exponential Distribution |
4.20.1. | The Chi-Square Distribution | |
4.20.2. | Computing Chi-Square Probabilities From the Normal Distribution | |
4.20.3. | The Student's T-Distribution |
4.21.1. | The Gamma Function | |
4.21.2. | The Gamma Distribution | |
4.21.3. | Modeling With the Gamma Distribution |
5.25.1. | Linear Combinations of Binomial Random Variables | |
5.25.2. | Linear Combinations of Poisson Random Variables | |
5.25.3. | Linear Combinations of Chi-Square Random Variables | |
5.25.4. | Combining Two Normally Distributed Random Variables | |
5.25.5. | Combining Multiple Normally Distributed Random Variables | |
5.25.6. | I.I.D Normal Random Variables |
5.27.1. | The Change-of-Variables Method for Two Random Variables | |
5.27.2. | The Beta Distribution | |
5.27.3. | The F-Distribution |
5.28.1. | The Uniqueness Property of MGFs | |
5.28.2. | MGFs of Linear Combinations of Random Variables |
5.29.1. | Convergence in Distribution | |
5.29.2. | Convergence in Probability | |
5.29.3. | Almost Sure Convergence |
6.31.1. | The Sample Mean | |
6.31.2. | Statistics and Sampling Distributions | |
6.31.3. | The Sample Variance | |
6.31.4. | Variance of Sample Means | |
6.31.5. | Sample Means From Normal Populations | |
6.31.6. | The Central Limit Theorem | |
6.31.7. | Sampling Proportions From Finite Populations | |
6.31.8. | Point Estimates of Population Proportions | |
6.31.9. | Finite Population Correction Factors | |
6.31.10. | Applications of the Central Limit Theorem | |
6.31.11. | Distributions of Sample Variances |
6.35.1. | Posterior Distributions Under the Non-Informative Prior | |
6.35.2. | Posterior Distributions Under an Informative Prior | |
6.35.3. | Maximum a Posteriori Estimation |
6.36.1. | Spearman's Rank Correlation Coefficient | |
6.36.2. | Multiple Regression |
6.37.1. | Confidence Intervals for One Mean: Known Population Variance | |
6.37.2. | Confidence Intervals for One Mean: Unknown Population Variance | |
6.37.3. | Confidence Intervals for Two Means: Known Population Variances | |
6.37.4. | Confidence Intervals for Two Means With Unknown but Equal Population Variance | |
6.37.5. | Confidence Intervals for Two Means With Unknown and Unequal Population Variance | |
6.37.6. | Confidence Intervals for Variances | |
6.37.7. | Confidence Intervals for Proportions | |
6.37.8. | Confidence Intervals for Proportions With Finite Populations | |
6.37.9. | Confidence Intervals for Slope Parameters in Linear Regression | |
6.37.10. | Confidence Intervals for Intercept Parameters in Linear Regression |
6.38.1. | Estimating a Mean | |
6.38.2. | Estimating a Proportion for a Large Population | |
6.38.3. | Estimating a Proportion for a Small Population |
6.39.1. | One-Tailed Hypothesis Tests | |
6.39.2. | Two-Tailed Hypothesis Tests | |
6.39.3. | Type I and Type II Errors in Hypothesis Testing |
6.40.1. | Hypothesis Tests For the Proportion of a Binomial Distribution | |
6.40.2. | Hypothesis Tests For the Rate of a Poisson Distribution | |
6.40.3. | Hypothesis Tests for One Mean: Known Population Variance | |
6.40.4. | Hypothesis Tests For the Proportion of a Binomial Distribution Using Critical Regions | |
6.40.5. | Hypothesis Tests For the Rate of a Poisson Distribution Using Critical Regions |
6.41.1. | The Size and Power of a Test | |
6.41.2. | The Power Function | |
6.41.3. | The Quality of Estimators |
6.42.1. | Hypothesis Tests Using the Poisson Approximation of the Binomial Distribution | |
6.42.2. | Hypothesis Tests Using the Normal Approximation of the Binomial Distribution | |
6.42.3. | Hypothesis Tests Using the Normal Approximation of the Poisson Distribution | |
6.42.4. | Hypothesis Testing With Correlation Coefficients |
6.43.1. | The Paired-Sample Z-Test | |
6.43.2. | The Paired-Sample T-Test | |
6.43.3. | Comparing Two Proportions | |
6.43.4. | Hypothesis Tests for Two Means: Known Population Variances | |
6.43.5. | Hypothesis Tests for One Mean: Unknown Population Variance | |
6.43.6. | Testing for the Equality of Two Means | |
6.43.7. | Hypothesis Tests for the Difference Between Two Means With Unknown Yet Equal Population Variances | |
6.43.8. | Hypothesis Tests for the Difference Between Two Means With Unknown and Unequal Population Variances |
6.44.1. | Hypothesis Tests for One Variance | |
6.44.2. | Hypothesis Tests for Two Variances | |
6.44.3. | One-Factor Analysis of Variance | |
6.44.4. | Two-Factor Analysis of Variance |
7.45.1. | Introduction to Goodness of Fit | |
7.45.2. | Testing Discrete Uniform Distribution Models Using Goodness of Fit | |
7.45.3. | Testing Binomial Distribution Models Using Goodness-of-Fit: Part One | |
7.45.4. | Testing Binomial Distribution Models Using Goodness-of-Fit: Part Two | |
7.45.5. | Testing Poisson Distribution Models Using Goodness-of-Fit | |
7.45.6. | Testing Uniform Distribution Models Using Goodness-of-Fit | |
7.45.7. | Testing Normal Distribution Models Using Goodness-of-Fit |
7.46.1. | Tests for Homogeneity Using Contingency Tables | |
7.46.2. | Tests for Independence Using Contingency Tables |
7.47.1. | Order Statistics | |
7.47.2. | Confidence Intervals for Quantiles and Percentiles | |
7.47.3. | The Wilcoxon Tests | |
7.47.4. | Run Test and Test for Randomness | |
7.47.5. | Kolmogorov-Smirnov Goodness-of-Fit Test |