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Introduction to Probability and Statistics                                     5th Week (4/5)     1. Mathematical Expectat...
Mathematical Expectations*: Definition- Discrete- Continuous                                      *in Korean: 기대값
Mathematical Expectations: Example
Mathematical Expectations: Example
Functions of Random Variables
Functions of Random Variables
Functions of Random Variables
A Few Theorems on Expectation
The Variance and Standard Deviation
The Variance and Standard Deviation
The Variance and Standard Deviation
The Variance and Standard Deviation
A Few Theorems on Variance
Compare! Vs. is true for any random variables is true for only independent variables is true for only independent variable...
Standardized Random Variables
Moments
Moments
Moment Generating Functions
Some Theorems on Moment Generating            Functions
Characteristic Functions
Some Theorems on Characteristic          Functions
Importance of Characteristic Functions
Variance for Joint Distributions
Covariance
Some Theorems on Covariance
Correlation Coefficient
• Correlation coefficient                            ⇒ 양의 상관관계                            ⇒ 음의 상관관계                       ...
Conditional Expectation, Variance, and              Moments
Understanding Conditional Probability   FunctionWhen X=x, Conditional        When Y=y, ConditionalProbability Function for...
Chebyshev’s Inequality
Law of Large Numbers
Other Measures of Central Tendency
Percentiles
Other Measures of Dispersion
Skewness
Kurtosis
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5주차

  1. 1. Introduction to Probability and Statistics 5th Week (4/5) 1. Mathematical Expectations2. Special Probability Distributions
  2. 2. Mathematical Expectations*: Definition- Discrete- Continuous *in Korean: 기대값
  3. 3. Mathematical Expectations: Example
  4. 4. Mathematical Expectations: Example
  5. 5. Functions of Random Variables
  6. 6. Functions of Random Variables
  7. 7. Functions of Random Variables
  8. 8. A Few Theorems on Expectation
  9. 9. The Variance and Standard Deviation
  10. 10. The Variance and Standard Deviation
  11. 11. The Variance and Standard Deviation
  12. 12. The Variance and Standard Deviation
  13. 13. A Few Theorems on Variance
  14. 14. Compare! Vs. is true for any random variables is true for only independent variables is true for only independent variables Not “Var(X) – Var(Y)”
  15. 15. Standardized Random Variables
  16. 16. Moments
  17. 17. Moments
  18. 18. Moment Generating Functions
  19. 19. Some Theorems on Moment Generating Functions
  20. 20. Characteristic Functions
  21. 21. Some Theorems on Characteristic Functions
  22. 22. Importance of Characteristic Functions
  23. 23. Variance for Joint Distributions
  24. 24. Covariance
  25. 25. Some Theorems on Covariance
  26. 26. Correlation Coefficient
  27. 27. • Correlation coefficient ⇒ 양의 상관관계 ⇒ 음의 상관관계 ⇒ 무 상관관계
  28. 28. Conditional Expectation, Variance, and Moments
  29. 29. Understanding Conditional Probability FunctionWhen X=x, Conditional When Y=y, ConditionalProbability Function for Y Probability Function for X
  30. 30. Chebyshev’s Inequality
  31. 31. Law of Large Numbers
  32. 32. Other Measures of Central Tendency
  33. 33. Percentiles
  34. 34. Other Measures of Dispersion
  35. 35. Skewness
  36. 36. Kurtosis

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