Probability Theory: Dealing with Uncertainties

1. Probability theory provides a mathematical framework for dealing with uncertainties. Appendix A: Probability Theory Many processes inevitably involves uncertainties, which may result from error, noise, imprecision, incompletion, distortion, etc.. A.1 Probabilistic Models -- A probabilistic model is a mathematical description of a process under an uncertain situation (i.e., random process). 1

2. Elements of probability model Random process : whose outcome o is not predictable Sample space S: the set of all possible outcomes Event E: any subset of S Probability law P : the collective probability of the outcomes of an event,e.g. 2

3. Probability :frequency of occurring o, or degree of belief in o (e.g., war) Example: A coin is tossed 3 times. Let the probability of a head on an individual toss be p. Sample space: Probability Law:

4. Properties of probability laws: Let A, B, C: events

5. Axioms of probability Conditional Probability The proportion of F in E 5

6. Bayes’ Formula • Total Probability Theorem: Partition: 6

7. Bayes’ formula becomes Multiplication Rule: 7

8. 8

9. “Disjoint” simplifies “Independent” simplifies 9 9

10. Example: Two successive rolls of a 4-sided die in which all 16 possible outcomes have probability 1/16. 10 10

11. ((1,2), (2,1), (2,2)) ((4,2), (2,4), (3,2), (2,3), (2,2)) 11 11

12. A.2 Random Variables -- A random variable is a function, which assigns to each outcome a number that may represent a label or a quantity. • Example (label): Coin tossing Define random variable • Example (quantity): A coin is tossed 3 times (head: +$1; tail: -$1) Define random variable X as the net earned. 12

13. A.2.1 Cumulative Distribution Function (CDF), Probability Density Function (PDF) and Probability Mass Function (PMF) The cumulative distribution function (CDF)F of a random variable X for a real number a is Continuous: probability density function (PDF) Discrete: probability mass function (PMF) 13

14. Example: A coin is tossed 3 times. PMF: 14 14

15. CDF: 15 15

16. A.2.2 Joint CDF and Joint PDF -- involving multiple random variables Example: A coin is tossed 3 times. RVs: X (head: +$1; tail: -$1); Y (head: +$2; tail: $0) 16 16

17. Joint CDF: : joint PDF Marginal CDF: Marginal PDF: • Marginal functions are to eliminate the influences of some random variables. 17

18. Conditional distribution: Given joint PDF , the marginal PDF can be calculated and in turn the conditional distribution can be computed. Multiplication rule: 18

19. Proof: 19 19

20. Bayes’ rule: Proof: • If X and Y are independent, 20

21. * What is the relationship between events and random variables (rv)? -- rvs are a special case of events, i.e., events containing only one outcome (singleton events). * What is the difference between ordinary variables (ov) and random variables (rv)? -- rv’s are associated with probability distributions while ovs are not. 21

22. A.2.5 Expectation Expectation: Example: 22 22

23. Properties of expectation: Proof: (Assignment) 23 23

24. A.2.6 Variance Variance: : standard deviation • Properties of variance: 24 24

25. Covariance: 25 25

26. Properties of covariance: 26 26

27. (Assignment) vi) If are independent, the covariance of any two rv’s is zero, Correlation: 27 27

28. A.3 Special Random Variables A.3.1 Discrete Distributions Bernoulli distribution • Random variable X takes 0/1, e.g., coin tossing. Let p be the probability that X = 1. • Bernoulli probability: 28 28

29. Binomial distribution -- N identical independent Bernoulli trials • Random variable X represents the number of 1s. • Binomial probability: Multinomial distribution -- N iid trials, each takes one of K states 29 29

30. Multinomial probability: 30 30

31. Geometric distribution -- rv X represents #Bernoulli tosses needed for a head to come up for the first time Geometric probability: Poisson distribution -- rv X represents, e.g., i) #typos in a book with a total of n words, ii) #cars involved in accidents in a city Poisson probability: 31 31

32. A.3.2 Continuous Distributions Uniform distribution: -- X takes values equably over the interval [a, b] Uniform density: 32 32

33. 33

34. Normal (Gaussian) distribution: Normal density: 34 34

35. Exponential distribution: Exponential density: Rayleigh distribution: Rayleigh density: 35

36. A.3.3 Hybrid Distributions Theorem 1:Let f be a differentiable strictly increasing or strictly decreasing function defined on I. Let X be a continuous random variable having density . Let having density . Then, Proof: Let the CDFs ofrandom variables X and Y. 36

37. (a) f strictly increasing 34

38. (b) f strictly decreasing 34

39. Example:Z-normalization Proof: 39 39

40. 40 40

41. Theorem 2: Let be independent random variables having the respective normal densities . Then, has the normal density Proof: Assume that , then Since are independent, 41

42. 42

43. Example: 43

44. Theorem 3: Central Limit Theorem Example: 44

45. Chi-square distribution Chi-square density: Properties: 45 45

46. Gamma density: Chi-square density 46 46

47. Erlang density: Beta density: Dirichlet density: 47 47

48. t density: 48 48

49. F density: 49 49

50. Summary • Probability Model • Properties of probability laws: 50