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.

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

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 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

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 of
Functions 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