1. 1 Pertemuan 05 Peubah Acak Kontinu dan Fungsi Kepekatannya Matakuliah: I0262 – Statistik Probabilitas Tahun: 2007 Versi: Revisi

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

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

4. 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. 5 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

6. 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. 7 Probability Density Function is given by the area of the shaded region. ba

8. 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. 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. 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. 11 Exponential distribution X is said to have the exponential distribution if for some

12. 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. 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. 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. 15 Variance and Standard Deviation The variance of continuous rv X with pdf f(x) and mean is The standard deviation is

16. 16 Short-cut Formula for Variance

17. 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. 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. 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. 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. 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. 22 Median The median of a continuous distribution, denoted by, is the 50 th percentile. So satisfies That is, half the area under the density curve is to the left of

23. 23 Selamat Belajar Semoga Sukses.