peubah acak kontinu pertemuan 09 matakuliah: l0104 / statistika psikologi tahun : 2008

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Peubah Acak KontinuPertemuan 09

Matakuliah : L0104 / Statistika PsikologiTahun : 2008

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Learning Outcomes

3

Pada akhir pertemuan ini, diharapkan mahasiswa

akan mampu :

• Mahasuswa akan dapat menghitung sifat-sifat peluang peubah acak kontinu.

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Outline Materi

4

• Fungsi kepekatan peubah acak kontinu• Fungsi distribusi peubah acak kontinu• Nilai harapan peubah acak kontinu• Varians dan simpangan baku peubah

acak kontinu

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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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Probability Density Function is given by the area of the shaded region.

( )P a X b

ba

( )y f x

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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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Uniform Distribution

A continuous rv X is said to have a uniform distribution on the interval [a, b] if the pdf of X is

otherwise

bxaabbaxf

0

1),;(

X ~ U (a,b)

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Exponential distribution

• X is said to have the exponential distribution • if for some

00

01

)(

,0

x

xexf

x

)(~ ExpX

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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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Expected Value of h(X)

• If X is a continuous rv with pdf f(x) and h(x) is any function of X, then

( )( ) ( ) ( )h XE h x h x f x dx

[ ( )] ( ) ( )D

E h X h x p x

• If X is a discrete rv with pmf f(x) and h(x) is any function of X, then

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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

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Short-cut Formula for Variance

22( ) ( )V X E 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

( ) ( )x

F x P X x f y dy

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

( ) ( )P a X b F b F a

Let X be a continuous rv with pdf f(x) and cdf F(x). Then for any number a,

and for any numbers a and b with a < b,

1 ( )P X a F a

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Ex 6 (Continue). X = length of time in remission, and

30,9

1)( 2 xxxf

What is the probability that a malaria patient’s remission lasts long than one year?

%29.96)127(27

1

1

3

39

1

9

1)1(

3

1

32

xdxxXP

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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 f x

( ) exists, F x

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• Selamat Belajar Semoga Sukses.