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8/14/2019 Lec 14 Discrete Prob Dist SLIDE
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Copyright 2004 by The McGraw-Hill Companies, Inc. All rights reserved.
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TerminologyRandom Variable
is a numerical value determined by
the outcome of an experiment.
is a numerical value determined by
the outcome of an experiment.
Copyright 2004 by The McGraw-Hill Companies, Inc. All rights reserved.
Probability Distribution
is the listing ofall possible outcomes
of an experimentand thecorresponding probability.
is the listing ofall possible outcomes
of an experimentand thecorresponding probability.
Example
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Tree Diagrams
This is a useful device to show all the possible outcomes
of the experimentand their corresponding probabilities
Copyright 2004 by The McGraw-Hill Companies, Inc. All rights reserved.
ons er ng e ran om exper men o pp ng a co n r ce.
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In a random experiment in which
a coin is tossed three times:
In a random experiment in which
a coin is tossed three times:
HeadsLetx be the number of
Copyright 2004 by The McGraw-Hill Companies, Inc. All rights reserved.
TailsLet T represent the outcome of
LetHrepresent the outcome of a Head
Determine the probability distributionDetermine the probability distribution
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Tree DiagramsOrigin First
Flip
HH
SecondFlip
H
HT HHT
HTH
HHH
ThirdFlip
Copyright 2004 by The McGraw-Hill Companies, Inc. All rights reserved.
T
TH
T
TTT
H
H
T
T
T
THH
THTTTH
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6 - 6Listing the possibilitiesListing the possibilities
Heads Heads Heads TailsHeads Heads
TailsHeads Heads Tails Heads Heads
Copyright 2004 by The McGraw-Hill Companies, Inc. All rights reserved.
TailsHeads Tails TailsHeadsTails
TailsTails Tails
the possible values ofx
(number of heads) are 0,1,2,3.
the possible values ofx
(number of heads) are 0,1,2,3.
Tails HeadsTails
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DiscreteDiscrete
Types ofProbability Distributions
ContinuousContinuous
Copyright 2004 by The McGraw-Hill Companies, Inc. All rights reserved.
the random variablehas a
countable numberof possible outcomes
the random variablehas a
countable numberof possible outcomes
the random variablehas an
infinite numberof possible outcomes
the random variablehas an
infinite numberof possible outcomes
ExamplesExamples
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DiscreteDiscrete ContinuousContinuous
ExamplesExamplesStudents in a classStudents in a class
Types ofProbability Distributions
Distance driven by an
Copyright 2004 by The McGraw-Hill Companies, Inc. All rights reserved.
Number ofchildren
in a family
Mortgage Loan
Number of Mortgages
approved in a month
executive to get to work
The length of time of a
particular phone call
The length of
time of an
afternoon nap!
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Consider a
random
experiment in
which acoin is
tossed three times.
Consider a
random
experiment in
which acoin is
tossed three times.
Probability DistributionProbability Distribution
x # ofOutcomes
P(x)
0 1 1/8
Copyright 2004 by The McGraw-Hill Companies, Inc. All rights reserved.
Determine theprobability
distribution.
Determine theprobability
distribution.
1
2
3
3
3
1
8
3/8
3/8
1/8
8/8 = 1
What is theprobability of
tossing 2 heads in
3 flips?
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reports thecentral location of the data
Mean of a Discrete
ProbabilityDistribution
Mean of a Discrete
ProbabilityDistribution
is denoted by the Greek symbol ,mu
Copyright 2004 by The McGraw-Hill Companies, Inc. All rights reserved.
is the long- un average value ofthe random variable
also referred to as its expected value,E(X),
in a probability distribution
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xP(x)
Mean of a Discrete
ProbabilityDistribution
Mean of a Discrete
ProbabilityDistribution
Fli a c in thr tim
)]([xxP=
)(
xxPFormulaFormula
x P(x)
Copyright 2004 by The McGraw-Hill Companies, Inc. All rights reserved.
Letx be thenumber of heads
Letx be thenumber of heads
3/8
3/8
6/8
12/8=1.5
1
2
3
3/8
3/8
1/8
8/8 = 1
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Varianceof a Discrete
ProbabilityDistribution
Varianceof a Discrete
ProbabilityDistribution
Flip a coin three times.
Letx be the number of heads
Flip a coin three times.
Letx be the number of heads
)]()[(2
xPx 2 =
(x)Px FormulaFormula
Copyright 2004 by The McGraw-Hill Companies, Inc. All rights reserved.
(X- )2X-- 1.5
- 0.5
1.50.5
0.25
2.25
0.25
(X- )2 P(x).28125
.09375
0.75
.09375
.28125
xP(x)0
3/8
3/8
6/8
x0
1
2
3
P(x)1/8
3/8
3/8
1/8
8/8 = 1 =1.5
2.25
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Dan Desch, owner
of College Painters,
studied his recordsfor the past
20 weeks
and re orted the
Dan Desch, owner
of College Painters,
studied his recordsfor the past
20 weeks
and re orted the
P(x)
5/20 = 0.25
6/20 = 0.30
# of Painted
Houses
10
11
Weeks
5
6
x
Copyright 2004 by The McGraw-Hill Companies, Inc. All rights reserved.
followingnumber of houses
painted per week:
followingnumber of houses
painted per week:
7/20 = 0.35
2/20 = 0.10
20/20 = 1.0
Computing theComputing the
Determine theProbability distribution and itsmean and variance.Determine theProbability distribution and itsmean and variance.
12
13
7
2
20
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)]([ xxP= )(xxPxP(x)
2.5
Computing theComputing the
FormulaFormula
P(x)# of PaintedHouses
5/20 = 0.2510
Copyright 2004 by The McGraw-Hill Companies, Inc. All rights reserved.
3.3
1.3
4.2
11.3
Computing theComputing the2
6/20 = 0.30
7/20 = 0.35
2/20 = 0.10
20/20 = 1.0
11
12
13
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)]()[(2
xPx 2 =
(x - )2
1.69
(x - )2 P(x)
.4225
(x)P
Computing theComputing the 2
P(x)
0.25
x
10
xP(x)
2.5
)2
(x FormulaFormula
(10-11.3)2(10-11.3)2
Copyright 2004 by The McGraw-Hill Companies, Inc. All rights reserved.
0.09
2.89
0.49
.0270
0.910.91
.1715
.2890
0.30
0.35
0.10
1.0
11
12
13
3.3
1.3
4.2
11.3