distribusi binomial, poisson, dan hipergeometrik · distribusi binomial, poisson, dan...
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DistribusiBinomial,
Poisson, danHipergeometrik
STATISTIK & PROBABILITASCopyright © 2017 By. Ir. Arthur Daniel Limantara, MM, MT.
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CHAPTER TOPICS
The Probability of a Discrete Random Variable Covariance and Its Applications in Finance Binomial Distribution Poisson Distribution Hypergeometric Distribution
STATISTIK & PROBABILITASCopyright © 2017 By. Ir. Arthur Daniel Limantara, MM, MT.
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RANDOM VARIABLE
Random Variable Outcomes of an experiment expressed numerically E.g., Toss a die twice; count the number of times the
number 4 appears (0, 1 or 2 times)
E.g., Toss a coin; assign $10 to head and -$30 to a tail
= $10
T = -$30
STATISTIK & PROBABILITASCopyright © 2017 By. Ir. Arthur Daniel Limantara, MM, MT.
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DISCRETE RANDOMVARIABLE
Discrete Random Variable Obtained by counting (0, 1, 2, 3, etc.) Usually a finite number of different values E.g., Toss a coin 5 times; count the number of
tails (0, 1, 2, 3, 4, or 5 times)
STATISTIK & PROBABILITASCopyright © 2017 By. Ir. Arthur Daniel Limantara, MM, MT.
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Probability DistributionValues Probability
0 1/4 = .251 2/4 = .502 1/4 = .25
DISCRETE PROBABILITYDISTRIBUTION EXAMPLE
Event: Toss 2 Coins Count # Tails
T
T
T T This is using the A Priori ClassicalProbability approach.
STATISTIK & PROBABILITASCopyright © 2017 By. Ir. Arthur Daniel Limantara, MM, MT.
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DISCRETE PROBABILITYDISTRIBUTION
List of All Possible [Xj , P(Xj) ] Pairs Xj = Value of random variable
P(Xj) = Probability associated with value
Mutually Exclusive (Nothing in Common)
Collective Exhaustive (Nothing Left Out)
0 1 1j jP X P X
STATISTIK & PROBABILITASCopyright © 2017 By. Ir. Arthur Daniel Limantara, MM, MT.
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SUMMARY MEASURES
Expected Value (The Mean) Weighted average of the probability distribution
E.g., Toss 2 coins, count the number of tails, computeexpected value:
j jj
E X X P X
0 .25 1 .5 2 .25 1
j jj
X P X
STATISTIK & PROBABILITASCopyright © 2017 By. Ir. Arthur Daniel Limantara, MM, MT.
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SUMMARY MEASURES
Variance Weighted average squared deviation about the
mean
E.g., Toss 2 coins, count number of tails, computevariance:
(continued)
222j jE X X P X
22
2 2 2 0 1 .2 5 1 1 .5 2 1 .2 5
.5
j jX P X
STATISTIK & PROBABILITASCopyright © 2017 By. Ir. Arthur Daniel Limantara, MM, MT.
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COVARIANCE AND ITSAPPLICATION
1
th
th
th
: discrete random variable
: outcome of
: discrete random variable
: outcome of
: probability of occurrence of the
outcome of an
N
XY i i i ii
i
i
i i
X E X Y E Y P X Y
X
X i X
Y
Y i Y
P X Y i
X
thd the outcome of Yi
STATISTIK & PROBABILITASCopyright © 2017 By. Ir. Arthur Daniel Limantara, MM, MT.
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COMPUTING THE MEAN FORINVESTMENT RETURNSReturn per $1,000 for two types of investments
100 .2 100 .5 250 .3 $105XE X
200 .2 50 .5 350 .3 $90YE Y
P(Xi) P(Yi) Economic Condition Dow Jones Fund X Growth Stock Y
.2 .2 Recession -$100 -$200
.5 .5 Stable Economy + 100 + 50
.3 .3 Expanding Economy + 250 + 350
Investment
STATISTIK & PROBABILITASCopyright © 2017 By. Ir. Arthur Daniel Limantara, MM, MT.
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COMPUTING THE VARIANCEFOR INVESTMENT RETURNS
2 2 22 .2 100 105 .5 100 105 .3 250 105
14, 725 121.35X
X
2 2 22 .2 200 90 .5 50 90 .3 350 90
37,900 194.68Y
Y
P(Xi) P(Yi) Economic Condition Dow Jones Fund X Growth Stock Y
.2 .2 Recession -$100 -$200
.5 .5 Stable Economy + 100 + 50
.3 .3 Expanding Economy + 250 + 350
Investment
STATISTIK & PROBABILITASCopyright © 2017 By. Ir. Arthur Daniel Limantara, MM, MT.
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COMPUTING THE COVARIANCEFOR INVESTMENT RETURNS
P(XiYi) Economic Condition Dow Jones Fund X Growth Stock Y
.2 Recession -$100 -$200
.5 Stable Economy + 100 + 50
.3 Expanding Economy + 250 + 350
Investment
100 105 200 90 .2 100 105 50 90 .5
250 105 350 90 .3 23,300
XY
The covariance of 23,000 indicates that the two investments arepositively related and will vary together in the same direction.
STATISTIK & PROBABILITASCopyright © 2017 By. Ir. Arthur Daniel Limantara, MM, MT.
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COMPUTING THE COEFFICIENT OFVARIATION FOR INVESTMENT RETURNS
Investment X appears to have a lower risk (variation)per unit of average payoff (return) than investmentY
Investment X appears to have a higher averagepayoff (return) per unit of variation (risk) thaninvestment Y
121.351.16 116%
105X
X
CV X
194.682.16 216%
90Y
Y
CV Y
STATISTIK & PROBABILITASCopyright © 2017 By. Ir. Arthur Daniel Limantara, MM, MT.
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SUM OF TWO RANDOMVARIABLES
The expected value of the sum is equal to the sumof the expected values
The variance of the sum is equal to the sum of thevariances plus twice the covariance
The standard deviation is the square root of thevariance
E X Y E X E Y
2 2 2 2X Y X Y XYVar X Y
2X Y X Y
STATISTIK & PROBABILITASCopyright © 2017 By. Ir. Arthur Daniel Limantara, MM, MT.
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PORTFOLIO EXPECTEDRETURN AND RISK
The portfolio expected return for a two-assetinvestment is equal to the weighted averageof the two assets
Portfolio risk
1
where
portion of the portfolio value assigned to asset
E P wE X w E Y
w X
22 2 21 2 1P X Y XYw w w w
STATISTIK & PROBABILITASCopyright © 2017 By. Ir. Arthur Daniel Limantara, MM, MT.
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COMPUTING THE EXPECTED RETURN ANDRISK OF THE PORTFOLIO INVESTMENT
P(XiYi) Economic Condition Dow Jones Fund X Growth Stock Y
.2 Recession -$100 -$200
.5 Stable Economy + 100 + 50
.3 Expanding Economy + 250 + 350
Investment
Suppose a portfolio consists of an equal investment in each ofX and Y:
0.5 105 0.5 90 97.5E P
2 20.5 14725 0.5 37900 2 0.5 0.5 23300 157.5P
STATISTIK & PROBABILITASCopyright © 2017 By. Ir. Arthur Daniel Limantara, MM, MT.
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DOING IT IN PHSTAT
PHStat | Decision Making | Covariance and PortfolioAnalysis Fill in the “Number of Outcomes:” Check the “Portfolio Management Analysis” box Fill in the probabilities and outcomes for investment X and Y Manually compute the CV using the formula in the previous
slide
Here is the Excel spreadsheet that contains theresults of the previous investment example:
Microsoft ExcelWorksheet
STATISTIK & PROBABILITASCopyright © 2017 By. Ir. Arthur Daniel Limantara, MM, MT.
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IMPORTANT DISCRETEPROBABILITY DISTRIBUTIONS
Discrete ProbabilityDistributions
Binomial Hypergeometric Poisson
STATISTIK & PROBABILITASCopyright © 2017 By. Ir. Arthur Daniel Limantara, MM, MT.
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BINOMIAL PROBABILITYDISTRIBUTION
‘n’ Identical Trials E.g., 15 tosses of a coin; 10 light bulbs taken from
a warehouse 2 Mutually Exclusive Outcomes on Each Trial E.g., Heads or tails in each toss of a coin;
defective or not defective light bulb Trials are Independent The outcome of one trial does not affect the
outcome of the other
STATISTIK & PROBABILITASCopyright © 2017 By. Ir. Arthur Daniel Limantara, MM, MT.
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BINOMIAL PROBABILITYDISTRIBUTION
Constant Probability for Each Trial E.g., Probability of getting a tail is the same each
time we toss the coin 2 Sampling Methods Infinite population without replacement Finite population with replacement
(continued)
STATISTIK & PROBABILITASCopyright © 2017 By. Ir. Arthur Daniel Limantara, MM, MT.
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BINOMIAL PROBABILITYDISTRIBUTION FUNCTION
!1
! !
: probability of successes given and
: number of "successes" in sample 0,1, ,
: the probability of each "success"
: sample size
n XXnP X p p
X n X
P X X n p
X X n
p
n
Tails in 2 Tosses of Coin
X P(X)0 1/4 = .25
1 2/4 = .50
2 1/4 = .25
STATISTIK & PROBABILITASCopyright © 2017 By. Ir. Arthur Daniel Limantara, MM, MT.
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BINOMIAL DISTRIBUTIONCHARACTERISTICS
Mean E.g.,
Variance andStandard Deviation
E.g.,
E X np 5 .1 .5np
n = 5 p = 0.1
0.2.4.6
0 1 2 3 4 5X
P(X)
1 5 .1 1 .1 .6708np p
2 1
1
np p
np p
STATISTIK & PROBABILITASCopyright © 2017 By. Ir. Arthur Daniel Limantara, MM, MT.
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BINOMIAL DISTRIBUTION INPHSTAT
PHStat | Probability & Prob. Distributions |Binomial
Example in Excel Spreadsheet
Microsoft ExcelWorksheet
STATISTIK & PROBABILITASCopyright © 2017 By. Ir. Arthur Daniel Limantara, MM, MT.
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EXAMPLE: BINOMIALDISTRIBUTION
A mid-term exam has 30 multiple choicequestions, each with 5 possible answers. What isthe probability of randomly guessing the answerfor each question and passing the exam (i.e.,having guessed at least 18 questions correctly)?
Microsoft ExcelWorksheet
Are the assumptions for the binomial distribution met?
6
30 0 .2
18 1 .84245 10
n p
P X
Yes, the assumptions are met.Using results from PHStat:
STATISTIK & PROBABILITASCopyright © 2017 By. Ir. Arthur Daniel Limantara, MM, MT.
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POISSON DISTRIBUTION
Siméon Poisson
STATISTIK & PROBABILITASCopyright © 2017 By. Ir. Arthur Daniel Limantara, MM, MT.
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POISSON DISTRIBUTION Discrete events (“successes”) occurring in a given
area of opportunity (“interval”) “Interval” can be time, length, surface area, etc.
The probability of a “success” in a given “interval” isthe same for all the “intervals”
The number of “successes” in one “interval” isindependent of the number of “successes” in other“intervals”
The probability of two or more “successes”occurring in an “interval” approaches zero as the“interval” becomes smaller E.g., # customers arriving in 15 minutes E.g., # defects per case of light bulbs
STATISTIK & PROBABILITASCopyright © 2017 By. Ir. Arthur Daniel Limantara, MM, MT.
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POISSON PROBABILITYDISTRIBUTION FUNCTION
!
: probability of "successes" given
: num ber of "successes" per unit
: expected (average) num ber of "successes"
: 2 .71828 (base of natural logs)
XeP X
XP X X
X
e
E.g., Find the probability of 4customers arriving in 3 minuteswhen the mean is 3.6.
3.6 43.6
.19124!
eP X
STATISTIK & PROBABILITASCopyright © 2017 By. Ir. Arthur Daniel Limantara, MM, MT.
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POISSON DISTRIBUTION INPHSTAT
PHStat | Probability & Prob.Distributions | Poisson
Example in Excel Spreadsheet
Microsoft ExcelWorksheet
P X x
x
x( |
!
e-
STATISTIK & PROBABILITASCopyright © 2017 By. Ir. Arthur Daniel Limantara, MM, MT.
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POISSON DISTRIBUTIONCHARACTERISTICS
Mean
Standard Deviationand Variance
1
N
i ii
E X
X P X
= 0.5
= 6
0.2.4.6
0 1 2 3 4 5X
P(X)
0.2.4.6
0 2 4 6 8 10X
P(X)
2
STATISTIK & PROBABILITASCopyright © 2017 By. Ir. Arthur Daniel Limantara, MM, MT.
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HYPERGEOMETRICDISTRIBUTION
“n” Trials in a Sample Taken from a FinitePopulation of Size N
Sample Taken Without Replacement Trials are Dependent Concerned with Finding the Probability of “X”
Successes in the Sample Where There are “A”Successes in the Population
STATISTIK & PROBABILITASCopyright © 2017 By. Ir. Arthur Daniel Limantara, MM, MT.
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HYPERGEOMETRICDISTRIBUTION FUNCTION
: probability that successes given , , and
: sam ple size
: population size
: num ber of "successes" in population
: num ber of "successes" in sam ple
0,1, 2,
A N A
X n XP X
N
n
P X X n N A
n
N
A
X
X
, n
E.g., 3 Light bulbs were selectedfrom 10. Of the 10, there were 4defective. What is the probabilitythat 2 of the 3 selected aredefective?
4 6
2 12 .30
10
3
P
STATISTIK & PROBABILITASCopyright © 2017 By. Ir. Arthur Daniel Limantara, MM, MT.
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HYPERGEOMETRIC DISTRIBUTIONCHARACTERISTICS
Mean
Variance and Standard Deviation
AE X n
N
22
2
1
1
nA N A N n
N N
nA N A N n
N N
FinitePopulationCorrectionFactor
STATISTIK & PROBABILITASCopyright © 2017 By. Ir. Arthur Daniel Limantara, MM, MT.
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HYPERGEOMETRIC DISTRIBUTIONIN PHStat
PHStat | Probability & Prob. Distributions |Hypergeometric …
Example in Excel Spreadsheet
Microsoft ExcelWorksheet
DistribusiNormal