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Peubah Acak Kontinu Pertemuan 09 Matakuliah: L0104 / Statistika Psikologi Tahun : 2008

Bina Nusantara Learning Outcomes 3 Pada akhir pertemuan ini, diharapkan mahasiswa akan mampu : Mahasuswa akan dapat menghitung sifat- sifat peluang peubah acak kontinu.

Bina Nusantara 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

Bina Nusantara 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).

Bina Nusantara 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

Bina Nusantara 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.

Bina Nusantara Probability Density Function is given by the area of the shaded region. b a

Bina Nusantara 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.

Bina Nusantara 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.

Bina Nusantara 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)

Bina Nusantara Exponential distribution X is said to have the exponential distribution if for some

Bina Nusantara 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,

Bina Nusantara 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

Bina Nusantara 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

Bina Nusantara Variance and Standard Deviation The variance of continuous rv X with pdf f(x) and mean is The standard deviation is

Bina Nusantara Short-cut Formula for Variance

Bina Nusantara 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.

Bina Nusantara 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,

Bina Nusantara 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?

Bina Nusantara 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

Bina Nusantara Selamat Belajar Semoga Sukses.

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