Matakuliah
Tahun
: L0104 / Statistika Psikologi
: 2008
Fungsi Kepekatan Peluang Khusus
Pertemuan 10
Learning Outcomes
Pada akhir pertemuan ini, diharapkan mahasiswa
akan mampu :
• Mahasiswa akan dapat menghitung
peluang, nilai harapan dan varians fungsi
kepekatan seragam dan eksponensial.
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Outline Materi
• Fungsi kepekatan seragam
• Fungsi distribusi seragam
• Nilai harapan dan varians fungsi kepekatan
seragam
• Fungsi kepekatan eksponensial
• Fungsi distribusi eksponensial
• Nilai harapan dan varians peubah acak
eksponensial
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Uniform Distribution
A continuous rv X is said to have a
uniform distribution on the interval [a, b]
if the pdf of X is
 1

f ( x; a, b)   b  a
0
X ~ U (a,b)
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a xb
otherwise
Exponential distribution
X ~ Exp( )
• X is said to have the exponential distribution
• if for some
  0,
1
 e
f ( x)   
 0
x


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x0
x0
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

• The expected or mean value of a discrete rv X
with pmf f (x) is
E( X )   X 
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 x  p ( x)
xD
Expected Value of h(X)
• If X is a continuous rv with pdf f(x) and h(x) is any
function of X, then
E  h( x )    h ( X ) 

 h( x)  f ( x)dx

• If X is a discrete rv with pmf f(x) and h(x) is any
function of X, then
E[h( X )]   h( x)  p( x)
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D
Variance and Standard Deviation
The variance of continuous rv X with
pdf f(x) and mean  is
2
X

 V ( x) 
 (x  )

2
 f ( x)dx
 E[ X    ]
2
The standard deviation is  X  V ( x).
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Short-cut Formula for Variance
    E ( X )
V (X )  E X
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The Cumulative Distribution Function
The cumulative distribution function,
F(x) for a continuous rv X is defined for
every number x by
F ( x)  P  X  x    f ( y)dy
x

For each x, F(x) is the area under the
density curve to the left of x.
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Using F(x) to Compute Probabilities
Let X be a continuous rv with pdf f(x)
and cdf F(x). Then for any number a,
P  X  a   1  F (a )
and for any numbers a and b with a < b,
P  a  X  b   F (b)  F (a)
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Ex 6 (Continue). X = length of time in
remission, and
1 2
f ( x)  x , 0  x  3
9
What is the probability that a malaria
patient’s remission lasts long than one year?
P( X  1)  
3
1
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3
1 2
1x 3 1
x dx 
 (27  1)  96.29%
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9 3 1 27
Obtaining f(x) from F(x)
If X is a continuous rv with pdf f(x)
and cdf F(x), then at every number x
for which the derivative F ( x) exists,
F ( x)  f ( x).
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• Selamat Belajar Semoga Sukses.
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