1. Statistics for Financial Engineering Part1: Probability Instructor: Youngju Lee MFE, Haas Business School University of California, Berkeley

2. Overview of Class Part1: Probability – March 23 rd, 2006 Part2: Statistics – March 25 th, 2006 Class will be organized as  Definitions  Some comments about from definition  Problems  Applications in financial engineering – I will give short examples how I apply these concepts in my real life and practice since I assume you do not have any idea about financial engineering as of now.

3. Probability 1. Probability 2. Random Variables – Discrete and Continuous 3. Distribution and Probability Density 4. Moments and Moments Generating Function 5. Stochastic Independence 6. Basic Limit Theorem

4. Probability Definition1

5. Probability Some consequences of definition 1

6. Probability Try this

7. Probability In finance world? This is the very basic concept of everything. – States, Monte-Carlo simulations and Binomial Trees, etc.

8. Conditional Probability Definition 2

9. Conditional Probability Some consequences from definition 2

10. Conditional Probability Try this.

11. Conditional Probability In finance world? Fancy empirical model – Regime Switch Model

12. Independence Definition 3

13. Independence Some consequences from definition 3

14. Independence Try this. – Easy! Six fair dice are tossed once. What is the probability that all six faces appear? Seven fair dice are tossed once. What is the probability that every face appears at least once?

15. Independence In finance world? Is there any independent event in the financial world or at least in practice?

16. Random Variables Definition 4

17. Discrete Random Variable Definition 5: Binomial distribution is associated with binomial experiments – success or fail

18. Discrete Random Variables Definition 6: Poisson distribution

19. Discrete Random Variables Definition 7: Discrete uniform distribution

20. Discrete Random Variables Definition 8: Hyper-geometric distribution

21. Discrete Random Variables Definition 9: Negative binomial distribution

22. Discrete Random Variables Definition 10: Multi-nominal distribution

23. Continuous Random Variables Definition 11: Normal distribution

24. Continuous Random Variables Some consequences from definition 11 Normal distribution is symmetric. Normal distribution has maximum value at mean.

25. Continuous Random Variables Try this.

26. Continuous Random Variables In finance world? Everything is assumed normal distribution in financial engineering. To check normality, Use K-S test or Normal Probability Plot. I will cover this later.

27. Continuous Random Variables Definition 12: Gamma distribution

28. Continuous Random Variables Definition 13: Chi-square distribution

29. Continuous Random Variables Definition 14: Negative exponential distribution

30. Continuous Random Variables Definition 15: Continuous uniform distribution

31. Continuous Random Variables Definition 16: Beta distribution

32. Continuous Random Variables Definition 17: Cauchy distribution

33. Continuous Random Variables Definition 18: Lognormal distribution

34. Continuous Random Variables Definition 19: Bi-variate normal distribution

35. D.F. and P.D.F. Definition 20: The distribution function

36. D.F. and P.D.F. Some consequences from definition 20

37. D.F. and P.D.F. Try this. – It is better to know what logistic distribution is.

38. D.F. and P.D.F. In finance world? You probably want to remember some consequences from last slide. We use this all the time to make trading signals.

39. D.F. and P.D.F. Definition 21: Joint distribution function

40. D.F. and P.D.F. Definition 22: Quantile of a distribution

41. D.F. and P.D.F. Definition 23: Mode

42. D.F. and P.D.F. Try this. Let X be an r.v. with p.d.f. f symmetric about a constant c then show c is a median of f.

43. D.F. and P.D.F. In finance world?

44. Moments Definition 24: Moments of random variables

45. Moments Some consequences from definition 24

46. Moments Try this. A roulette wheel has 38 slots of which 18 are red, 18 black, and 2 green. Suppose a gambler is placing a bet of $M on red. What is the gambler’s expected gain or loss and what is the standard deviation?

47. Moments Try this. But do not calculate! Let X be an r.v. taking on the values -2,-1,1,2 each with probability 0.25. Set Y=X*X and compute the following quantities. EX, Var(X), EY and Var(Y).

48. Moments In finance world? I do not think you can be in finance industry without talking about Sharpe ratio a lot. (mean/sd) We also need to look at skewness and kurtosis.

49. Stochastic Independence Definition 25: Stochastic independence

50. Stochastic Independence Some consequences from definition 25

51. Stochastic Independence Some consequences from definition 25

52. The Central Limit Theorem Definition 26: Central Limit Theorem