Pertemuan 05 Peubah Acak Kontinu dan Fungsi Kepekatannya

1. Pertemuan 05Peubah Acak Kontinu dan Fungsi Kepekatannya Matakuliah : I0272 – Statistik Probabilitas Tahun : 2005 Versi : Revisi

2. Learning Outcomes Pada akhir pertemuan ini, diharapkan mahasiswa akan mampu : • Mahasiswa akan dapat menghitung nilai harapan, dan ragam peubah acak kontinu.

3. Outline Materi • Konsep dasar • Nilai harapan dan ragam • Sebaran normal • Hampiran normal terhadap Binomial • Sebaran khusus : Eksponensial, Gamma, Beta, dst.

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).

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

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.

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

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.

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.

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)

11. Exponential distribution X is said to have the exponential distribution if for some

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,

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

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

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

16. Short-cut Formula for Variance

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.

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,

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?

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

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

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

23. Selamat Belajar Semoga Sukses.