How It Works

Probability & Statistics

This course is currently under construction. The target release date for this course is March.
Learn the mathematics of chance and use it to draw precise conclusions about possible outcomes of uncertain events. Analyze real-world data using mathematically rigorous techniques.



Univariate Distributions

Joint Distributions


Hypothesis Testing

Probability & Random Variables
15 topics
1.1. Introduction to Discrete Random Variables
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. Introduction to Continuous Random Variables
1.2.1. Probability Density Functions of Continuous Random Variables
1.2.2. Cumulative Distribution Functions for Continuous Random Variables
1.2.3. Median, Quartiles and Percentiles of Continuous Random Variables
1.2.4. Finding the Mode of a Continuous Random Variable
1.3. Functions of Random Variables
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 Distribution Function Method With Many-to-One Transformations
1.3.5. The Change-of-Variables Method for Continuous Random Variables
1.3.6. Conditional Expectation for Continuous Random Variables
1.4. Bayes' Theorem
1.4.1. Introduction to Bayes' Theorem
1.4.2. Generalizing Bayes' Theorem
Expectations of Random Variables
17 topics
2.5. Expectation
2.5.1. Expected Values of Discrete Random Variables
2.5.2. Expected Values of Transformed Discrete Random Variables
2.5.3. Moments of Discrete Random Variables
2.6. Variance
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. Expectations of Continuous Random Variables
2.7.1. Expected Values of Continuous Random Variables
2.7.2. Variance of Continuous Random Variables
2.7.3. Skewness of Continuous Random Variables
2.7.4. The Rule of the Lazy Statistician
2.8. Moment-Generating Functions
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
Discrete Random Variables
23 topics
3.9. The Discrete Uniform Distribution
3.9.1. The Discrete Uniform Distribution
3.9.2. The Mean and Variance of the Discrete Uniform Distribution
3.9.3. Modeling With the Discrete Uniform Distribution
3.10. The Bernoulli Distribution
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. The Binomial 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. The Poisson 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. The Geometric 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. The Negative Binomial 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. The Hypergeometric Distribution
3.15.1. The Hypergeometric Distribution
3.15.2. Modeling With the Hypergeometric Distribution
Continuous Random Variables
21 topics
4.16. The Continuous Uniform Distribution
4.16.1. The Continuous Uniform Distribution
4.16.2. The Mean and Variance of the Continuous Uniform Distribution
4.16.3. Modeling With the Continuous Uniform Distribution
4.17. The Normal Distribution
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. Normal Approximations
4.18.1. Approximating Discrete Random Variables as Continuous
4.18.2. The Normal Approximation of the Binomial Distribution
4.18.3. The Normal Approximation of the Poisson Distribution
4.19. The Exponential 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. The Chi-Square 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. The Gamma Distribution
4.21.1. The Gamma Function
4.21.2. The Gamma Distribution
4.21.3. Modeling With the Gamma Distribution
Combining Random Variables
18 topics
5.25. Distributions of Linear Combinations of Random Variables
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
5.26. The Correlation Coefficient
5.26.1. The Covariance of Two Random Variables
5.26.2. The Correlation Coefficient of Two Random Variables
5.26.3. Interpreting the Correlation Coefficient
5.26.4. The Sample Covariance Matrix
5.27. Functions of Two 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. The Moment-Generating Function Method
5.28.1. The Uniqueness Property of MGFs
5.28.2. MGFs of Linear Combinations of Random Variables
5.29. Convergence of Random Variables
5.29.1. Convergence in Distribution
5.29.2. Convergence in Probability
5.29.3. Almost Sure Convergence
Parametric Inference
52 topics
6.31. Point Estimation
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. Sampling Distributions of Sample Means From Normal Populations
6.31.6. Distributions of Sample Means With Unknown Population Variance
6.31.7. The Central Limit Theorem
6.31.8. Applications of the Central Limit Theorem
6.31.9. Distributions of Sample Variances
6.35. Bayesian Inference
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. Linear Regression and Correlation
6.36.1. Spearman's Rank Correlation Coefficient
6.36.2. Multiple Regression
6.37. Confidence Intervals
6.37.1. Confidence Intervals for Population Means: Known Population Variance
6.37.2. Confidence Intervals for Population Means: Unknown Population Variance
6.37.3. Confidence Intervals for Two Means
6.37.4. Confidence Intervals for Variances
6.37.5. Confidence Intervals for Proportions
6.37.6. Confidence Intervals With Linear Regression Parameters
6.38. Sample Size
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. Introduction to Hypothesis Testing
6.39.1. Hypothesis Testing
6.39.2. One-Tailed and Two-Tailed Hypothesis Tests
6.39.3. The Significance Level of a One-Tailed Hypothesis Test
6.39.4. The Significance Level of a Two-Tailed Hypothesis Test
6.39.5. Critical Regions and Critical Values in Hypothesis Testing
6.39.6. Type I and Type II Errors in Hypothesis Testing
6.40. Hypothesis Tests for Discrete Random Variables
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 the Mean of a Normal Distribution
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. Quality of Tests and Estimators
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. Hypothesis Testing Using Approximations
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. Further Hypothesis Testing
6.43.1. Hypothesis Tests Using the Central Limit Theorem
6.43.2. P-Values in Hypothesis Testing
6.43.3. Comparing Two Proportions
6.43.4. The Z-Test
6.43.5. The T-Test
6.43.6. Testing for the Equality of Two Means
6.43.7. The Paired T-Test
6.44. Analysis of Variance
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
Nonparametric Inference
14 topics
7.45. Chi-Square Goodness-of-Fit Tests
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. Contingency Tables
7.46.1. Tests for Homogeneity Using Contingency Tables
7.46.2. Tests for Independence Using Contingency Tables
7.47. Other Nonparametric Methods
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