1. Stats 241.3 Probability Theory

2. Instructor:W.H.Laverty Office:235 McLean Hall Phone:966-6096 Lectures: M W F 2:30pm - 3:20am Arts 133 Lab: M 3:30 - 4:20 Arts 133 Evaluation:Assignments, Labs, Term tests - 40% Final Examination - 60%

3. Text: Wackerly, Mendenhall & Scheaffer, Mathematical Statistics, with applications 6 th edition. I will provide lecture notes (power point slides). I will provide tables. The assignments will not come from the textbook. This means that the purchasing of the text is optional.

4. Course Outline

5. Introduction Chapter 1

6. Probability Counting techniques Rules of probability Conditional probability and independence –Multiplicative Rule –Bayes Rule, Simpson’s paradox Chapter 2

7. Random variables Discrete random variables - their distributions Continuous random variables - their distributions Expectation –Rules of expectation –Moments – variance, standard deviation, skewness, kurtosis –Moment generating functions Chapters 3 and 4

8. Multivariate probability distributions Discrete and continuous bivariate distributions Marginal distributions, Conditional distributions Expectation for multivariate distributions Regression and Correlation Chapter 5

9. Functions of random variables Distribution function method, moment generating function method, transformation method Law of large numbers, Central Limit theorem Chapter 5, 7

10. Introduction to Probability Theory Probability – Models for random phenomena

11. Phenomena Deterministic Non-deterministic

12. Deterministic Phenomena There exists a mathematical model that allows “perfect” prediction the phenomena’s outcome. Many examples exist in Physics, Chemistry (the exact sciences). Non-deterministic Phenomena No mathematical model exists that allows “perfect” prediction the phenomena’s outcome.

13. Non-deterministic Phenomena may be divided into two groups. 1.Random phenomena –Unable to predict the outcomes, but in the long- run, the outcomes exhibit statistical regularity. 2.Haphazard phenomena –unpredictable outcomes, but no long-run, exhibition of statistical regularity in the outcomes.

14. Phenomena Deterministic Non-deterministic Haphazard Random

15. Haphazard phenomena –unpredictable outcomes, but no long-run, exhibition of statistical regularity in the outcomes. –Do such phenomena exist? –Will any non-deterministic phenomena exhibit long-run statistical regularity eventually?

16. Random phenomena –Unable to predict the outcomes, but in the long- run, the outcomes exhibit statistical regularity. Examples 1.Tossing a coin – outcomes S ={Head, Tail} Unable to predict on each toss whether is Head or Tail. In the long run can predict that 50% of the time heads will occur and 50% of the time tails will occur

17. 2.Rolling a die – outcomes S ={,,,,, } Unable to predict outcome but in the long run can one can determine that each outcome will occur 1/6 of the time. Use symmetry. Each side is the same. One side should not occur more frequently than another side in the long run. If the die is not balanced this may not be true.

18. 3.Rolling a two balanced dice – 36 outcomes

19. 4.Buffoon’s Needle problem –A needle of length l, is tossed and allowed to land on a plane that is ruled with horizontal lines a distance, d, apart d l A typical outcome

20. 5.Stock market performance –A stock currently has a price of $125.50. We will observe the price for the next 100 days typical outcomes

21. Definitions

22. The sample Space, S The sample space, S, for a random phenomena is the set of all possible outcomes. The sample space S may contain 1.A finite number of outcomes. 2.A countably infinite number of outcomes, or 3.An uncountably infinite number of outcomes.

23. A countably infinite number of outcomes means that the outcomes are in a one-one correspondence with the positive integers {1, 2, 3, 4, 5, …} This means that the outcomes can be labeled with the positive integers. S = {O 1, O 2, O 3, O 4, O 5, …}

24. A uncountably infinite number of outcomes means that the outcomes are can not be put in a one-one correspondence with the positive integers. Example: A spinner on a circular disc is spun and points at a value x on a circular disc whose circumference is 1. 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 x S = {x | 0 ≤ x <1} = [0,1) 0.0 1.0 [) S

25. Examples 1.Tossing a coin – outcomes S ={Head, Tail} 2.Rolling a die – outcomes S ={,,,,, } ={1, 2, 3, 4, 5, 6}

26. 3.Rolling a two balanced dice – 36 outcomes

27. S ={ (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)} outcome (x, y), x = value showing on die 1 y = value showing on die 2

28. 4.Buffoon’s Needle problem –A needle of length l, is tossed and allowed to land on a plane that is ruled with horizontal lines a distance, d, apart d l A typical outcome

29. An outcome can be identified by determining the coordinates (x,y) of the centre of the needle and , the angle the needle makes with the parallel ruled lines. (x,y)(x,y)  S = {(x, y,  )| -∞ < x < ∞, - ∞ < y < ∞, 0 ≤  ≤  }

30. An Event, E The event, E, is any subset of the sample space, S. i.e. any set of outcomes (not necessarily all outcomes) of the random phenomena S E

31. The event, E, is said to have occurred if after the outcome has been observed the outcome lies in E. S E

32. Examples 1.Rolling a die – outcomes S ={,,,,, } ={1, 2, 3, 4, 5, 6} E = the event that an even number is rolled = {2, 4, 6} ={,, }

33. 2.Rolling a two balanced dice – 36 outcomes

34. S ={ (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)} outcome (x, y), x = value showing on die 1 y = value showing on die 2

35. E = the event that a “7” is rolled ={ (6, 1), (5, 2), (4, 3), (3, 4), (3, 5), (1, 6)}

36. Special Events The Null Event, The empty event -   = { } = the event that contains no outcomes The Entire Event, The Sample Space - S S = the event that contains all outcomes The empty event, , never occurs. The entire event, S, always occurs.

37. Set operations on Events Union Let A and B be two events, then the union of A and B is the event (denoted by A  B) defined by: A  B = {e| e belongs to A or e belongs to B} A  B AB

38. The event A  B occurs if the event A occurs or the event and B occurs. A  B AB

39. Intersection Let A and B be two events, then the intersection of A and B is the event (denoted by A  B) defined by: A  B = {e| e belongs to A and e belongs to B} A  B AB

40. AB The event A  B occurs if the event A occurs and the event and B occurs.

41. Complement Let A be any event, then the complement of A (denoted by ) defined by: = {e| e does not belongs to A} A

42. The event occurs if the event A does not occur A

43. In problems you will recognize that you are working with: 1.Union if you see the word or, 2.Intersection if you see the word and, 3.Complement if you see the word not.

44. DeMorgan’s laws  =  =

45. DeMoivre’s laws (in words) The event A or B does not occur if the event A does not occur and the event B does not occur = The event A and B does not occur if the event A does not occur or the event B does not occur

46. Another useful rule =  In words The event A occurs if A occurs and B occurs or A occurs and B doesn’t occur.

47. Rules involving the empty set, , and the entire event, S.

48. Definition: mutually exclusive Two events A and B are called mutually exclusive if: A B

49. If two events A and B are are mutually exclusive then: A B 1.They have no outcomes in common. They can’t occur at the same time. The outcome of the random experiment can not belong to both A and B.