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.

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Pertemuan 05 Peubah Acak Kontinu dan Fungsi Kepekatannya

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Pertemuan 05Peubah 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.


Outline Materi

  • Konsep dasar

  • Nilai harapan dan ragam

  • Sebaran normal

  • Hampiran normal terhadap Binomial

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


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).


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


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.


Probability Density Function

is given by the area of the shaded region.

a

b


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.


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.


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 distribution

X is said to have the exponential distribution

if for some


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,


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


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


Variance and Standard Deviation

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

The standard deviation is


Short-cut Formula for Variance


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.


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,


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?


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


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


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


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