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- 1. Introduction to Probability and Statistics 4th Week (3/29) 1. Bayer’s Theorem 2. Random Variables 3. Probability Distributions4. Mathematical Expectations (intro)
- 2. What would you do…..IF a medical test (tumor marker) inform you that yougot an incurable disease (i.e. Pancreases Cancer)1.Cry2.Use your remaining time for some important thing3.Invent a new iphone
- 3. Baye’s Theorem: Definition
- 4. Baye’s Theorem: Proof
- 5. Baye’s Theorem: When do we need?• Why do we care??• Why is Bayes’ Rule useful??• It turns out that sometimes it is very useful to be able to “flip” conditional probabilities. That is, we may know the probability of A given B, but the probability of B given A may not be obvious.
- 6. Baye’s Theorem: Example
- 7. Random Variables
- 8. Las Vegas 777(Jack Pot) => 1 million dollars (1) Others: Bam => 0 dollars (0) How often do you get “1”? How much do you put money to get 1 million dollars?
- 9. Discrete Probability Distributions
- 10. Discrete Probability Distributions
- 11. Distribution Function
- 12. Distribution Function for Discrete Random Variables
- 13. Distribution Function for Random Variable
- 14. Distribution Function for Discrete Random Variables Distribution Function
- 15. Continuous Probability Distributions
- 16. Example
- 17. Example
- 18. Joint Distribution
- 19. Joint Distribution: An ExampleX: Get A+ for P&SY: Get a great boy/girl friend X A+ Others - Dependent? - Independent? Get a friend Y No friend
- 20. Discrete Joint Probability Function
- 21. Discrete Joint Distribution Function Probability Function (it’s like a point)Understand the difference between Distribution Function (it’s like an area)
- 22. Continuous Joint Distribution Function/DistributionProbability SurfaceProbability Function
- 23. Marginal Distribution FunctionWe call them the marginal distribution functions, or simply the distributionfunctions, of X and Y, respectively. Density Function
- 24. Independent Random Variables
- 25. Independent Random Variables
- 26. Changes of Variables
- 27. Changes of Variables
- 28. Changes of Variables: Example
- 29. Changes of Variables: Example
- 30. Probability Distributions ofFunctions of Random Variables
- 31. Convolutions
- 32. Conditional Distributions: Discrete
- 33. Conditional Distributions: Continuous
- 34. Conditional Distributions: Example
- 35. Applications to Geometric Probability
- 36. Mathematical Expectations*: Definition- Discrete- Continuous *in Korean: 기대값
- 37. Mathematical Expectations: Example
- 38. Mathematical Expectations: Example
- 39. Functions of Random Variables
- 40. Functions of Random Variables
- 41. Functions of Random Variables
- 42. A Few Theorems on Expectation
- 43. The Variance and Standard Deviation
- 44. The Variance and Standard Deviation
- 45. The Variance and Standard Deviation
- 46. The Variance and Standard Deviation
- 47. A Few Theorems on Variance
- 48. 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)”
- 49. Standardized Random Variables

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