Stats 241.3 Probability Theory
Instructor: Office: W.H.Laverty 235 McLean Hall Phone: 966-6096 Lectures: Evaluation: M T W Th F 1:30pm - 2:50pm Thorv 105 Lab: T W Th 3:00-3:50 Thorv 105 Assignments, Labs, Term tests - 40% Final Examination - 60% Test: Every Thursday in lab
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.
Course Outline
Chapter 1 Introduction
Probability Counting techniques Rules of probability Conditional probability and independence Multiplicative Rule Bayes Rule, Simpson s paradox Chapter 2
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
Multivariate probability distributions Discrete and continuous bivariate distributions Marginal distributions, Conditional distributions Expectation for multivariate distributions Regression and Correlation Chapter 5
Functions of random variables Distribution function method, moment generating function method, transformation method Law of large numbers, Central Limit theorem Chapters 5, 6
Introduction to Probability Theory Probability Models for random phenomena
Phenomena Deterministic Non-deterministic
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.
Non-deterministic Phenomena may be divided into two groups. 1. Random phenomena Unable to predict the outcomes, but in the longrun, the outcomes exhibit statistical regularity. 2. Haphazard phenomena unpredictable outcomes, but no long-run, exhibition of statistical regularity in the outcomes.
Phenomena Non-deterministic Deterministic Haphazard Random
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?
Random phenomena Unable to predict the outcomes, but in the longrun, 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
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.
3. Rolling a two balanced dice 36 outcomes
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 A typical outcome d l
5. Stock market performance A stock currently has a price of $125.50. We will observe the price for the next 100 days 250 typical outcomes 200 price 150 100 50 0 0 20 40 60 80 100 day
Definitions
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.
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, }
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. 0.8 0.7 0.9 0.0 0.1 x S = {x 0 x <1} = [0,1) 0.2 0.3 0.0 S 1.0 [ ) 0.6 0.5 0.4
Examples 1. Tossing a coin outcomes S ={Head, Tail} 2. Rolling a die outcomes S ={,,,,, } ={1, 2, 3, 4, 5, 6}
3. Rolling a two balanced dice 36 outcomes
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
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 A typical outcome d l
An outcome can be identified by determining the coordinates (x,y) of the centre of the needle and θ, the angle the needle makes with the parallel ruled lines. (x,y) θ S = {(x, y, θ) - < x <, - < y <, 0 θ π }
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
The event, E, is said to have occurred if after the outcome has been observed the outcome lies in E. S E
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} ={,, }
2. Rolling a two balanced dice 36 outcomes
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
E = the event that a 7 is rolled ={ (6, 1), (5, 2), (4, 3), (3, 4), (3, 5), (1, 6)}
Special Events The Null Event, The empty event - φ φ = { } = the event that contains no outcomes The Entire Event, The Sample Space - S S = the event that contains all outcomes The empty event, φ, never occurs. The entire event, S, always occurs.
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 A B= {e e belongs to A or e belongs to B} A(B A B A B
The event A(B A Boccurs if the event A occurs or the event and B occurs. A(B A B A B
Intersection Let A and B be two events, then the intersection of A and B is the event (denoted by A A'B B ) defined by: A ' B = {e e belongs to A and e belongs to B} A B A'B A B A B
The event A'B A B occurs if the event A occurs and the event and B occurs. A'B A B A B
Complement Let A be any event, then the complement of A (denoted by A) defined by: A = {e e does not belongs to A} A A
The event A occurs if the event A does not occur A A
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.
DeMorgan s laws 1. A B = A B = 2. A B = A B =
DeMorgan s laws (in words) 1. A B = A B The event A or B does not occur if the event A does not occur and the event B does not occur 2. A B = A B The event A and B does not occur if the event A does not occur = or the event B does not occur
Another useful rule A = ( A B) ( A B) = In words The event A occurs if A occurs and B occurs or A occurs and B doesn t occur.
Rules involving the empty set, φ, and the entire event, S. 1. A φ = A 2. A φ = φ 3. A S = S 4. A S = A
Definition: mutually exclusive Two events A and B are called mutually exclusive if: A B= φ A B
If two events A and B are are mutually exclusive then: 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. A B
Some other set notation We will use the notation e A to mean that e is an element of A. We will use the notation e A to mean that e is not an element of A.
Thus { or } A B = e e A e B { and } A B = e e A e B A = { ee A}
We will use the notation A B (or B A) to mean that A is a subset B. (B is a superset of A.) ie.. if e A then e B. B A
Union and Intersection more than two events
k Union: E = E E E E i 1 2 3 i= 1 i= 1 E = E E E i 1 2 3 k E 2 E 1 E 3
k Intersection: E = E E E E i 1 2 3 i= 1 i= 1 E = E E E i 1 2 3 k E 2 E 1 E 3
DeMorgan s laws = 1. Ei Ei i i = 2. Ei Ei i i =
Probability
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 1. P[E] 0, for each E. 2. P[S] = 1. P Ei = P Ei Ei Ei =φ i j i i 3. [ ] if for all, [ ] [ ] [ ] P E E = P E + P E + 1 2 1 2
= i i i [ ] P E P E i P[E 1 ] P[E 2 ] P[E 3 ] P[E 4 ] P[E 5 ] P[E 6 ]
Example: Finite uniform probability space In many examples the sample space S = {o 1, o 2, o 3, o N } has a finite number, N, of oucomes. Also each of the outcomes is equally likely (because of symmetry). Then P[{o i }] = 1/N and for any event E [ ] P E = Note n( E) n( E) no. of outcomes in E n S ( ) = = N total no. of outcomes : n( A) = no. of elements of A
Note: with this definition of P[E], i.e. [ ] P E = n( E) n( E) no. of outcomes in E n S ( ) 1. P[ E] 0. n( S) 2. PS= 1 3. [ ] P = = N total no. of outcomes n( S ) = n Ei n( E ) n( E ) i E i i N = = [ ] [ ] + + 1 2 = P E + P E + E E = φ N 1 2 if i j
Thus this definition of P[E], i.e. [ ] P E = n( E) n( E) no. of outcomes in E n S ( ) = = N total no. of outcomes satisfies the axioms of a probability measure 1. P[E] 0, for each E. 2. P[S] = 1. P Ei = P Ei Ei Ei =φ i j i i 3. [ ] if for all, [ ] [ ] [ ] P E E = P E + P E + 1 2 1 2
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 S = set of all points in the target = {(x,y) x 2 + y 2 R 2 } R E, any event is a sub region (subset) of S.
E, any event is a sub region (subset) of S. E [ ] Define : P E = Area ( E) Area E = 2 Area S π R ( ) ( )
[ ] P Bullseye 2 R π 4 = = 2 π R 1 16 [ ] P White ring 2 2 3 2 π R π R 4 4 9 4 5 = = = 2 π R 16 16
Thus this definition of P[E], i.e. [ ] P E = Area ( E) Area E = 2 Area S π R ( ) ( ) satisfies the axioms of a probability measure 1. P[E] 0, for each E. 2. P[S] = 1. P Ei = P Ei Ei Ei =φ i j i i 3. [ ] if for all, [ ] [ ] [ ] P E E = P E + P E + 1 2 1 2
Finite uniform probability space Many examples fall into this category 1. Finite number of outcomes 2. All outcomes are equally likely 3. P[ E] = Note n( E) n( E) no. of outcomes in E n S ( ) = = N total no. of outcomes : n( A) = no. of elements of To handle problems in case we have to be able to count. Count n(e) and n(s). A
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