V15_rv.ppt

1
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.

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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 m, or the mean

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Law of Large Numbers

• Probability Long-run
  relative frequency
• Expected Long-run Value
  average

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Learn how to leave a Casinowith more moneythan
you ever thoughtyou could leave with!!!
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Properties for Discrete r.v.
2. Sum of all probabilities must equal
________.
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Mean of a Discrete R.V.

• pi P( X xi )
• mx S xi pi
• x1p1 x2p2 ... xkpk

7
Roll one die many, many times.What do you expect
the averageto be?
mx

• 1(1/6) 2(1/6) ... 6(1/6)

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A Fair Bet one in which the expected value
of the winnings is zero.

• How much wouldyou be willingto pay to play?

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Payoff s
1 2 3
Probability
3/6 2/6 1/6

• Dont pay any more than this to play.

mx
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Variance of a Discrete R.V.

• s2 S (xi - mX)2 pi
• Variance is a measure of RISK.
• Standard Deviation

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The Wheel problem
Payoff s
1 2 3
Probability
3/6 2/6 1/6

• s2

12
Find the mean and variancefor the following
discrete r.v.
Payoff s
1 2 5
Probability
1/2 1/3 1/6
mx
s2
s
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Bigger variance
MoreRisk
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Which r.v. has the greater risk?
mx
Xi
1.9 2.1
s2
pi
.5 .5
s
my
Yi
-100 104
s2
pi
.5 .5
s
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Roulette Wheel
18 Red, 18 Black, 2 Green
38 slots
12 low 12 middle12 high
18 odd18 even
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Roulette
RED
13 to 24
19 to 36
1 to 12
25 to 36
EVEN
THIRDCOL.
21
24
33
36
3
9
27
30
6
12
15
18
00
SECCOL.
8
23
26
2
5
17
20
29
35
11
14
32
0
FIRSTCOL.
31
7
25
1
4
13
16
22
10
19
28
34
25 to 36
1 to 12
1 to 18
13 to 24
ODD
BLACK
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Bet 10 on Red many, many times.
Let X Net amount won per play
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Bet 10 on Red many, many times.
Let X Net amount won per play
10 -10
In the long run, you will lose an average of
____ per spin of the wheel.
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Bet 10 on 12 many, many times.
Let X Net amount won per play
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Bet 10 on 12 many, many times.
Let X Net amount won per play
350 -10
mx
In the long run, you will lose an average of
____ per spin of the wheel.
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Summary
mx
s2
Strategy
10 on Red 10 on 12
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Simulated Roulette Play
1000 individual plays. A one dollar bet on
Red and a one dollar bet on 12.
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Simulated Roulette Play
Results for 500 players, each playing 1000 times.
J16_Simul means roulette.doc used in MINITAB on
the Command Line.
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