Stat 155, Section 2, Last Time

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Stat 155, Section 2, Last Time

• Big Rules of Probability
• Not Rule ( 1 Popposite)
• Or Rule (glasses football)
• And rule (multiply conditional probs)
• Use in combination for real power
• Bayes Rule
• Turn around conditional probabilities
• Write hard ones in terms of easy ones
• Recall surprising disease testing result

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Reading In Textbook

• Approximate Reading for Todays Material
• Pages 266-271, 311-323, 277-286
• Approximate Reading for Next Class
• Pages 291-305, 334-351

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Midterm I

• Coming up Tuesday, Feb. 27
• Material HW Assignments 1 6
• Extra Office Hours
• Mon. Feb. 26, 830 1200, 200 330
• (Instead of Review Session)
• Bring Along
• 1 8.5 x 11 sheet of paper with formulas

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Recall Pepsi Challenge

• In class taste test
• Removed bias with randomization
• Double blind approach
• Asked which was
• Better
• Sweeter
• which

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Recall Pepsi Challenge

• Results summarized in spreadsheet
• Eyeball impressions
• a. Perhaps no consensus preference between Pepsi
  and Coke?
• Is 54 "significantly different from 50? (will
  develop methods to understand this)
• Result of "marketing research"???

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Recall Pepsi Challenge

• b. Perhaps no consensus as to which is sweeter?
• Very different from the past, when Pepsi was
  noticeably sweeter
• This may have driven old Pepsi challenge
  phenomenon
• Coke figured this out, and matched Pepsi in
  sweetness

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Recall Pepsi Challenge

• c. Most people believe they know
• Serious cola drinkers, because now flavor driven
• In past, was sweetness driven, and there were
  many advertising caused misperceptions!
• d. People tend to get it right or not??? (less
  clear)
• Overall 71 right. Seems like it, but again is
  that significantly different from 50?

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Recall Pepsi Challenge

• e. Those who think they know tend to be right???
• People who thought they knew right 71 of the
  time
• f. Those who don't think they know seem to right
  as well. Wonder why?
• People who didn't also right 70 of time?
  Why? "Natural sampling variation"???
• Any difference between people who thought they
  knew, and those who did not think so?

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Recall Pepsi Challenge

• g. Coin toss was fair (or is 57 heads
  significantly different from 50?)
• How accurate are those ideas?
• Will build tools to assess this
• Called hypo tests and P-values
• Revisit this example later

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Independence

• (Need one more major concept at this level)
• An event A does not depend on B, when
• Knowledge of B does not change
• chances of A
• PA B PA

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Independence

• E.g. I Toss a Coin, and somebody on South Pole
  does too.
• PH(me) T(SP) PH(me) ½.
• (no way that can matter, i.e. independent)

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Independence

• E.g. I Toss a Coin twice
• (toss number indicated with subscript)
• Is it lt ½?
• What if have 5 Heads in a row?
• (isnt it more likely to get a Tail?)
• (Wanna bet?!?)

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Independence

• E.g. I Toss a Coin twice,
• Rational approach
• Look at Sample Space
• Model all as equally likely
• Then
• So independence is good model for coin tosses

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New Ball Urn Example

• H ? R R R R G G T ? R R G
• Again toss coin, and draw ball
• Same, so R H are independent events
• Not true above, but works here, since proportions
  of R G are same

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Independence

• Note, when A is independent of B
• so
• And thus
• i.e. B is independent of A

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Independence

• Note, when A in independent of B
• It follows that B is independent of A
• I.e. independence is symmetric in A and B
• (as expected)
• More formal treatments use symmetric version as
  definition
• (to avoid hassles with 0 probabilities)

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Independence

• HW
• 4.31

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Special Case of And Rule

• For A and B independent
• PA B PA B PB PB A PA
• PA PB
• i.e. When independent, just multiply
  probabilities
• Textbook Call this another rule
• Me Only learn one, this is a special case

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Independent And Rule

• E.g. Toss a coin until the 1st Head appears,
  find P3 tosses
• Model tosses are independent
• (saw this was reasonable last time, using equally
  likely sample space ideas)
• P3 tosses
• When have 3 group with parentheses

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Independent And Rule

• E.g. Toss a coin until the 1st Head appears,
  find P3 tosses
• (by indep)
• I.e. just multiply

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Independent And Rule

• E.g. Toss a coin until the 1st Head appears, P3
  tosses
• Multiplication idea holds in general
• So from now on will just say
• Since Independent, multiply probabilities
• Similarly for Exclusive Or rule,
• Will just add probabilities

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Independent And Rule

• HW
• 4.29 (hint Calculate
• PG1G2G3G4G5G6G7)
• 4.33

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Overview of Special Cases

• Careful these can be tricky to keep separate
• OR works like adding,
• for mutually exclusive
• AND works like multiplying,
• for independent

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Overview of Special Cases

• Caution special cases are different
• Mutually exclusive independent
• For A and B mutually exclusive
• PA B 0 PA
• Thus not independent

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Overview of Special Cases

• HW C15 Suppose events A, B, C all have
  probability 0.4, A B are independent, and A
  C are mutually exclusive.
• Find PA or B (0.64)
• Find PA or C (0.8)
• Find PA and B (0.16)
• Find PA and C (0)

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Random Variables

• Text, Section 4.3 (we are currently jumping)
• Idea take probability to next level
• Needed for probability structure of political
  polls, etc.

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Random Variables

• Definition
• A random variable, usually denoted as X,
• is a quantity that
• takes on values at random

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Random Variables

• Two main types
• (that require different mathematical models)
• Discrete, i.e. counting
• (so look only at counting numbers, 1,2,3,)
• Continuous, i.e. measuring
• (harder math, since need all fractions, etc.)

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Random Variables

• E.g X for Candidate A in a randomly
  selected political poll discrete
• (recall
  all that means)
• Power of the random variable idea
• Gives something to get a hold of
• Similar in spirit to high school algebra

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High School Algebra

• Recall Main Idea?
• Rules for solving equations???
• No, major breakthrough is
• Give unknown(s) a name
• Find equation(s) with unknown
• Solve equation(s) to find unknown(s)

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Random Variables

• E.g X that comes up, in die rolling
• Discrete
• But not very interesting
• Since can study by simple methods
• As done above
• Dont really need random variable concept

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Random Variables

• E.g Measurement error
• Let X measurement
• Continuous
• How to model probabilities???

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Random Variables

• HW on discrete vs. continuous
• 4.40 ((b) discrete, (c) continuous, (d)
  could be either, but discrete is more common)

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And now for something completely different

• My idea about visualization last time
• 30 really liked it
• 70 less enthusiastic
• Depends on mode of thinking
• Visual thinkers loved it
• But didnt connect with others
• So hadnt planned to continue that

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And now for something completely different

• But here was another viewpoint
• Professor Marron,
• Could you focus on something more intelligent in
  your "And now for something completely different"
  section once every two weeks, perhaps, instead of
  completely abolishing it? I really enjoyed your
  discussion of how to view three dimensions in 2-D
  today.

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And now for something completely different

• A fun example
• Faces as data
• Each data point is a digital image
• Data from U. Carlos, III in Madrid
• (hard to do here for confidentiality reasons)
• Q What distinguishes men from women?

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And now for something completely different
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And now for something completely different

• Context statistical problem of
  classification, i.e. discrimination
• Basically automatic disease diagnosis
• Have measurmts on sick healthy cases
• Given new person, make measmts
• Closest to sick or healthy populations?

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And now for something completely different

• Approach Distance Weight Discrimination
• (Marron Todd)
• Idea find best separating direction in high
  dimensional data space
• Here
• Data are images
• Classes Male Females
• Given new image classify make - female

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And now for something completely different

• Fun visualization
• March through point clouds
• Along separating direction
• Captures Femaleness Maleness
• Note relation to training data

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And now for something completely different
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Random Variables

• A die rolling example
• (where random variable concept is useful)
• Win 9 if 5 or 6, Pay 4, if 1, 2 or 3,
  otherwise (4) break even
• Notes
• Dont care about number that comes up
• Random Variable abstraction allows focusing on
  important points
• Are you keen to play? (will calculate)

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Random Variables

• Die rolling example
• Win 9 if 5 or 6, Pay 4, if 1, 2 or 4
• Let X net winnings
• Note X takes on values 9, -4 and 0
• Probability Structure of X is summarized by
• PX 9 1/3 PX -4 1/2 PX 0 1/6
• (should you want to play?, study later)

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Random Variables

• Die rolling example, for X net winnings
• Win 9 if 5 or 6, Pay 4, if 1, 2 or 4
• Probability Structure of X is summarized by
• PX 9 1/3 PX -4 1/2 PX 0 1/6
• Convenient form a table

Winning 9 -4 0
Prob. 1/3 1/2 1/6
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Summary of Prob. Structure

• In general for discrete X, summarize
  distribution (i.e. full prob. Structure) by a
  table
• Where
• All are between 0 and 1
• (so get a prob. functn as above)

Values x1 x2 xk
Prob. p1 p2 pk
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Summary of Prob. Structure

• Summarize distribution, for discrete X,
• by a table
• Power of this idea
• Get probs by summing table values
• Special case of disjoint OR rule

Values x1 x2 xk
Prob. p1 p2 pk
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Summary of Prob. Structure

• E.g. Die Rolling game above
• PX 9 1/3
• PX lt 2 PX 0 PX -4 1/61/2 2/3
• PX 5 0 (not in table!)

Winning 9 -4 0
Prob. 1/3 1/2 1/6
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Summary of Prob. Structure

• E.g. Die Rolling game above

Winning 9 -4 0
Prob. 1/3 1/2 1/6
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Summary of Prob. Structure

• HW
• 4.41 (c) Find PX 3 X gt 2 (3/7)
• 4.52 (0.144, , 0.352)

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Probability Histogram

• Idea Visualize probability distribution using a
  bar graph
• E.g. Die Rolling game above

Winning 9 -4 0
Prob. 1/3 1/2 1/6
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Probability Histogram

• Construction in Excel
• Very similar to bar graphs (done before)
• Bar heights probabilities
• Example Class Example 18

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Probability Histogram

• HW
• 4.43

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Random Variables

• Now consider continuous random variables
• Recall for measurements (not counting)
• Model for continuous random variables
• Calculate probabilities as areas,
• under probability density curve, f(x)

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Continuous Random Variables

• Model probabilities for continuous random
  variables, as areas under probability density
  curve, f(x)
• Area(
  )
• 
  a b
• 
  (calculus notation)

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Continuous Random Variables

• Note
• Same idea as idealized distributions above
• Recall discussion from
• Page 8, of Class Notes, Jan. 23

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Continuous Random Variables

• e.g. Uniform Distribution
• Idea choose random number from 0,1
• Use constant density f(x) C
• Models equally likely
• To choose C, want
  Area
• 1 PX in 0,1 C
• So want C 1. 0
  1

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Uniform Random Variable

• HW
• 4.54 (0.73, 0, 0.73, 0.2, 0.5)
• 4.56 (1, ½, 1/8)

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Continuous Random Variables

• e.g. Normal Distribution
• Idea Draw at random from a normal population
• f(x) is the normal curve (studied above)
• Review some earlier concepts

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Normal Curve Mathematics

• The normal density curve is
• usual function of
• circle constant 3.14
• natural number 2.7

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Normal Curve Mathematics

• Main Ideas
• Basic shape is
• Shifted to mu
• Scaled by sigma
• Make Total Area 1 divide by
• as , but never

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Computation of Normal Areas

• EXCEL Computation
• works in terms of lower areas
• E.g. for
• Area lt 1.3

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Computation of Normal Probs

• EXCEL Computation
• probs given by lower areas
• E.g. for X N(1,0.5)
• PX lt 1.3 0.73

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Normal Random Variables

• As above, compute probabilities as areas,
• In EXCEL, use NORMDIST NORMINV
• E.g. above X N(1,0.5)
• PX lt 1.3 NORMDIST(1.3,1,0.5,TRUE)
• 0.73 (as in pic
  above)

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Normal Random Variables

• HW
• 4.57, 4.58 (0.965, 0)