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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
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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
nXXX ,...,, 21
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Discrete Prob. Distributions
Recall table summary of distribution:
Taken on by random variable X,
Probabilities: P{X = xi} = pi
(note: big difference between X and
x!)
Values x1 x2 … xk
Prob. p1 p2 … pk
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Discrete Prob. Distributions
Table summary of distribution:
Recall power of this:
Can compute any prob., by summing pi
Values x1 x2 … xk
Prob. p1 p2 … pk
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Mean of Discrete Distributions
Frequentist approach to mean:
kkii x
nxX
xnxX ##
11
n
XXX n1
i
k
iikk xpxpxp
111
n
xxXxxX kkii ## 11
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Mean of Discrete Distributions
Frequentist approach to mean:
a weighted average of values
where weights are probabilities
i
k
iixpX
1
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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
Winning 9 -4 0
Prob. 1/3 1/2 1/6
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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 P{win} = 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/ P{win} = 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/ P{win} = 0.48
Expected Value analysis:
E(Winnings) = P{lose} x $0 + P{win} 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 ValueE.g. Suppose have $5000, and need $10,000
and can make even bets, w/ P{win} = 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 ValueE.g. Suppose have $5000, and need $10,000
and can make even bets, w/ P{win} = 0.48
Another view:
Strategy P{get $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 ValueE.g. Suppose have $5000, and need $10,000
and can make even bets, w/ P{win} = 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 don’t have long run)
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Expected ValueReal life decisions against Expected Value:
1. 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 ValueReal life decisions against Expected Value:
1. 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 ValueReal 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 ValueReal 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 Co’s 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 Men’s 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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Frequency of Seeds Reaching Final Four
0
10
20
30
40
50
1 2 3 4 5 6 7 8 9 10 11 12M
ore
Seed in Tourney
Fre
qu
ency
Bin Frequency
1 43
2 23
3 13
4 8
5 4
6 6
7 0
8 5
9 1
10 0
11 1
12 0
Probability of a #1 Seed Reaching the Final Four
Final Four Data:
2004-1979
P{FF} = 43/104 =0.413
http://cbs.sportsline.com/collegebasketball/mayhem/history/finalfourseeds
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How many of these #1 seeds actually win the Tourney?
NCAA Men's Basketball Champions
0
2
4
6
8
10
12
14
1 2 3 4 5 6 7 8 9
Seed Number
Fre
qu
en
cy
P{Champ} = 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’ RPI’s compare to the rest of the Top 25?
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RPI vs Rank of Top 25 Teams
R2 = 0.54
05
10152025303540
0 5 10 15 20 25 30
Rank
RP
I
As expected, teams with higher rankings have higher ranking RPI’s. This indicates that the best teams are going to be at the bottom left corner of the graph.
BUT… RPI’s are not an entirely accurate way of measuring team’s ability (as seen with mediocre R^2)
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 Steve’s Stat. 31 class: 92 (from class exam data)
92*0.628 ≈ 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/2004/2004090202ms.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?
P{A} = 0.628, P{S} = 0.87P {A and S} = P{A}*P{S} = 0.628*0.87 = 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 ValueImportant Properties of the Mean:i. Linearity:
Why?
i. e. mean “preserves linear transformations”
i i i
iiiiibaX bpxapbaxp
ba XbaX
bapbxpa Xi
ii
ii
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Functions of Expected Value
Important Properties of the Mean:
ii. summability:
Why is harder, so won’t do here
i. e. can add means to get mean of sums
i. e. mean “preserves sums”
YXYX
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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
Winning 9 -4 0
Prob. 1/3 1/2 1/6
2$22 XX
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Functions of Expected ValueE. g. above game:
If we “play twice”, then have
Same as above?
But isn’t playing twice different from doubling
stake?
Yes, but not in means
Winning 9 -4 0
Prob. 1/3 1/2 1/6
2$1$1$2121
XXXX
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Functions of Expected ValueHW:
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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IndependenceApplication: 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)
nXX ,...,1
X
n"lim"
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IndependenceApplication: 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,
Values x1 x2 … xk
Prob. p1 p2 … pk
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Variance of Random Variables
Again connect via frequentist approach:
n
iin XX
nXX
1
21 1
1,...,var
1
222
21
nXXXXXX n
1## 2
111
nXxxXXxxX kii
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Variance of Random Variables
Again connect via frequentist approach:
2211 XxpXxp kk
n
iin XX
nXX
1
21 1
1,...,var
22
11
1#
1#
Xxn
xXXx
nxX
kkii
k
iii Xxp
1
2
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Variance of Random VariablesSo define:
Variance of a distribution
As:
random variable
k
jXjjX xp
1
22
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Variance of Random Variables
E. g. above game:
=(1/2)*5^2+(1/6)*1^2+(1/3)*8^2
Note: one acceptable Excel form,
e.g. for exam (but there are many)
Winning 9 -4 0
Prob. 1/3 1/2 1/6
2222 1931
1061
1421 X
X
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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)*5^2+(1/6)*1^2+(1/3)*8^2)
Winning 9 -4 0
Prob. 1/3 1/2 1/6
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Variance of Random VariablesHW:
C14: Find the variance and standard
deviation of the distribution in 4.60.
(1.21, 1.10)
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Properties of Variancei. 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)
222XbaX a
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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:
222YXYX
22YXX
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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)
Winning 9 -4 0
Prob. 1/3 1/2 1/6
222 4 XX
2222 22121 XXXXX
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Variance of Random VariablesHW:
4.74 ((a) 550, 5.7, (b) 0, 5.7, (c) 1022, 10.3)
4.75