Transcript values - The University of North Carolina at Chapel Hill

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
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,
X 1 , X 2 ,..., X n
from the distribution
Discrete Prob. Distributions
Recall table summary of distribution:
Values
x1
x2
…
xk
Prob.
p1
p2
…
pk
Taken on by random variable X,
Probabilities:
P{X = xi} = pi
(note: big difference between X and x!)
Discrete Prob. Distributions
Table summary of distribution:
Values
x1
x2
…
xk
Prob.
p1
p2
…
pk
Recall power of this:
Can compute any prob., by summing pi
Mean of Discrete Distributions
Frequentist approach to mean:
X1   X n
X

n
#  X i  x1   x1   #  X i  xk   xk


n
#  X i  x1 
#  X i  xk 

 x1   
 xk 
n
n
k
 p1 x1   pk xk   pi xi
i 1
Mean of Discrete Distributions
Frequentist approach to mean:
k
X   pi xi
i 1
a weighted average of values
where weights are probabilities
Mean of Discrete Distributions
E.g. Above Die Rolling Game:
Winning
Prob.
9
-4
0
1/3 1/2 1/6
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
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)
Expected Value
HW:
4.57
4.60
4.61
(2.45)
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
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)
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?
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
Expected Value
E.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?
Expected Value
E.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”
Expected Value
E.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….
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)
Expected Value
Real 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
Expected Value
Real 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…
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)
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 Co’s play many times (Expected
Value works for them)
So they are an evening out mechanism
And now for something
completely different
Interesting Suggestion / Request
By Katie Baer
Well supported with Data / Analysis!
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
WHY SHOULD STEVE
RESCHEDULE THE
EXAM?
STATISTICAL EVIDENCE:
Probability of a #1 Seed Reaching the Final Four
Final Four Data:
2004-1979
Frequency of Seeds Reaching Final Four
Frequency
6
7
0
8
5
9
1
10
0
11
1
12
0
20
10
0
12
M
or
e
6
30
10
11
4
9
5
8
8
7
4
40
6
13
5
3
50
4
23
3
2
2
43
1
1
Frequency
Bin
Seed in Tourney
P{FF} = 43/104 =0.413
http://cbs.sportsline.com/collegebasketball/mayhem/history/finalfourseeds
How many of these #1 seeds
actually win the Tourney?
NCAA Men's Basketball Champions
14
12
P{Champ} =
12/25 = 0.48
Frequency
10
8
6
4
48 %
2
0
1
2
3
4
5
6
Seed Number
7
8
9
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
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
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?
RPI
RPI vs Rank of Top 25 Teams
40
35
30
25
20
15
10
5
0
R2 = 0.54
0
5
10
15
20
25
30
Rank
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.
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
• 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} = 0.87
P {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
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
PLEASE MOVE THE
TEST, STEVE!
GO HEELS!!!
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
Functions of Expected Value
Important Properties of the Mean:
i. Linearity:
aX b  a X  b
Why?
aX b   pi axi  b   api xi   pi b
i
i
i




 a   pi xi   b  pi   a X  b
 i
  i 
i. e. mean “preserves linear transformations”
Functions of Expected Value
Important Properties of the Mean:
ii.
summability:
 X Y   X  Y
Why is harder, so won’t do here
i. e. can add means to get mean of sums
i. e. mean “preserves sums”
Functions of Expected Value
E. g. above game:
Winning
Prob.
9
-4
0
1/3 1/2 1/6
If we “double the stakes”, then want:
“mean of 2X”  2 X  2   X  $2
Recall $1 before
i.e. have twice the expected value
Functions of Expected Value
E. g. above game:
Winning
Prob.
9
-4
0
1/3 1/2 1/6
If we “play twice”, then have
 X  X   X   X  $1  $1  $2
1
2
1
2
Same as above?
But isn’t playing twice different from doubling
stake?
Yes, but not in means
Functions of Expected Value
HW:
4.67
4.68
(70)
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
Indep. Of Random Variables
HW:
4.64
4.65
(Indep., Dep., Dep.)
Independence
Application: Law of Large Numbers
IF X 1 ,..., X n are independent draws from the
same distribution, with mean  ,
THEN:
" lim" X  
n
(needs more mathematics to make precise,
but this is the main idea)
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….
Variance of Random Variables
Again consider discrete random variables:
Where distribution is summarized by a table,
Values
x1
x2
…
xk
Prob.
p1
p2
…
pk
Variance of Random Variables
Again connect via frequentist approach:
1 n
2


var  X 1 ,..., X n  
X

X


i
n  1 i 1

X1  X   X 2  X     X n  X 
2
2
n 1
2

#  X i  x1  xi  X     #  X 1  x1  xk  X 


n 1
2
Variance of Random Variables
Again connect via frequentist approach:
1 n
2
X i  X  
var  X 1 ,..., X n  

n  1 i 1
#  X i  x1 
#  X i  xk 
2
 x1  X    
 xk  X 2 

n 1
n 1
 p1  x1  X     pk  xk  X  
2
k
2
  pi  xi  X 
i 1
2
Variance of Random Variables
So define:
Variance of a distribution
As:
   p j x j   X 
2
X
k
2
j 1
random variable
Variance of Random Variables
E. g. above game:
Winning
Prob.
9
-4
0
1/3 1/2 1/6
1
1
1
2
2
2
   4  1  0  1  9  1
2
6
3
2
X
=(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)
X
Standard Deviation
Recall standard deviation is square root of
variance (same units as data)
E. g. above game:
Winning
Prob.
9
-4
0
1/3 1/2 1/6
Standard Deviation
=sqrt((1/2)*5^2+(1/6)*1^2+(1/3)*8^2)
Variance of Random Variables
HW:
C14:
Find the variance and standard
deviation of the distribution in 4.60.
(1.21, 1.10)
Properties of Variance
i.
Linear transformation
I.e. “ignore shifts” var(

2
aX  b
a 
) = var (
2
2
X
)
(makes sense)
And scales come through squared
(recall s.d. on scale of data, var is square)
Properties of Variance
ii.
For X and Y independent (important!)

2
X Y
  
2
X
2
Y
I. e. Variance of sum is sum of variances
Here is where variance is “more natural”
than standard deviation:
 X   
2
X
2
Y
Properties of Variance
Winning
E. g. above game:
9
Prob.
-4
0
1/3 1/2 1/6
Recall “double the stakes”, gave same mean,
as “play twice”, but seems different
Doubling: 
2
2X
 4 
2
X
2
2
2
2






2

Play twice, independently: X  X
X
X
X
1
2
1
2
Note: playing more reduces uncertainty
(var quantifies this idea, will do more later)
Variance of Random Variables
HW:
4.74
4.75
((a) 550, 5.7, (b) 0, 5.7, (c) 1022, 10.3)