Math Academy

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.

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 Change-of-Variables Method for Continuous Random Variables
	1.3.5.	The Distribution Function Method With Many-to-One Transformations

1.4. Bayes' Theorem

	1.4.1.	Bayes' Theorem
	1.4.2.	Extending Bayes' Theorem
	1.4.3.	The Law of Total Probability (Extended)

Expectations of Random Variables

18 topics

2.5. Expectation

	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. 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.	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. 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.	Mean and Variance of Discrete Uniform Distributions
	3.9.3.	Modeling With Discrete Uniform Distributions

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.	Mean and Variance of Continuous Uniform Distributions
	4.16.3.	Modeling With Continuous Uniform Distributions

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.	Normal Approximations of Binomial Distributions
	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

14 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

5.27. 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

56 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.	Distributions of 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. 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 for Proportions With Finite Populations
	6.37.7.	Confidence Intervals for Slope Parameters in Linear Regression
	6.37.8.	Confidence Intervals for Intercept Parameters in Linear Regression

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: 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. 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.	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 the Difference Between Two Means: Known Population Variances
	6.43.5.	Hypothesis Tests for the Mean of a Normal Distribution: Unknown Population Variance
	6.43.6.	Testing for the Equality of Two Means
	6.43.7.	Hypothesis Tests for the Difference Between Two Means: Unknown Population Variances

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

Probability & Statistics

Outcomes

Content

Univariate Distributions

Joint Distributions

Estimators

Hypothesis Testing