45733: lecture 2 chapter 3 - PowerPoint PPT Presentation

1 / 58
About This Presentation
Title:

45733: lecture 2 chapter 3

Description:

Elements of Venn diagram. Large rectangle representing the sample space ... Intersection: Venn diagram. Example Intersection: A='4,5,3' B='3,2,6' C=A B='3' C. 1 ... – PowerPoint PPT presentation

Number of Views:59
Avg rating:3.0/5.0
Slides: 59
Provided by: Andre7
Category:

less

Transcript and Presenter's Notes

Title: 45733: lecture 2 chapter 3


1
45-733 lecture 2 (chapter 3)
  • Probability

2
What do we do with statistics?
  • Describe a sample well
  • Analyze the sample to estimate properties of the
    population
  • Analyze the analysis to describe how sure we are
    of it

3
Why do we need probability?
  • Utility outside statistics
  • Gambling, physics, chemistry, asset pricing,
    insurance, etc

4
Why do we need probability?
  • Utility within statistics
  • When we are describing how sure we are that our
    analysis of the population is right, probability
    gives us a precise language in which to speak.
  • We will want to say things like
  • I am more than 95 sure that US household income
    is greater than 30,000
  • I am 99 certain that the mean time to failure of
    our light bulbs is between 100 and 120 hours
  • I am 80 sure that GDP growth will be between
    1.2 and 3.5 next year

5
Experiment, sample space, event
  • Consider an experiment
  • A process which can have one of several possible
    outcomes
  • Which outcome will occur is unknown to the
    experimenter or observer
  • Examples
  • Coin toss, die throw
  • Light bulb tested to failure
  • Economy evolves for one year

6
Experiment, sample space, event
  • The sample space
  • A list of all the possible outcomes of an
    experiment
  • Examples
  • Coin toss sample space heads,tails
  • Die throw sample space1,2,3,4,5,6
  • Light bulb failure time Sall positive real
    numbers
  • Economy growth Sall real numbers gt -100

7
Experiment, sample space, event
  • A basic event
  • One point in the sample space
  • Examples
  • Coin toss heads
  • Die throw 3
  • Light bulb 400 hours
  • Economy growth 1.7

8
Experiment, sample space, event
  • An event
  • A collection of one or more basic events
  • A collection of one or more points in sample
    space
  • Examples
  • Coin toss heads tails heads,tails
  • Die throw 3 3,6 1,2,3,4,5,6
  • Light bulb 400 hours between 5 and 18 hours
  • Economy growth 1.7 2.1,between 1 and 1

9
Experiment, sample space, event
  • Notation
  • We often write the sample space as S
  • We often denote basic events as s
  • We often write events as A, B, C, etc

10
Experiment, sample space, event
  • Venn diagram
  • A Venn diagram is a way of representing sample
    space, events, and operations
  • Elements of Venn diagram
  • Large rectangle representing the sample space
  • Circles or other shapes representing events
  • (optional) points representing basic events

11
Experiment, sample space, event
  • Venn diagram
  • Example the sample space of the die throw

1
4
6
3
5
2
12
Experiment, sample space, event
B
A
  • Venn diagram
  • Example
  • A4,5,3
  • B3,2,6

1
4
6
3
5
2
13
Experiment, sample space, event
  • Notice
  • All basic events are events
  • The sample space is an event
  • There is a special event called the null set or
    the null event or the empty event. It is ?

14
Experiment, sample space, event
  • Membership
  • A basic event may either belong to an event or
    not
  • We will write s?A when the basic event s is in
    the event A
  • We will write s?A when the basic event s is not
    in the event A

15
Experiment, sample space, event
  • Membership
  • Examples
  • heads? heads,tails
  • 1 ? 1,4,5
  • 1 ? 3,6
  • 1 ? between 0 and 3
  • 140 hours ? between 50 and 100 hours

16
Experiment, sample space, event
  • Membership Venn diagram
  • Example
  • A4,5,3
  • 3 ? A
  • 1 ? A

1
4
6
3
5
2
A
17
Experiment, sample space, event
  • Sub-event (subset)
  • We say that an event B is a sub-event of A if
    every member of B is also in A, and we write B?A
  • Examples
  • heads ? heads,tails
  • 3,4,5 ? 1,2,3,4,5
  • between 1 and 1.3 ? between 0.5 and 4
  • 3,4,5 ? 1,2,3,4

18
Experiment, sample space, event
B
  • Sub-event Venn diagram
  • Example
  • A4,5,3
  • B4,5
  • B ? A

1
4
6
3
5
2
A
19
Experiment, sample space, event
  • Intersection
  • A way of making a new event from two events
  • The intersection of events A and B is the event
    consisting of all the basic events A and B have
    in common.
  • CA?B means C is the intersection of A and B

20
Experiment, sample space, event
  • Intersection
  • Examples
  • 1 1,2,3 ? 1,4
  • between 1 and 1.5 btw 1 and 2 ? btw
    0.8 and 1.5
  • ?1 ? 3,4,5

21
Experiment, sample space, event
  • Intersection Venn diagram
  • Example Intersection
  • A4,5,3
  • B3,2,6
  • CA ? B3

C
1
4
6
5
3
2
22
Experiment, sample space, event
  • Union
  • A way of making a new event from two events
  • The union of two events is the event which
    contains all the basic events which are in
    either.
  • CA?B says C is the union of A and B --- C
    contains all the basic events in either A or B

23
Experiment, sample space, event
  • Union
  • Examples
  • 1,2,3 1,2 ? 2,3
  • btw 1 and 3 btw 1 and 1.5 ? btw 1.1
    and 3
  • heads heads ? ?

24
Experiment, sample space, event
  • Union Venn diagram
  • Example Union
  • A4,5,3
  • B3,2,6
  • DA ? B 2,3,4,5,6

D
1
4
6
3
5
2
25
Experiment, sample space, event
  • Mutual Exclusivity
  • A and B share no basic events in common
  • A ?B?
  • Example A1,4 B3,2

1
4
6
B
3
5
2
A
26
Experiment, sample space, event
  • Collective exhaustivity
  • A bunch of events are collectively exhaustive if
    their union is the sample space
  • Example E11,4 E23,2,6 E33,4,5
  • E1 ? E2 ? E31,2,3,4,5,6

27
Experiment, sample space, event
  • Collective exhaustivity
  • A bunch of events are collectively exhaustive if
    their union is the sample space

1
4
6
3
5
2
28
Experiment, sample space, event
  • Partitioning
  • A bunch of events partition the sample space if
    they are mutually exclusive and collectively
    exhaustive

1
4
6
3
5
2
29
Experiment, sample space, event
  • Complement
  • A complement is all the basic events in the
    sample space which are not in A
  • Complements are partitioning

6
1
4
3
5
2
30
Experiment, sample space, event
  • Some useful rules

6
1
4
3
5
2
31
Experiment, sample space, event
  • Some useful rules

32
Experiment, sample space, event
  • Some useful rules
  • If E1,E2,E3,,Ek are partition the sample space,
    then
  • E1 ?A, E2 ?A, E3 ?A,,Ek ?A, are mutually
    exclusive
  • (E1 ?A) ?(E2 ?A) ?(E3 ?A) ? ?(Ek ?A)A

33
Experiment, sample space, event
  • Some useful rules

34
What is probability?
  • Probability is a language within which to
    describe uncertainty
  • Uncertainty about which event will occur
  • Some events are more likely than others
  • When one event occurs, that may make other events
    more/less likely to occur
  • Since it is a language it has rules
  • Since it is a mathematical language, the rules
    are precise
  • Since the rules are precise, the statements it
    can make are correspondingly precise

35
What is probability?
  • Probability is a number between 0 and 1
  • When we say the probability of an event is 0,
    that means it is impossible for the event to
    occur
  • When we say the probability of an event is 1,
    that means it is certain that the event will
    occur
  • If the probability of A occurring is greater than
    the probability of B occurring, that means that A
    is more likely than B

36
What is probability?
  • There are differing interpretations of what this
    number between 0 and 1 means (in terms of the
    external world)
  • Frequentist
  • Imagine doing an experiment many independent
    times
  • Each time, we record whether or not the event A
    occurred
  • As N (number of experiments) goes to infinity
  • P(A) NA/N

37
What is probability?
  • There are differing interpretations of what this
    number between 0 and 1 means (in terms of the
    external world)
  • Subjectivist
  • The probability of an event A occurring exists
    only in our minds, reflecting our
    uncertainty/ignorance
  • When I say P(A)0.5 that means I think a 11 bet
    on whether A occurs is a fair bet
  • When I say P(A)0.33 that means I think a 21 bet
    on whether A occurs is a fair bet

38
What is probability?
  • Venn diagram
  • It is often useful to think of probability as
    area in a Venn diagram

A
B
39
Postulates of probability
  • For any event A, 0?P(A) ?1
  • For any event A, P(A)?s?AP(s)
  • The probability of an event is just the sum of
    the probabilities of the basic events which make
    it up
  • P(1,2,3)P(1)P(2)P(3)
  • P(S)1 and P(?)0

40
Consequences
  • If S has n equally likely basic events, each one
    has probability 1/n
  • If S has n equally likely basic events and nA of
    them are in A, then A has probability nA/n

41
Consequences
  • If A and B are mutually exclusive events, then
    P(A ?B)P(A)P(B)

1
4
6
3
A
B
5
2
42
Consequences
  • In general P(A ?B)?P(A)P(B)

1
4
6
3
A
B
5
2
43
Consequences
  • If B ? A, then P(B) ?P(A)

B
A
44
Rules of probability

45
Rules of probability

A
B
46
Conditional probability
  • Conditional probability is used to deal with
    partial information
  • Suppose there are two events, A and B and we wish
    to know the probability of A occurring
  • Estimate this probability somehow
  • Now we learn that B has occurred
  • How should we change our assessment of the
    probability of A occurring given that we know for
    sure B has occurred?

47
Conditional probability
  • Example, die throw
  • A1,2,3,5
  • B3,4
  • P(A)1/61/61/61/62/3
  • The probability of A given that we know B has
    occurred is ½
  • Two equally likely outcomes in B
  • Only one of them is also in A

48
Conditional probability
  • Notation
  • We write P(AB)
  • This is said The probability of A given Bor
    The probability of A conditional on B
  • So, we could write P(AB)1/2 in our previous
    example

49
Conditional probability
  • Rule for calculating

A
B
50
Conditional probability
  • Rule for calculating
  • In die throw, A1,2,3,5 B3,4

51
Conditional probability
  • Multiplication rule

52
Conditional probability
  • Independence
  • Two events A,B are independent if

53
Conditional probability
  • Independence, consequences
  • If two events A,B are independent

54
Conditional probability
  • Bayes rule

55
Conditional probability
  • Bayes rule, example
  • Bob is a stock analyst
  • A Stock price increases
  • B Bob recommends the stock
  • We know
  • 80 of stock prices rise
  • When a stock price rises, bob picked it 15 of
    the time
  • When a stock price falls, bob picked it 10 of
    the time
  • We want to know, what is the probability that the
    stock price rises, given that Bob picked it?

56
Conditional probability
  • Bayes rule, example
  • We want
  • We know

57
Conditional probability
  • Bayes rule, example

58
Conditional probability
  • Bayes rule, example
Write a Comment
User Comments (0)
About PowerShow.com