1. Introduction to Probability and Statistics
5th Week (4/5)
1. Mathematical Expectations
2. Special Probability Distributions

2. Mathematical Expectations*: Definition
- Discrete
- Continuous
*in Korean: 기대값

3. Mathematical Expectations: Example

4. Mathematical Expectations: Example

5. Functions of Random Variables

6. Functions of Random Variables

7. Functions of Random Variables

8. A Few Theorems on Expectation

9. The Variance and Standard Deviation

10. The Variance and Standard Deviation

11. The Variance and Standard Deviation

12. The Variance and Standard Deviation

13. A Few Theorems on Variance

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. Standardized Random Variables

16. Moments

17. Moments

18. Moment Generating Functions

19. Some Theorems on Moment Generating
Functions

20. Characteristic Functions

21. Some Theorems on Characteristic
Functions

22. Importance of Characteristic Functions

23. Variance for Joint Distributions

24. Covariance

25. Some Theorems on Covariance

26. Correlation Coefficient

27. • Correlation coefficient
⇒ 양의 상관관계
⇒ 음의 상관관계
⇒ 무 상관관계

28. Conditional Expectation, Variance, and
Moments

29. Understanding Conditional Probability
Function
When X=x, Conditional When Y=y, Conditional
Probability Function for Y Probability Function for X

30. Chebyshev’s Inequality

31. Law of Large Numbers

32. Other Measures of Central Tendency

33. Percentiles

34. Other Measures of Dispersion

35. Skewness

36. Kurtosis