1 M14 Expected Value, Discrete Department of ISM, University of Alabama, ’95,2002 Lesson...
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1 M14 Expected Value, Discrete partment of ISM, University of Alabama, ’95,2002 Lesson Objectives Understand the meaning of “expected value.” (Know what it is NOT !) Know what is meant by “a fair game.” Find the Expected Value for discrete random variables. Find the variance and standard deviatio for discrete random variables. Know how to use these values for making decisions.
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Transcript of 1 M14 Expected Value, Discrete Department of ISM, University of Alabama, ’95,2002 Lesson...
1M14 Expected Value, Discrete Department of ISM, University of Alabama, ’95,2002
Lesson Objectives
Understand the meaning of “expected value.” (Know what it is NOT!)
Know what is meant by “a fair game.”
Find the Expected Value for discrete random variables.
Find the variance and standard deviation for discrete random variables.
Know how to use these values for making decisions.
2M14 Expected Value, Discrete Department of ISM, University of Alabama, ’95,2002
Expected Value
If we repeat an experiment many, many times under the same conditions, and if we average the results, thenthis average is called theexpected value.
We call it , or “the mean”
3M14 Expected Value, Discrete Department of ISM, University of Alabama, ’95,2002
“Law of Large Numbers”
Probability = Long-run relative frequency
Expected = Long-run Value average
4M14 Expected Value, Discrete Department of ISM, University of Alabama, ’95,2002
Learn how to Learn how to
leave a Casinoleave a Casino
with more moneywith more money
than you ever thought
than you ever thought
you could leave with!!!
you could leave with!!!
Learn how to Learn how to
leave a Casinoleave a Casino
with more moneywith more money
than you ever thought
than you ever thought
you could leave with!!!
you could leave with!!!
5M14 Expected Value, Discrete Department of ISM, University of Alabama, ’95,2002
Properties for Discrete r.v.1. 0 < pi < 1, for each i
2. Sum of all probabilities
must equal ________.
6M14 Expected Value, Discrete Department of ISM, University of Alabama, ’95,2002
Mean of a Discrete R.V.
pi = P( X = xi )
x = xi pi
= x1p1 + x2p2 +...+ xkpk
7M14 Expected Value, Discrete Department of ISM, University of Alabama, ’95,2002
Roll one die many, many times.What do you expect the averageto be?
xi
pi
1 2 3 4 5 6
1/6 1/6 1/6 1/6 1/6 1/6
x = 1(1/6) + 2(1/6) + ... + 6(1/6)
=
8M14 Expected Value, Discrete Department of ISM, University of Alabama, ’95,2002
A “Fair Bet”: one in which
the expected value of the winnings is zero.
1
2 3
How much wouldyou be willingto pay to play?
How much wouldyou be willingto pay to play?
9M14 Expected Value, Discrete Department of ISM, University of Alabama, ’95,2002
Don’t pay any more than this to play.
Payoff $s
Probability 3/6 2/6 1/6
1 2 3
x =
10M14 Expected Value, Discrete Department of ISM, University of Alabama, ’95,2002
Variance of a Discrete R.V.
2 = (xi - X)2 pi
Variance is a measure of RISK.
Standard Deviation: = 2
11M14 Expected Value, Discrete Department of ISM, University of Alabama, ’95,2002
The “Wheel” problem:
Payoff $s
Probability 3/6 2/6 1/6
1 2 3
2 =
12M14 Expected Value, Discrete Department of ISM, University of Alabama, ’95,2002
Find the mean and variancefor the following discrete r.v.
Payoff $s
Probability 1/2 1/3 1/6
1 2 5
2 =x =
=
13M14 Expected Value, Discrete Department of ISM, University of Alabama, ’95,2002
Bigger Bigger variancevariance
MoreMoreRiskRisk
14M14 Expected Value, Discrete Department of ISM, University of Alabama, ’95,2002
Which r.v. has the greater risk?
Xi
pi
1.9 2.1
.5 .5
Yi
pi
-100 104
.5 .5
x =
y =
2 =
2 =
=
=
15M14 Expected Value, Discrete Department of ISM, University of Alabama, ’95,2002
Roulette Wheel
18 Red, 18 Black, 2 Green
18 Red, 18 Black, 2 Green
18 odd18 even18 odd18 even
12 low12 middle12 high
12 low12 middle12 high
38 slots:
16M14 Expected Value, Discrete Department of ISM, University of Alabama, ’95,2002
RoulettRoulettee
12
3
45
6
78
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
32
31
33
34
35
36
13 to 2419 to 36 1 to 1225 to 361 to 121 to 18 13 to 24
25 to 36 EVENODD
000
FIR
STC
OL.
SEC
CO
L.TH
IRD
CO
L.
REDBLACK
17M14 Expected Value, Discrete Department of ISM, University of Alabama, ’95,2002
Bet $10 on Red many, many times.Let X = Net amount won per play:
18M14 Expected Value, Discrete Department of ISM, University of Alabama, ’95,2002
Bet $10 on Red many, many times.Let X = Net amount won per play:
Xi
pi
+10 -10
In the long run, you will lose anaverage of $____ per spin of the wheel.
19M14 Expected Value, Discrete Department of ISM, University of Alabama, ’95,2002
Bet $10 on #12 many, many times.Let X = Net amount won per play:
20M14 Expected Value, Discrete Department of ISM, University of Alabama, ’95,2002
Bet $10 on #12 many, many times.Let X = Net amount won per play:
Xi
pi
+350 -10
x =
In the long run, you will lose anaverage of $____ per spin of the wheel.
21M14 Expected Value, Discrete Department of ISM, University of Alabama, ’95,2002
Strategy x 2
$10 on Red
$10 on #12
Summary:
22M14 Expected Value, Discrete Department of ISM, University of Alabama, ’95,2002
Simulated Roulette Play
1000 individual plays.
A one dollar bet on “Red”and a one dollar bet on “12”.
23M14 Expected Value, Discrete Department of ISM, University of Alabama, ’95,2002200 400 600 800 1000
-250
-200
-150
-100
-50
0
Play Number
Win
ning
sBet on or ?"Red" "#12"
Standard DeviationsEach Play, 1000 Plays
.9986 31.575.7626 182.23
Mean = -52.63
24M14 Expected Value, Discrete Department of ISM, University of Alabama, ’95,2002200 400 600 800 1000
-100
0
100
200
Play Number
Win
ning
sBet on or ?"Red" "#12"
Standard DeviationsEach Play, 1000 Plays
.9986 31.575.7626 182.23
Mean = -52.63
25M14 Expected Value, Discrete Department of ISM, University of Alabama, ’95,2002200 400 600 800 1000
-200
-100
0
100
200
Play Number
Win
ning
sBet on or ?"Red" "#12"
Standard DeviationsEach Play, 1000 Plays
.9986 31.575.7626 182.23
Mean = -52.63
26M14 Expected Value, Discrete Department of ISM, University of Alabama, ’95,2002200 400 600 800 1000
-100
0
100
200
300
400
500
Play Number
Win
ning
sBet on or ?"Red" "#12"
Standard DeviationsEach Play, 1000 Plays
.9986 31.575.7626 182.23
Mean = -52.63
27M14 Expected Value, Discrete Department of ISM, University of Alabama, ’95,2002
Simulated Roulette Play
Results for 500 players, each playing 1000 times.
J16_Simul means roulette.doc used in MINITAB on the Command Line.
28M14 Expected Value, Discrete Department of ISM, University of Alabama, ’95,2002
-800 -600 -400 -200 0 200 400 600 800
0
40
80
120
160
200
Ran 12
Fre
quen
cy
Betting on 'Red'500 people, 1000 bets each
-800 -600 -400 -200 0 200 400 600 8000
40
80
120
Winnings
Fre
qu
en
cy
Betting on '12'500 people, 1000 bets each
