Stat 31, Section 1, Last Time

1
Stat 31, Section 1, Last Time

• Independence
• Special Case of And Rule
• Relation to Mutually Exclusive
• Random Variables
• Discrete vs. Continuous
• Tables of Probabilities for Discrete R.V.s
• Areas as Probabilities for Continuous R.V.s

2
Means and Variances

• (of random variables) Text, Sec. 4.4
• Idea Above population summaries, extended from
  populations to probability distributions
• Connection frequentist view
• Make repeated draws,
• from the distribution

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Discrete Prob. Distributions

• Recall table summary of distribution
• Taken on by random variable X,
• Probabilities PX xi pi
• (note big difference between X and x!)

4
Discrete Prob. Distributions

• Table summary of distribution
• Recall power of this
• Can compute any prob., by summing pi

5
Mean of Discrete Distributions

• Frequentist approach to mean

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Mean of Discrete Distributions

• Frequentist approach to mean
• a weighted average of values
• where weights are probabilities

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Mean of Discrete Distributions

• E.g. Above Die Rolling Game
• Mean of distribution
• (1/3)(9) (1/6)(0) (1/2)(-4) 3 - 2
  1
• Interpretation on average (over large number of
  plays) winnings per play 1
• Conclusion should be very happy to play

8
Mean of Discrete Distributions

• Terminology mean is also called
• Expected Value
• E.g. in above game expect 1 (per play)
• (caution on average over many plays)

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Expected Value

• HW
• 4.57
• 4.60 (2.45)
• 4.61

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Expected Value

• An application of Expected Value
• Assess fairness of games (e.g. gambling)
• Major Caution Expected Value is not what is
  expected on one play, but instead is average over
  many plays.
• Cannot say what happens in one or a few plays,
  only in long run average

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Expected Value

• E.g. Suppose have 5000, and need 10,000
• (e.g. you owe mafia 5000, clean out safe at
  work. If you give to mafia, you go to jail, so
  decide to try to raise additional 5000 by
  gambling)
• And can make even bets, where Pwin 0.48
• (can really do this, e.g. bets on Red in
  roulette at a casino)

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Expected Value

• E.g. Suppose have 5000, and need 10,000 and
  can make even bets, w/ Pwin 0.48
• Pressing Practical Problem
• Should you make one large bet?
• Or many small bets?
• Or something in between?

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Expected Value

• E.g. Suppose have 5000, and need 10,000 and
  can make even bets, w/ Pwin 0.48
• Expected Value analysis
• E(Winnings) Plose x 0 Pwin x 2
• 0.52 x 0 0.48 x
  2
• 0.96
• Thus expect to lose 0.04 for every dollar bet

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Expected Value

• E.g. Suppose have 5000, and need 10,000 and
  can make even bets, w/ Pwin 0.48
• Expect to lose 0.04 for every dollar bet
• This is why gambling is very profitable
• (for the casinos, been to Las Vegas?)
• They play many times
• So expected value works for them
• And after many bets, you will surely lose
• So should make fewer, not more bets?

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Expected Value

• E.g. Suppose have 5000, and need 10,000 and
  can make even bets, w/ Pwin 0.48
• Another view
• Strategy Pget
  10,000
• one 5000 bet 0.48 1/2
• two 2500 bets (0.48)2 1/4
  
• four 1250 bets (0.48)2
  1/16
• many no
  chance

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Expected Value

• E.g. Suppose have 5000, and need 10,000 and
  can make even bets, w/ Pwin 0.48
• Surprising (?) answer
• Best to make one big bet
• Not much fun
• But best chance at winning
• Casino Folklore
• This really happens
• Folks walk in, place one huge bet.

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Expected Value

• Warning about Expected Value
• Excellent for some things, but not all decisions
• e.g. if will play many times
• e.g. if only play once
• (so dont have long
  run)

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Expected Value

• Real life decisions against Expected Value
• State Lotteries
• State sells tickets
• Keeps about half of
• Gives rest to one (randomly chosen) player
• So Expected Value is clearly negative
• Why do people play? Totally irrational?
• Players buy faint hope of humongous gain
• Could be worth joy of thinking about it

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Expected Value

• Real life decisions against Expected Value
• State Lotteries
• Want one in North Carolina?
• You will be asked to decide
• Interesting (and deep) philosophical balances
• Only totally voluntary tax
• Yet tax burden borne mostly by poor
• Is that fair?
• But we lose revenue to other states

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Expected Value

• Real life decisions against Expected Value
• 2. Casino Gambling
• Always lose in long run (expected value)
• Yet people do it. Are they nuts?
• Depends on how many times they play
• If really enjoy being ahead sometimes
• Then could be worth price paid for the thrill
• Serious societal challenge
• (some are totally consumed by thrill)

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Expected Value

• Real life decisions against Expected Value
• 3. Insurance
• Everyone pays about 2 x Expected Loss
• Insurance Company keeps the rest!
• So very profitable.
• But e.g. car insurance is required by law!
• Sensible, since if lose, can lose very big
• Yet purchase is totally against Expected Value
• OK, since you only play once (not many times)
• Insurance Cos play many times (Expected Value
  works for them)
• So they are an evening out mechanism

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And now for something completely different

• Interesting Suggestion / Request
• By Katie Baer
• Well supported with Data / Analysis!

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SIMPLE MATH

• Date of the 2005 NCAA Mens Basketball Tournament
  Final Monday, April 4th, 2005
• Date of the Stat 31 Midterm 2 Tuesday, April
  5th, 2005

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• WHY SHOULD STEVE RESCHEDULE THE EXAM?
• STATISTICAL EVIDENCE

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Probability of a 1 Seed Reaching the Final Four
Final Four Data 2004-1979
PFF 43/104 0.413 http//cbs.sportsline.com/c
ollegebasketball/mayhem/history/finalfourseeds
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How many of these 1 seeds actually win the
Tourney?
PChamp 12/25 0.48 48
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However, this assumes that North Carolina has an
equal probability of winning the Tourney as the
other predicted 1 Seeds (Illinois, Wake Forest,
and Boston College)NBC Sports, msnbc.com
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So we all know that

• Illinois is undefeated
• Illinois beat Wake Forest 91-78 and is ranked 1
  in the Big 10
• Wake Forest beat North Carolina 95-82
• North Carolina is ranked 1 in the ACC and is 4-2
  versus ranked teams
• Boston College has lost only one game and is 1
  in the Big Least, I mean East

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How do we determine which team is better?

• RPI is derived from three component factors Div.
  I winning percentage (25), schedule strength
  (50) and opponent's schedule strength (25).
• How do the 1 Seeds RPIs compare to the rest of
  the Top 25?

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As expected, teams with higher rankings have
higher ranking RPIs. This indicates that the
best teams are going to be at the bottom left
corner of the graph. BUT RPIs are not an
entirely accurate way of measuring teams ability
(as seen with mediocre R2) RPI does not take
into account factors such as margin of victory,
location of game, etc.
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A different approach

• A study found that approximately 62.8 of all
  college students consume alcohol on a regular
  basis
• http//www.ftc.gov/reports/alcohol/appendixa.htm
• Considering that this percentage does not take
  into account specific drinking statistics at UNC
  nor the fact that a national championship is at
  stake, this is a conservative figure
• Number of students in Steves Stat. 31 class 92
  (from class exam data)
• 920.628 58 people
• This number estimates the number of people
  enrolled in Stat 31, section 1 that consume
  alcohol on a regular basis

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• A study by the NCAA showed that 87 of university
  students strongly believe that supporting
  collegiate sports is an integral part of college
  life
• http//www.ncaa.org/releases/miscellaneous/2004/20
  04090202ms.htm
• Taking into account that watching sports and
  drinking alcohol are major aspects of college
  students lives, what is the probability that a
  college student will support college sports AND
  consume alcohol at the same time?
• PA 0.628, PS 0.87
• P A and S PAPS 0.6280.87 0.546
  (54.6)
• THUS, over half the class (approx. 50 people)
  will probably drink alcohol the night of the
  final game of the NCAA Tourney

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Conclusions

• Carolina has a considerable chance of reaching
  the Final Four and winning the NCAA tourney as a
  1 seed as seen in past tournament data
• They have fierce competition, as seen with in the
  graph of RPI vs. Rank, for the title
• Over half of the class will probably consume
  alcohol the night of April 4th, resulting in
  difficulty in studying for a midterm scheduled
  the next day
• Note that these figures are very conservative
  percentages, given that students will most likely
  drink more when their team is in the final game
  and especially if it is a close, exciting match-up

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• PLEASE MOVE THE TEST, STEVE!
• GO HEELS!!!

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And now for something completely different

• Now about that exam change request
• It is possible
• But we all need to agree
• Some choices
• Thursday, April 7 or Tuesday, April 12
• Please email objections to either

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Functions of Expected Value

• Important Properties of the Mean
• Linearity
• Why?
• i. e. mean preserves linear transformations

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Functions of Expected Value

• Important Properties of the Mean
• ii. summability
• Why is harder, so wont do here
• i. e. can add means to get mean of sums
• i. e. mean preserves sums

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Functions of Expected Value

• E. g. above game
• If we double the stakes, then want
• mean of 2X
• Recall 1 before
• i.e. have twice the expected value

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Functions of Expected Value

• E. g. above game
• If we play twice, then have
• Same as above?
• But isnt playing twice different from doubling
  stake?
• Yes, but not in means

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Functions of Expected Value

• HW
• 4.67
• 4.68 (70)

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Indep. Of Random Variables

• Independence Random Variables X Y are
  independent when knowledge of value of X does not
  change chances of values of Y

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Indep. Of Random Variables

• HW
• 4.64 (Indep., Dep., Dep.)
• 4.65

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Independence

• Application Law of Large Numbers
• IF are independent draws from
  the same distribution, with mean ,
• THEN
• (needs more mathematics to make precise, but this
  is the main idea)

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Independence

• Application Law of Large Numbers
• Note this is the foundation of the
• frequentist view of probability
• Underlying thought experiment is based on many
  replications, so limit works.

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Variance of Random Variables

• Again consider discrete random variables
• Where distribution is summarized by a table,

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Variance of Random Variables

• Again connect via frequentist approach

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Variance of Random Variables

• Again connect via frequentist approach

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Variance of Random Variables

• So define
• Variance of a distribution
• As
• random
  variable

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Variance of Random Variables

• E. g. above game
• (1/2)52(1/6)12(1/3)82
• Note one acceptable Excel form,
  e.g. for exam (but there are many)

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Standard Deviation

• Recall standard deviation is square root of
  variance (same units as data)
• E. g. above game
• Standard Deviation
• sqrt((1/2)52(1/6)12(1/3)82)

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Variance of Random Variables

• HW
• C14 Find the variance and standard deviation
  of the distribution in 4.60. (1.21, 1.10)

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Properties of Variance

• Linear transformation
• I.e. ignore shifts var( ) var (
  )
• (makes sense)
• And scales come through squared
• (recall s.d. on scale of data, var is
  square)

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Properties of Variance

• ii. For X and Y independent (important!)
• I. e. Variance of sum is sum of variances
• Here is where variance is more natural than
  standard deviation

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Properties of Variance

• E. g. above game
• Recall double the stakes, gave same mean, as
  play twice, but seems different
• Doubling
• Play twice, independently
• Note playing more reduces uncertainty
• (var quantifies this idea, will do more later)

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Variance of Random Variables

• HW
• 4.74 ((a) 550, 5.7, (b) 0, 5.7, (c) 1022,
  10.3)
• 4.75