Statistics%20270%20-%20Lecture%205 - PowerPoint PPT Presentation

View by Category
About This Presentation
Title:

Statistics%20270%20-%20Lecture%205

Description:

Statistics 270 - Lecture 5 – PowerPoint PPT presentation

Number of Views:29
Avg rating:3.0/5.0
Slides: 29
Provided by: Derek188
Learn more at: http://people.stat.sfu.ca
Category:

less

Write a Comment
User Comments (0)
Transcript and Presenter's Notes

Title: Statistics%20270%20-%20Lecture%205


1
Statistics 270 - Lecture 5
2
  • Last class measures of spread and box-plots
  • Last Day - Began Chapter 2 on probability.
    Section 2.1
  • These Notes more Chapter 2Section 2.2 and 2.3
  • Assignment 2 2.8, 2.12, 2.18, 2.24, 2.30, 2.36,
    2.40
  • Due Friday, January 27
  • Suggested problems 2.26, 2.28, 2.39

3
Probability
  • Probability of an event is the long-term
    proportion of times the event would occur if the
    experiment is repeated many times
  • Read page 59-60 on Interpreting probability

4
Probability
  • Probability of event, A is denoted P(A)
  • Axioms of Probability
  • For any event, A,
  • P(S) 1
  • If A1, A2, , Ak are mutually exclusive events,
  • These imply that

5
Discrete Uniform Distribution
  • Sample space has k possible outcomes
    Se1,e2,,ek
  • Each outcome is equally likely
  • P(ei)
  • If A is a collection of distinct outcomes from S,
    P(A)

6
Example
  • A coin is tossed 1 time
  • S
  • Probability of observing a heads or tails is

7
Example
  • A coin is tossed 2 times
  • S
  • What is the probability of getting either two
    heads or two tails?
  • What is the probability of getting either one
    heads or two heads?

8
Example
  • Inherited characteristics are transmitted from
    one generation to the next by genes
  • Genes occur in pairs and offspring receive one
    from each parent
  • Experiment was conducted to verify this idea
  • Pure red flower crossed with a pure white flower
    gives
  • Two of these hybrids are crossed. Outcomes
  • Probability of each outcome

9
Note
  • Sometimes, not all outcomes are equally likely
    (e.g., fixed die)
  • Recall, probability of an event is long-term
    proportion of times the event occurs when the
    experiment is performed repeatedly
  • NOTE Probability refers to experiments or
    processes, not individuals

10
Probability Rules
  • Have looked at computing probability for events
  • How to compute probability for multiple events?
  • Example 65 of SFU Business School Professors
    read the Wall Street Journal, 55 read the
    Vancouver Sun and 45 read both. A randomly
    selected Professor is asked what newspaper they
    read. What is the probability the Professor
    reads one of the 2 papers?

11
  • Addition Rules
  • If two events are mutually exclusive
  • Complement Rule

12
  • Example 65 of SFU Business School Professors
    read the Wall Street Journal, 55 read the
    Vancouver Sun and 45 read both. A randomly
    selected Professor is asked what newspaper they
    read. What is the probability the Professor
    reads one of the 2 papers?

13
Counting and Combinatorics
  • In the equally likely case, computing
    probabilities involves counting the number of
    outcomes in an event
  • This can be hardreally
  • Combinatorics is a branch of mathematics which
    develops efficient counting methods
  • These methods are often useful for computing
    probabilites

14
Combinatorics
  • Consider the rhyme
  • As I was going to St. Ives
  • I met a man with seven wives
  • Every wife had seven sacks
  • Every sack had seven cats
  • Every cat had seven kits
  • Kits, cats, sacks and wives
  • How many were going to St. Ives?
  • Answer

15
Example
  • In three tosses of a coin, how many outcomes are
    there?

16
Product Rule
  • Let an experiment E be comprised of smaller
    experiments E1,E2,,Ek, where Ei has ni outcomes
  • The number of outcome sequences in E is (n1 n2 n3
    nk )
  • Example (St. Ives re-visited)

17
Example
  • In a certain state, automobile license plates
    list three letters (A-Z) followed by four digits
    (0-9)
  • How many possible license plates are there?

18
Tree Diagram
  • Can help visualize the possible outcomes
  • Constructed by listing the posbilites for E1 and
    connecting these separately to each possiblility
    for E2, and so on

19
Example
  • In three tosses of a coin, how many outcomes are
    there?

20
Example - Permuatation
  • Suppose have a standard deck of 52 playing cards
    (4 suits, with 13 cards per suit)
  • Suppose you are going to draw 5 cards, one at a
    time, with replacement (with replacement means
    you look at the card and put it back in the deck)
  • How many sequences can we observe

21
Permutations
  • In previous examples, the sample space for Ei
    does not depend on the outcome from the previous
    step or sub-experiment
  • The multiplication principle applies only if the
    number of outcomes for Ei is the same for each
    outcome of Ei-1
  • That is, the outcomes might change change
    depending on the previous step, but the number of
    outcomes remains the same

22
Permutations
  • When selecting object, one at a time, from a
    group of N objects, the number of possible
    sequences is
  • The is called the number of permutations of n
    things taken k at a time
  • Sometimes denoted Pk,n
  • Can be viewed as number of ways to select k
    things from n objects where the order matters

23
Permutations
  • The number of ordered sequences of k objects
    taken from a set of n distinct objects (I.e.,
    number of permutations) is
  • Pk,nn(n-1) (n-k1)

24
Example
  • Suppose have a standard deck of 52 playing cards
    (4 suits, with 13 cards per suit)
  • Suppose you are going to draw 5 cards, one at a
    time, without replacement
  • How many permutations can we observe

25
Combinations
  • If one is not concerned with the order in which
    things occur, then a set of k objects from a set
    with n objects is called a combination
  • Example
  • Suppose have 6 people, 3 of whom are to be
    selected at random for a committee
  • The order in which they are selected is not
    important
  • How many distinct committees are there?

26
Combinations
  • The number of distinct combinations of k objects
    selected from n objects is
  • n choose k
  • Note n!n(n-1)(n-2)1
  • Note 0!1
  • Can be viewed as number of ways to select mthings
    taken k at a time where the order does not matter

27
Combinations
  • Example
  • Suppose have 6 people, 3 of whom are to be
    selected at random for a committee
  • The order in which they are selected is not
    important
  • How many distinct committees are there?

28
Example
  • A committee of size three is to be selected from
    a group of 4 Conservatives, 3 Liberals and 2 NDPs
  • How many committees have a member from each
    group?
  • What is the probability that there is a member
    from each group on the committee?
About PowerShow.com