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Probability & Statistics

Math Academy official course import: Probability & Statistics

180 节课

课程大纲

  1. 1.Probability & Random Variables
    1. 1.1.1. Probability
      1. 1
        The Law of Total Probability
      2. 2
        Extending the Law of Total Probability
      3. 3
        Bayes' Theorem
      4. 4
        Extending Bayes' Theorem
    2. 2.1.2. Combinatorics
      1. 1
        Permutations With Repetition
      2. 2
        K Permutations of N With Repetition
      3. 3
        Combinations With Repetition
    3. 3.1.3. Random Variables
      1. 1
        Probability Density Functions of Continuous Random Variables
      2. 2
        Calculating Probabilities With Continuous Random Variables
      3. 3
        Continuous Random Variables Over Infinite Domains
      4. 4
        Cumulative Distribution Functions for Continuous Random Variables
      5. 5
        Median, Quartiles and Percentiles of Continuous Random Variables
      6. 6
        Finding the Mode of a Continuous Random Variable
      7. 7
        Approximating Discrete Random Variables as Continuous
      8. 8
        Simulating Random Observations
    4. 4.1.4. Functions of Random Variables
      1. 1
        One-to-One Transformations of Discrete Random Variables
      2. 2
        Many-to-One Transformations of Discrete Random Variables
      3. 3
        The Distribution Function Method
      4. 4
        The Change-of-Variables Method for Continuous Random Variables
      5. 5
        The Distribution Function Method With Many-to-One Transformations
  2. 2.Expectation
    1. 1.2.1. Expectation of Random Variables
      1. 1
        Expected Values of Discrete Random Variables
      2. 2
        Properties of Expectation for Discrete Random Variables
      3. 3
        Variance of Discrete Random Variables
      4. 4
        Moments of Discrete Random Variables
      5. 5
        Properties of Variance for Discrete Random Variables
      6. 6
        Moments of Continuous Random Variables
      7. 7
        Expected Values of Continuous Random Variables
      8. 8
        Variance of Continuous Random Variables
      9. 9
        The Rule of the Lazy Statistician
    2. 2.2.2. Moment-Generating Functions
      1. 1
        Moment-Generating Functions
      2. 2
        Calculating Moments Using Moment-Generating Functions
      3. 3
        Calculating Variance and Standard Deviation Using Moment-Generating Functions
      4. 4
        Constructing Moment-Generating Functions for Discrete Probability Distributions
      5. 5
        Constructing Moment-Generating Functions for Continuous Probability Distributions
      6. 6
        Properties of Moment-Generating Functions
      7. 7
        The Uniqueness Property of MGFs
  3. 3.Discrete Probability Distributions
    1. 1.3.1. The Discrete Uniform Distribution
      1. 1
        The Discrete Uniform Distribution
      2. 2
        Mean and Variance of the Discrete Uniform Distribution
      3. 3
        Modeling With Discrete Uniform Distributions
    2. 2.3.2. The Bernoulli Distribution
      1. 1
        The Bernoulli Distribution
      2. 2
        Mean and Variance of the Bernoulli Distribution
    3. 3.3.3. The Binomial Distribution
      1. 1
        The Binomial Distribution
      2. 2
        Modeling With the Binomial Distribution
      3. 3
        Mean and Variance of the Binomial Distribution
      4. 4
        The CDF of the Binomial Distribution
    4. 4.3.4. The Poisson Distribution
      1. 1
        The Poisson Distribution
      2. 2
        Modeling With the Poisson Distribution
      3. 3
        Mean and Variance of the Poisson Distribution
      4. 4
        The CDF of the Poisson Distribution
      5. 5
        The Poisson Approximation of the Binomial Distribution
    5. 5.3.5. The Geometric Distribution
      1. 1
        The Geometric Distribution
      2. 2
        Modeling With the Geometric Distribution
      3. 3
        Mean and Variance of the Geometric Distribution
    6. 6.3.6. The Negative Binomial Distribution
      1. 1
        The Negative Binomial Distribution
      2. 2
        Modeling With the Negative Binomial Distribution
      3. 3
        Mean and Variance of the Negative Binomial Distribution
  4. 4.Continuous Probability Distributions
    1. 1.4.1. The Normal Distribution
      1. 1
        The Z-Score
      2. 2
        The Standard Normal Distribution
      3. 3
        Symmetry Properties of the Standard Normal Distribution
      4. 4
        The Normal Distribution
      5. 5
        Mean and Variance of the Normal Distribution
      6. 6
        Percentage Points of the Standard Normal Distribution
      7. 7
        Modeling With the Normal Distribution
      8. 8
        The Empirical Rule for the Normal Distribution
      9. 9
        Normal Approximations of Binomial Distributions
      10. 10
        The Normal Approximation of the Poisson Distribution
    2. 2.4.2. The Continuous Uniform Distribution
      1. 1
        The Continuous Uniform Distribution
      2. 2
        Mean and Variance of the Continuous Uniform Distribution
      3. 3
        Modeling With Continuous Uniform Distributions
    3. 3.4.3. The Exponential Distribution
      1. 1
        The Exponential Distribution
      2. 2
        Modeling With the Exponential Distribution
      3. 3
        Mean and Variance of the Exponential Distribution
    4. 4.4.4. Other Continous Distributions
      1. 1
        The Gamma Function
      2. 2
        The Gamma Distribution
      3. 3
        The Chi-Square Distribution
      4. 4
        The Student's T-Distribution
      5. 5
        The F-Distribution
  5. 5.Combining Random Variables
    1. 1.5.1. Distributions of Two Discrete Random Variables
      1. 1
        Joint Distributions for Discrete Random Variables
      2. 2
        The Joint CDF of Two Discrete Random Variables
      3. 3
        Marginal Distributions for Discrete Random Variables
      4. 4
        Independence of Discrete Random Variables
      5. 5
        Conditional Distributions for Discrete Random Variables
      6. 6
        The Trinomial Distribution
      7. 7
        The Multinomial Distribution
    2. 2.5.2. Distributions of Two Continuous Random Variables
      1. 1
        Joint Distributions for Continuous Random Variables
      2. 2
        Marginal Distributions for Continuous Random Variables
      3. 3
        Independence of Continuous Random Variables
      4. 4
        Conditional Distributions for Continuous Random Variables
      5. 5
        The Joint CDF of Two Continuous Random Variables
      6. 6
        Properties of the Joint CDF of Two Continuous Random Variables
      7. 7
        The Bivariate Normal Distribution
    3. 3.5.3. Linear Combinations of Random Variables
      1. 1
        Linear Combinations of Binomial Random Variables
      2. 2
        Linear Combinations of Poisson Random Variables
      3. 3
        Combining Two Normally Distributed Random Variables
      4. 4
        Combining Multiple Normally Distributed Random Variables
      5. 5
        I.I.D Normal Random Variables
    4. 4.5.4. Expectation for Multivariate Distributions
      1. 1
        Expected Values of Sums and Products of Random Variables
      2. 2
        Variance of Sums of Independent Random Variables
      3. 3
        Computing Expected Values From Joint Distributions
      4. 4
        Conditional Expectation for Discrete Random Variables
      5. 5
        Conditional Variance for Discrete Random Variables
      6. 6
        The Rule of the Lazy Statistician for Two Random Variables
    5. 5.5.5. Covariance of Random Variables
      1. 1
        The Covariance of Two Random Variables
      2. 2
        Variance of Sums of Random Variables
      3. 3
        The Covariance Matrix
      4. 4
        The Correlation Coefficient for Two Random Variables
      5. 5
        The Sample Covariance Matrix
  6. 6.Parametric Inference
    1. 1.6.1. Mean, Variance, and Proportion
      1. 1
        The Sample Mean
      2. 2
        Sampling Distributions
      3. 3
        The Sample Variance
      4. 4
        Pooled Variance
      5. 5
        Variance of Sample Means
      6. 6
        Sample Means From Normal Populations
      7. 7
        Sampling Proportions From Finite Populations
      8. 8
        The Method of Moments
      9. 9
        The Method of Moments: Two-Parameter Distributions
    2. 2.6.2. The Central Limit Theorem
      1. 1
        The Central Limit Theorem
      2. 2
        Applications of the Central Limit Theorem
      3. 3
        Finite Population Corrections for Sample Means
      4. 4
        Point Estimates of Population Proportions
      5. 5
        Finite Population Corrections for Sample Proportions
    3. 3.6.3. Maximum Likelihood
      1. 1
        Product Notation
      2. 2
        Logarithmic Differentiation
      3. 3
        Likelihood Functions for Discrete Probability Distributions
      4. 4
        Log-Likelihood Functions for Discrete Probability Distributions
      5. 5
        Likelihood Functions for Continuous Probability Distributions
      6. 6
        Log-Likelihood Functions for Continuous Probability Distributions
      7. 7
        Maximum Likelihood Estimation
  7. 7.Confidence Intervals
    1. 1.7.1. One-Sample Procedures
      1. 1
        Confidence Intervals for One Mean: Known Population Variance
      2. 2
        Confidence Intervals for One Mean: Unknown Population Variance
      3. 3
        Confidence Intervals for One Means: Finite Population Correction
      4. 4
        Confidence Intervals for One Proportion
      5. 5
        Confidence Intervals for One Proportion: Finite Population Corrections
      6. 6
        Confidence Intervals for One Variance
    2. 2.7.2. Two-Sample Procedures
      1. 1
        Confidence Intervals for Two Means: Known and Unequal Population Variances
      2. 2
        Confidence Intervals for Two Means: Equal and Unknown Population Variance
      3. 3
        Confidence Intervals for Two Means: Unequal and Unknown Population Variance
      4. 4
        Confidence Intervals for Two Proportions
      5. 5
        Confidence Intervals for Paired Samples: Known Variances
      6. 6
        Confidence Intervals for Paired Samples: Unknown Variances
    3. 3.7.3. Sample Size
      1. 1
        Estimating Sample Sizes for Means
      2. 2
        Estimating Sample Sizes for Proportions
      3. 3
        Estimating Sample Sizes for Proportions: Finite Population Correction
  8. 8.Hypothesis Testing
    1. 1.8.1. One-Sample Procedures
      1. 1
        Introduction to Hypothesis Testing
      2. 2
        Hypothesis Tests for the Rate of a Poisson Distribution
      3. 3
        Critical Regions for Left-Tailed Hypothesis Tests
      4. 4
        Critical Regions for Right-Tailed Hypothesis Tests
      5. 5
        Two-Tailed Hypothesis Tests
      6. 6
        Type I and Type II Errors
      7. 7
        Hypothesis Tests for One Mean: Known Population Variance
      8. 8
        Hypothesis Tests for One Mean: Unknown Population Variance
      9. 9
        Hypothesis Tests for One Variance
    2. 2.8.2. Two-Sample Procedures
      1. 1
        Hypothesis Tests for Two Means: Known Population Variances
      2. 2
        Hypothesis Tests for Two Means: Equal But Unknown Population Variances
      3. 3
        Hypothesis Tests for Two Means: Unequal and Unknown Population Variances
      4. 4
        Hypothesis Tests for Two Proportions
      5. 5
        Hypothesis Tests for Two Means: Paired-Sample Z-Test
      6. 6
        Hypothesis Tests for Two Means: Paired-Sample T-Test
      7. 7
        Hypothesis Tests for Two Variances
    3. 3.8.3. Analysis of Variance
      1. 1
        One-Factor Within Groups and Between Groups Variation
      2. 2
        The Relationship Between SSW, SSB, SST
      3. 3
        One-Factor Analysis of Variance
  9. 9.Regression
    1. 1.9.1. Correlation and Regression
      1. 1
        The Linear Correlation Coefficient
      2. 2
        Linear Regression
      3. 3
        Residuals and Residual Plots
      4. 4
        Spearman's Rank Correlation Coefficient
      5. 5
        Confidence Intervals for Linear Regression Slope Parameters
      6. 6
        Confidence Intervals for Linear Regression Intercept Parameters
    2. 2.9.2. Linear and Nonlinear Regression With Matrices
      1. 1
        The Least-Squares Solution of a Linear System (Without Collinearity)
      2. 2
        The Least-Squares Solution of a Linear System (With Collinearity)
      3. 3
        Linear Regression With Matrices
      4. 4
        Polynomial Regression With Matrices
      5. 5
        Multiple Linear Regression With Matrices
  10. 10.Nonparametric Inference
    1. 1.10.1. Goodness-of-Fit and Order Statistics
      1. 1
        Introduction to Chi-Square Goodness-of-Fit
      2. 2
        Testing Binomial Models Using Chi-Square Goodness-of-Fit
      3. 3
        Testing Poisson Models Using Chi-Square Goodness-of-Fit
      4. 4
        Testing Continuous Uniform Models Using Chi-Square Goodness-of-Fit
      5. 5
        Testing Normal Models Using Chi-Square Goodness-of-Fit
      6. 6
        Chi-Square Tests of Independence and Homogeneity
      7. 7
        Introduction to Order Statistics