1. Stats 241.3
Probability Theory

2. T and Th 11:30am - 12:50am GEOL 255 Lab: W 3:30 - 4:20 GEOL 155
Instructor:
W.H.Laverty
Office:
235 McLean Hall
Phone:
Lectures:
T and Th 11:30am - 12:50am GEOL 255 Lab: W 3:30 - 4:20 GEOL 155
Evaluation:
Assignments, Labs, Term tests - 40% Final Examination - 60%

3. Text:
Devore and Berk, Modern Mathematical Statistics with applications.
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
Non-deterministic
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.
Random phenomena
Unable to predict the outcomes, but in the long-run, the outcomes exhibit statistical regularity.
Haphazard phenomena
unpredictable outcomes, but no long-run, exhibition of statistical regularity in the outcomes.

14. Phenomena
Non-deterministic
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. Tossing a coin – outcomes S ={Head, Tail}
Random phenomena
Unable to predict the outcomes, but in the long-run, the outcomes exhibit statistical regularity.
Examples
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. 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. Rolling a two balanced dice – 36 outcomes

19. 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
A typical outcome d l

20. Stock market performance
A stock currently has a price of $ 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
A finite number of outcomes.
A countably infinite number of outcomes, or
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 = {O1, O2, O3, O4, O5, …}

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
0.9
S = {x | 0 ≤ x <1} = [0,1) x
0.2
0.8 S
1.0
0.0 [ )
0.3
0.7
0.4
0.6
0.5

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

26. 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. 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
A typical outcome d l

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

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.

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

33. 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 f = { } = the event that contains no outcomes
The Null Event, The empty event - f
f = { } = the event that contains no outcomes
The Entire Event, The Sample Space - S
S = the event that contains all outcomes
The empty event, f , 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 ) defined by:
AB = {e| e belongs to A or e belongs to B}
AB A B

38. The event AB occurs if the event A occurs or the event and B occurs .

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 A B

40. 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

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

44. DeMoivre’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 =
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, f, and the entire event, S.

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

49. If two events A and B are are mutually exclusive then:
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. B A

50. Some other set notation
We will use the notation
to mean that e is an element of A.
We will use the notation
to mean that e is not an element of A.

51. Thus

52. We will use the notation
to mean that A is a subset B. (B is a superset of A.) B A

53. Union and Intersection
more than two events

54. Union: E2 E3 E1

55. Intersection: E2 E3 E1

56. DeMorgan’s laws =

57. Probability

58. Suppose we are observing a random phenomena
Let S denote the sample space for the phenomena, the set of all possible outcomes.
An event E is a subset of S.
A probability measure P is defined on S by defining for each event E, P[E] with the following properties
P[E] ≥ 0, for each E.
P[S] = 1.

59. P[E1]
P[E2]
P[E3]
P[E4]
P[E5]
P[E6] …

60. Example: Finite uniform probability space
In many examples the sample space S = {o1, o2, o3, … oN} has a finite number, N, of oucomes.
Also each of the outcomes is equally likely (because of symmetry).
Then P[{oi}] = 1/N and for any event E

61. Note:
with this definition of P[E], i.e.

62. Thus this definition of P[E], i.e.
satisfies the axioms of a probability measure
P[E] ≥ 0, for each E.
P[S] = 1.

63. Another Example:
We are shooting at an archery target with radius R. The bullseye has radius R/4. There are three other rings with width R/4. We shoot at the target until it is hit R
S = set of all points in the target
= {(x,y)| x2 + y2 ≤ R2}
E, any event is a sub region (subset) of S.

64. E, any event is a sub region (subset) of S.

66. Thus this definition of P[E], i.e.
satisfies the axioms of a probability measure
P[E] ≥ 0, for each E.
P[S] = 1.

67. Finite uniform probability space
Many examples fall into this category
Finite number of outcomes
All outcomes are equally likely
To handle problems in case we have to be able to count. Count n(E) and n(S).