# Probability & Statistics

This course is currently under construction. The target release date for this course is September.

## Content

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.

Upon successful completion of this course, students will have mastered the following:

### Univariate Distributions

• Compute the probability of a random variable over a set given a distribution or cumulative distribution.
• Convert between distributions and cumulative distributions.
• Construct distributions of transformed random variables.
• Calculate the mean, variance, and moment-generating function of a random variable or a linear combination of multiple random variables.
• Use moment-generating functions to calculate mean and variance.
• Understand the properties of various families of probability distributions and apply appropriate distributions to solve real-world modeling problems.
• Approximate the probability of a normal random variable using binomial and Poisson distributions.

### Joint Distributions

• Generalize prior knowledge of univariate probability distributions to joint distributions, including computing probabilities and statistics, converting between distributions and cumulative distributions, and change of variables.
• Determine whether two variables are dependent both intuitively from real-life context and mathematically by analyzing their joint probability distribution.
• Compute and interpret the covariance of two random variables, and understand the relationship between covariance and dependence.
• Calculate the mean, variance, and moment-generating function of a linear combination of random variables in terms of the means, variances, and moment-generating functions of the random variables being combined.
• Construct and interpret multiple regression models and correlation coefficients.
• Use Bayes’ theorem and marginal distributions to compute conditional distributions, conditional means, and conditional variances.

### Estimators

• Understand the concept of a point estimator and what it means for an estimator to be unbiased.
• Compute unbiased estimators for the mean, variance, and standard error of sample means.
• Compute probabilities using the student’s t-distribution and understand its relationship with the normal distribution.
• Understand convergence of random variables from both perspectives of distribution and probability.
• Reason about the assumptions and conclusions of the law of large numbers and central limit theorem.
• Fit probability models to data using maximum likelihood estimation, method of moments, and Bayesian maximum a posteriori estimation.

### Hypothesis Testing

• Construct confidence intervals for means, variances, proportions, and linear regression coefficients.
• Carry out z-tests, t-tests, and paired t-tests: formulate null and alternative hypotheses, compute p-values, and accept or reject the null hypothesis at a desired level of significance.
• Understand the relationship between significance, power, and type I/II errors.
• Generalize knowledge of t-tests for two groups to analysis of variance (ANOVA) for three or more variables.
• Perform chi-square and Kolmogorov-Smirnov goodness-of-fit tests for categorical and continuous random variables.
1.
Probability & Random Variables
19 topics
1.1. Probability
 1.1.1. Bayes' Theorem 1.1.2. Extending Bayes' Theorem 1.1.3. The Law of Total Probability (Extended)
1.2. 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 1.2.7. Approximating Discrete Random Variables as Continuous 1.2.8. Simulating Random Observations
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 Change-of-Variables Method for Continuous Random Variables 1.3.5. The Distribution Function Method With Many-to-One Transformations
1.4. Convergence of Random Variables
 1.4.1. Convergence in Distribution 1.4.2. Convergence in Probability 1.4.3. Almost Sure Convergence
2.
Expectation
17 topics
2.5. Expectation of Random variables
 2.5.1. Expected Values of Discrete Random Variables 2.5.2. Variance of Discrete Random Variables 2.5.3. Properties of Expectation for Discrete Random Variables 2.5.4. Properties of Variance for Discrete Random Variables 2.5.5. Moments of Continuous Random Variables 2.5.6. Expected Values of Continuous Random Variables 2.5.7. Variance of Continuous Random Variables 2.5.8. The Rule of the Lazy Statistician 2.5.9. Skewness of Continuous Random Variables 2.5.10. Skewness of Discrete Random Variables
2.6. Moment-Generating Functions
 2.6.1. Moment-Generating Functions 2.6.2. Calculating Moments Using Moment-Generating Functions 2.6.3. Calculating Variance and Standard Deviation Using Moment-Generating Functions 2.6.4. Identifying Discrete Distributions From Moment Generating Functions 2.6.5. Identifying Continuous Distributions From Moment-Generating Functions 2.6.6. Properties of Moment-Generating Functions 2.6.7. Further Properties of Moment-Generating Functions
3.
Discrete Random Variables
23 topics
3.7. The Discrete Uniform Distribution
 3.7.1. The Discrete Uniform Distribution 3.7.2. Mean and Variance of Discrete Uniform Distributions 3.7.3. Modeling With Discrete Uniform Distributions
3.8. The Bernoulli Distribution
 3.8.1. The Bernoulli Distribution 3.8.2. Modeling With the Bernoulli Distribution 3.8.3. Mean and Variance of the Bernoulli Distribution
3.9. The Binomial Distribution
 3.9.1. The Binomial Distribution 3.9.2. Modeling With the Binomial Distribution 3.9.3. Mean and Variance of the Binomial Distribution 3.9.4. The CDF of the Binomial Distribution
3.10. The Poisson Distribution
 3.10.1. The Poisson Distribution 3.10.2. Modeling With the Poisson Distribution 3.10.3. Mean and Variance of the Poisson Distribution 3.10.4. The CDF of the Poisson Distribution 3.10.5. The Poisson Approximation of the Binomial Distribution
3.11. The Geometric Distribution
 3.11.1. The Geometric Distribution 3.11.2. Modeling With the Geometric Distribution 3.11.3. Mean and Variance of the Geometric Distribution
3.12. The Negative Binomial Distribution
 3.12.1. The Negative Binomial Distribution 3.12.2. Modeling With the Negative Binomial Distribution 3.12.3. Mean and Variance of the Negative Binomial Distribution
3.13. The Hypergeometric Distribution
 3.13.1. The Hypergeometric Distribution 3.13.2. Modeling With the Hypergeometric Distribution
4.
Continuous Random Variables
21 topics
4.14. The Continuous Uniform Distribution
 4.14.1. The Continuous Uniform Distribution 4.14.2. Mean and Variance of Continuous Uniform Distributions 4.14.3. Modeling With Continuous Uniform Distributions
4.15. The Normal Distribution
 4.15.1. The Standard Normal Distribution 4.15.2. Symmetry Properties of the Standard Normal Distribution 4.15.3. The Z-Score 4.15.4. The Normal Distribution 4.15.5. Mean and Variance of the Normal Distribution 4.15.6. Percentage Points of the Standard Normal Distribution 4.15.7. Modeling With the Normal Distribution
4.16. Normal Approximations
 4.16.1. Normal Approximations of Binomial Distributions 4.16.2. The Normal Approximation of the Poisson Distribution
4.17. The Exponential Distribution
 4.17.1. The Exponential Distribution 4.17.2. Modeling With the Exponential Distribution 4.17.3. Mean and Variance of the Exponential Distribution
4.18. The Chi-Square Distribution
 4.18.1. The Chi-Square Distribution 4.18.2. Computing Chi-Square Probabilities From the Normal Distribution 4.18.3. The Student's T-Distribution
4.19. The Gamma Distribution
 4.19.1. The Gamma Function 4.19.2. The Gamma Distribution 4.19.3. Modeling With the Gamma Distribution
5.
Combining Random Variables
39 topics
5.20. Distributions of Two Discrete Random Variables
 5.20.1. Joint Distributions for Discrete Random Variables 5.20.2. The Joint CDF of Two Discrete Random Variables 5.20.3. Marginal Distributions for Discrete Random Variables 5.20.4. Independence of Discrete Random Variables 5.20.5. Conditional Distributions for Discrete Random Variables 5.20.6. The Trinomial Distribution 5.20.7. The Multinomial Distribution
5.21. Distributions of Two Continuous Random Variables
 5.21.1. Joint Distributions for Continuous Random Variables 5.21.2. Marginal Distributions for Continuous Random Variables 5.21.3. Independence of Continuous Random Variables 5.21.4. Conditional Distributions for Continuous Random Variables 5.21.5. The Joint CDF of Two Continuous Random Variables 5.21.6. Properties of the Joint CDF of Two Continuous Random Variables 5.21.7. The Bivariate Normal Distribution 5.21.8. The Multivariate Normal Distribution
5.22. Linear Combinations of Random Variables
 5.22.1. Linear Combinations of Binomial Random Variables 5.22.2. Linear Combinations of Poisson Random Variables 5.22.3. Linear Combinations of Chi-Square Random Variables 5.22.4. Combining Two Normally Distributed Random Variables 5.22.5. Combining Multiple Normally Distributed Random Variables 5.22.6. I.I.D Normal Random Variables
5.23. Expectation for Multivariate Distributions
 5.23.1. Expected Values of Sums and Products of Random Variables 5.23.2. Variance of Sums of Independent Random Variables 5.23.3. Computing Expected Values From Joint Distributions 5.23.4. Conditional Expectation for Discrete Random Variables 5.23.5. The Law of Iterated Expectations 5.23.6. Conditional Variance for Discrete Random Variables 5.23.7. The Law of Total Variance
5.24. Covariance of Random Variables
 5.24.1. The Covariance of Two Random Variables 5.24.2. Variance of Sums of Random Variables 5.24.3. The Covariance Matrix 5.24.4. The Correlation Coefficient for Two Random Variables 5.24.5. Interpreting the Correlation Coefficient 5.24.6. The Sample Covariance Matrix
5.25. Functions of Two Random Variables
 5.25.1. The Change-of-Variables Method for Two Random Variables 5.25.2. The Beta Distribution 5.25.3. The F-Distribution
5.26. The Moment-Generating Function Method
 5.26.1. The Uniqueness Property of MGFs 5.26.2. MGFs of Linear Combinations of Random Variables
6.
Parametric Inference
27 topics
6.27. Point Estimation
 6.27.1. The Sample Mean 6.27.2. Statistics and Sampling Distributions 6.27.3. The Sample Variance 6.27.4. Variance of Sample Means 6.27.5. Sample Means From Normal Populations 6.27.6. The Central Limit Theorem 6.27.7. Sampling Proportions From Finite Populations 6.27.8. Point Estimates of Population Proportions 6.27.9. Finite Population Corrections Factor for the Mean Sample Distribution 6.27.10. Finite Population Corrections Factor for the Proportion Sample Distribution 6.27.11. Applications of the Central Limit Theorem 6.27.12. Distributions of Sample Variances
6.28. Estimators
 6.28.1. Biased vs. Unbiased Estimators 6.28.2. Consistent Estimators
6.29. Linear Regression and Correlation
 6.29.1. Spearman's Rank Correlation Coefficient 6.29.2. Multiple Regression
6.30. Sample Size
 6.30.1. Estimating a Mean 6.30.2. Estimating Samples Sizes for Proportions From Large Populations 6.30.3. Estimating a Proportion for a Small Population
6.31. The Method of Moments
 6.31.1. The Method of Moments
6.32. Maximum Likelihood
 6.32.1. Likelihood Functions for Discrete Probability Distributions 6.32.2. Log-Likelihood Functions for Discrete Probability Distributions 6.32.3. Likelihood Functions for Continuous Probability Distributions 6.32.4. Log-Likelihood Functions for Continuous Probability Distributions 6.32.5. Maximum Likelihood Estimation 6.32.6. Properties of Maximum Likelihood Estimators 6.32.7. Consistency of Maximum Likelihood Estimators
7.
Hypothesis Testing
22 topics
7.33. One-Sample Procedures
 7.33.1. One-Tailed Hypothesis Tests 7.33.2. Two-Tailed Hypothesis Tests 7.33.3. Type I and Type II Errors in Hypothesis Testing 7.33.4. Hypothesis Tests For the Rate of a Poisson Distribution 7.33.5. Hypothesis Tests For the Rate of a Poisson Distribution Using Critical Regions 7.33.6. Hypothesis Tests For the Proportion of a Binomial Distribution 7.33.7. Hypothesis Tests For the Proportion of a Binomial Distribution Using Critical Regions 7.33.8. Hypothesis Tests for One Mean: Known Population Variance 7.33.9. Hypothesis Tests for One Mean: Unknown Population Variance 7.33.10. Hypothesis Tests Using the Poisson Approximation of the Binomial Distribution 7.33.11. Hypothesis Tests Using the Normal Approximation of the Binomial Distribution 7.33.12. Hypothesis Tests Using the Normal Approximation of the Poisson Distribution
7.34. Quality of Tests and Estimators
 7.34.1. The Size and Power of a Test 7.34.2. The Power Function 7.34.3. The Quality of Estimators
7.35. Two-Sample Procedures
 7.35.1. Hypothesis Tests for Two Means: Known Population Variances 7.35.2. Hypothesis Tests for Two Means: Equal But Unknown Population Variances 7.35.3. Hypothesis Tests for Two Means: Unequal and Unknown Population Variances 7.35.4. Hypothesis Tests for Differences in Proportions 7.35.5. Hypothesis Tests for Two Means: Paired-Sample Z-Test 7.35.6. Hypothesis Tests for Two Means: Paired-Sample T-Test 7.35.7. Hypothesis Testing With Correlation Coefficients
8.
Confidence Intervals
14 topics
8.36. One-Sample Procedures
 8.36.1. Confidence Intervals for One Mean: Known Population Variance 8.36.2. Confidence Intervals for One Mean: Unknown Population Variance 8.36.3. Confidence Intervals for Proportions 8.36.4. Confidence Intervals for Proportions From Small Populations 8.36.5. Confidence Intervals for Variances 8.36.6. Confidence Intervals for Slope Parameters in Linear Regression 8.36.7. Confidence Intervals for Intercept Parameters in Linear Regression
8.37. Two-Sample Procedures
 8.37.1. Confidence Intervals for Two Means: Known and Unequal Population Variances 8.37.2. Pooled Variance 8.37.3. Confidence Intervals for Two Means: Equal and Unknown Population Variance 8.37.4. Confidence Intervals for Two Means: Unequal and Unknown Population Variance 8.37.5. Confidence Intervals for Differences in Proportions 8.37.6. Confidence Intervals for Two Means: Paired-Sample Z-Test 8.37.7. Confidence Intervals for Two Means: Paired-Sample T-Test
9.
Nonparametric Inference
14 topics
9.38. Chi-Square Goodness-of-Fit Tests
 9.38.1. Introduction to Goodness of Fit 9.38.2. Testing Discrete Uniform Distribution Models Using Goodness of Fit 9.38.3. Testing Binomial Distribution Models Using Goodness-of-Fit: Part One 9.38.4. Testing Binomial Distribution Models Using Goodness-of-Fit: Part Two 9.38.5. Testing Poisson Distribution Models Using Goodness-of-Fit 9.38.6. Testing Uniform Distribution Models Using Goodness-of-Fit 9.38.7. Testing Normal Distribution Models Using Goodness-of-Fit
9.39. Contingency Tables
 9.39.1. Tests for Homogeneity Using Contingency Tables 9.39.2. Tests for Independence Using Contingency Tables
9.40. Other Nonparametric Methods
 9.40.1. Order Statistics 9.40.2. Confidence Intervals for Quantiles and Percentiles 9.40.3. The Wilcoxon Tests 9.40.4. Run Test and Test for Randomness 9.40.5. Kolmogorov-Smirnov Goodness-of-Fit Test
10.
ANOVA
4 topics
10.41. Analysis of Variance
 10.41.1. Hypothesis Tests for One Variance 10.41.2. Hypothesis Tests for Two Variances 10.41.3. One-Factor Analysis of Variance 10.41.4. Two-Factor Analysis of Variance
11.
Bayesian Statistics
3 topics
11.42. Bayesian Inference
 11.42.1. Posterior Distributions Under the Non-Informative Prior 11.42.2. Posterior Distributions Under an Informative Prior 11.42.3. Maximum a Posteriori Estimation