4주차
The document discusses key concepts in probability and statistics covered during the 4th week, including: 1. Bayes' theorem, which defines how to calculate conditional probabilities and is useful for "flipping" conditional probabilities. 2. Random variables and different types of probability distributions (discrete, continuous, joint) that describe the probabilities of random variables. 3. Mathematical expectations, which define the average value of a random variable, and variance/standard deviation, which measure how spread out values are. 4. Theorems on expectation, variance and standardized random variables. Examples are provided to illustrate concepts like probability distributions, conditional probabilities, and functions of random variables.
4주차
- 1. Introduction to Probability and Statistics 4th Week (3/29) 1. Bayer’s Theorem 2. Random Variables 3. Probability Distributions 4. Mathematical Expectations (intro)
- 2. What would you do….. IF a medical test (tumor marker) inform you that you got an incurable disease (i.e. Pancreases Cancer) 1.Cry 2.Use your remaining time for some important thing 3.Invent a new iphone
- 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.
- 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?
- 12. Distribution Function for Discrete Random Variables
- 13. Distribution Function for Random Variable
- 14. Distribution Function for Discrete Random Variables Distribution Function
- 16. Example
- 17. Example
- 19. Joint Distribution: An Example X: Get A+ for P&S Y: 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/Distribution Probability Surface Probability Function
- 23. Marginal Distribution Function We call them the marginal distribution functions, or simply the distribution functions, of X and Y, respectively. Density Function
- 28. Changes of Variables: Example
- 29. Changes of Variables: Example
- 30. Probability Distributions of Functions of Random Variables
- 31. Convolutions
- 35. Applications to Geometric Probability
- 36. Mathematical Expectations*: Definition - Discrete - Continuous *in Korean: 기대값
- 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)”