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# 4주차

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### 4주차

1. 1. Introduction to Probability and Statistics 4th Week (3/29) 1. Bayer’s Theorem 2. Random Variables 3. Probability Distributions4. Mathematical Expectations (intro)
2. 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. 3. Baye’s Theorem: Definition
4. 4. Baye’s Theorem: Proof
5. 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. 6. Baye’s Theorem: Example
7. 7. Random Variables
8. 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. 9. Discrete Probability Distributions
10. 10. Discrete Probability Distributions
11. 11. Distribution Function
12. 12. Distribution Function for Discrete Random Variables
13. 13. Distribution Function for Random Variable
14. 14. Distribution Function for Discrete Random Variables Distribution Function
15. 15. Continuous Probability Distributions
16. 16. Example
17. 17. Example
18. 18. Joint Distribution
19. 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. 20. Discrete Joint Probability Function
21. 21. Discrete Joint Distribution Function Probability Function (it’s like a point)Understand the difference between Distribution Function (it’s like an area)
22. 22. Continuous Joint Distribution Function/DistributionProbability SurfaceProbability Function
23. 23. Marginal Distribution FunctionWe call them the marginal distribution functions, or simply the distributionfunctions, of X and Y, respectively. Density Function
24. 24. Independent Random Variables
25. 25. Independent Random Variables
26. 26. Changes of Variables
27. 27. Changes of Variables
28. 28. Changes of Variables: Example
29. 29. Changes of Variables: Example
30. 30. Probability Distributions ofFunctions of Random Variables
31. 31. Convolutions
32. 32. Conditional Distributions: Discrete
33. 33. Conditional Distributions: Continuous
34. 34. Conditional Distributions: Example
35. 35. Applications to Geometric Probability
36. 36. Mathematical Expectations*: Definition- Discrete- Continuous *in Korean: 기대값
37. 37. Mathematical Expectations: Example
38. 38. Mathematical Expectations: Example
39. 39. Functions of Random Variables
40. 40. Functions of Random Variables
41. 41. Functions of Random Variables
42. 42. A Few Theorems on Expectation
43. 43. The Variance and Standard Deviation
44. 44. The Variance and Standard Deviation
45. 45. The Variance and Standard Deviation
46. 46. The Variance and Standard Deviation
47. 47. A Few Theorems on Variance
48. 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. 49. Standardized Random Variables