1 Pertemuan 07 Variabel Acak Diskrit dan Kontinu Matakuliah: I0284 - Statistika Tahun: 2008 Versi:...
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Transcript of 1 Pertemuan 07 Variabel Acak Diskrit dan Kontinu Matakuliah: I0284 - Statistika Tahun: 2008 Versi:...
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Pertemuan 07Variabel Acak Diskrit dan Kontinu
Matakuliah : I0284 - Statistika
Tahun : 2008
Versi : Revisi
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Learning Outcomes
Pada akhir pertemuan ini, diharapkan mahasiswa
akan mampu :
• Mahasiswa akan dapat menghitung Peluang, nilai harapan, dan varians variabel acak diskrit dan kontinu.
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Outline Materi
• Definisi variabel acak
• Distribusi probabilitas diskrit
• Distribusi probabilitas kontinu
• Nilai harapan dan varians
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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
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Discrete Random Variable
• 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)
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Probability Distribution
Values Probability
0 1/4 = .25
1 2/4 = .50
2 1/4 = .25
Discrete Probability Distribution Example
Event: Toss 2 Coins Count # Tails
T
T
T T This is using the A Priori Classical Probability approach.
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Discrete Probability Distribution
• 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
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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, compute expected value:
j jj
E X X P X
0 .25 1 .5 2 .25 1
j jj
X P X
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Summary Measures
• Variance– Weighted average squared deviation about the
mean–
– E.g., Toss 2 coins, count number of tails, compute variance:
(continued)
222j jE X X P X
22
2 2 2 0 1 .25 1 1 .5 2 1 .25
.5
j jX P X
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Computing the Mean for Investment Returns
Return 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
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Computing the Variance for 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
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Continuous Random Variables
A random variable X is continuous if its set of possible values is an entire interval of numbers (If A < B, then any number x between A and B is possible).
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Probability Density Function
For f (x) to be a pdf
1. f (x) > 0 for all values of x.
2.The area of the region between the graph of f and the x – axis is equal to 1.
Area = 1
( )y f x
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Probability Distribution
Let X be a continuous rv. Then a probability distribution or probability density function (pdf) of X is a function f (x) such that for any two numbers a and b,
( )b
aP a X b f x dx
The graph of f is the density curve.
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Important difference of pmf and pdf
k
k
b
aA
dxxfkXP
dxxfdxxfAP
0)()(
)()()(
Y, a discrete r.v. with pmf f(y)X, a continuous r.v. with pdf f(x);
• f(y)=P(Y = k) = probability that the outcome is k.
• f(x) is a particular function with the property that for any event A (a,b), P(A) is the integral of f over A.
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Ex 1. (4.1) X = amount of time for which a book on 2-hour reserve at a college library is checked out by a randomly selected student and suppose that X has density function.
otherwise
xxxf
0
205.0)(
4375.05.05.1.
5.05.0)5.15.0(.
25.00
1
4
15.0)()1(.
2
5.1
5.1
5.0
1 1
0
2
xdxxPc
xdxxPb
xxdxdxxfxPa
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Probability for a Continuous rv
If X is a continuous rv, then for any number c, P(x = c) = 0. For any two numbers a and b with a < b,
( ) ( )P a X b P a X b
( )P a X b
( )P a X b
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Expected Value
• The expected or mean value of a continuous rv X with pdf f (x) is
( )X E X x f x dx
( ) ( )Xx D
E X x p x
• The expected or mean value of a discrete rv X with pmf f (x) is
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Variance and Standard Deviation
The variance of continuous rv X with pdf f(x) and mean is
2 2( ) ( ) ( )X V x x f x dx
2
[ ]E X
The standard deviation is
( ).X V x