Introduction to Probability and Statistics                                     4th Week (3/29)               1. Bayer’s Th...
What would you do…..IF a medical test (tumor marker) inform you that yougot an incurable disease (i.e. Pancreases Cancer)1...
Baye’s Theorem: Definition
Baye’s Theorem: Proof
Baye’s Theorem: When do we need?• Why do we care??• Why is Bayes’ Rule useful??• It turns out that sometimes it is very us...
Baye’s Theorem: Example
Random Variables
Las Vegas       777(Jack Pot) => 1 million dollars (1)       Others: Bam => 0 dollars (0)            How often do you get ...
Discrete Probability Distributions
Discrete Probability Distributions
Distribution Function
Distribution Function for Discrete Random Variables
Distribution Function for Random Variable
Distribution Function for Discrete Random Variables                              Distribution Function
Continuous Probability Distributions
Example
Example
Joint Distribution
Joint Distribution: An ExampleX: Get A+ for P&SY: Get a great boy/girl friend                                             ...
Discrete Joint Probability Function
Discrete Joint Distribution Function                                    Probability Function (it’s like a point)Understand...
Continuous Joint Distribution                  Function/DistributionProbability SurfaceProbability Function
Marginal Distribution FunctionWe call them the marginal distribution functions, or simply the distributionfunctions, of X ...
Independent Random Variables
Independent Random Variables
Changes of Variables
Changes of Variables
Changes of Variables: Example
Changes of Variables: Example
Probability Distributions ofFunctions of Random Variables
Convolutions
Conditional Distributions: Discrete
Conditional Distributions: Continuous
Conditional Distributions: Example
Applications to Geometric Probability
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
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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
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