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Pertemuan 05 Peubah Acak Kontinu dan Fungsi Kepekatannya






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Pertemuan 05 Peubah Acak Kontinu dan Fungsi Kepekatannya. Matakuliah : I0272 – Statistik Probabilitas Tahun : 2005 Versi : Revisi. Learning Outcomes. Pada akhir pertemuan ini, diharapkan mahasiswa akan mampu : Mahasiswa akan dapat menghitung nilai harapan, dan ragam peubah acak kontinu.
Pertemuan 05 Peubah Acak Kontinu dan Fungsi Kepekatannya

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Pertemuan 05 peubah acak kontinu dan fungsi kepekatannyaSlide 1

Pertemuan 05Peubah Acak Kontinu dan Fungsi Kepekatannya

Matakuliah : I0272 – Statistik Probabilitas

Tahun : 2005

Versi : Revisi

Learning outcomesSlide 2

Learning Outcomes

Pada akhir pertemuan ini, diharapkan mahasiswa

akan mampu :

  • Mahasiswa akan dapat menghitung nilai harapan, dan ragam peubah acak kontinu.

Outline materiSlide 3

Outline Materi

  • Konsep dasar

  • Nilai harapan dan ragam

  • Sebaran normal

  • Hampiran normal terhadap Binomial

  • Sebaran khusus : Eksponensial, Gamma, Beta, dst.

Pertemuan 05 peubah acak kontinu dan fungsi kepekatannya 1346311Slide 4

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).

Pertemuan 05 peubah acak kontinu dan fungsi kepekatannya 1346311Slide 5

Probability Density Function

For f (x) to be a pdf

  • f (x) > 0 for all values of x.

  • The area of the region between the graph of f and the x – axis is equal to 1.

Area = 1

Pertemuan 05 peubah acak kontinu dan fungsi kepekatannya 1346311Slide 6

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,

The graph of f is the density curve.

Pertemuan 05 peubah acak kontinu dan fungsi kepekatannya 1346311Slide 7

Probability Density Function

is given by the area of the shaded region.

a

b

Important difference of pmf and pdfSlide 8

Important difference of pmf and pdf

  • 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.

Pertemuan 05 peubah acak kontinu dan fungsi kepekatannya 1346311Slide 9

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.

Pertemuan 05 peubah acak kontinu dan fungsi kepekatannya 1346311Slide 10

Uniform Distribution

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

X ~ U (a,b)

Exponential distributionSlide 11

Exponential distribution

X is said to have the exponential distribution

if for some

Pertemuan 05 peubah acak kontinu dan fungsi kepekatannya 1346311Slide 12

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,

Pertemuan 05 peubah acak kontinu dan fungsi kepekatannya 1346311Slide 13

Expected Value

  • The expected or mean value of a continuous rv X with pdf f (x) is

  • The expected or mean value of a discrete rv X with pmf f (x) is

Pertemuan 05 peubah acak kontinu dan fungsi kepekatannya 1346311Slide 14

Expected Value of h(X)

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

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

Pertemuan 05 peubah acak kontinu dan fungsi kepekatannya 1346311Slide 15

Variance and Standard Deviation

The variance of continuous rv X with pdf f(x) and mean is

The standard deviation is

Pertemuan 05 peubah acak kontinu dan fungsi kepekatannya 1346311Slide 16

Short-cut Formula for Variance

Pertemuan 05 peubah acak kontinu dan fungsi kepekatannya 1346311Slide 17

The Cumulative Distribution Function

The cumulative distribution function, F(x) for a continuous rv X is defined for every number x by

For each x, F(x) is the area under the density curve to the left of x.

Pertemuan 05 peubah acak kontinu dan fungsi kepekatannya 1346311Slide 18

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,

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

Pertemuan 05 peubah acak kontinu dan fungsi kepekatannya 1346311Slide 19

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

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

Pertemuan 05 peubah acak kontinu dan fungsi kepekatannya 1346311Slide 20

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

Pertemuan 05 peubah acak kontinu dan fungsi kepekatannya 1346311Slide 21

Percentiles

Let p be a number between 0 and 1. The (100p)th percentile of the distribution of a continuous rv X denoted by , is defined by

Pertemuan 05 peubah acak kontinu dan fungsi kepekatannya 1346311Slide 22

Median

The median of a continuous distribution, denoted by , is the 50th percentile. So satisfies That is, half the area under the density curve is to the left of

Pertemuan 05 peubah acak kontinu dan fungsi kepekatannya 1346311Slide 23

  • Selamat Belajar Semoga Sukses.


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