1 Pertemuan 07 Variabel Acak Diskrit dan Kontinu Matakuliah: I Statistika Tahun: 2008 Versi: Revisi

2 Learning Outcomes Pada akhir pertemuan ini, diharapkan mahasiswa akan mampu : Mahasiswa akan dapat menghitung Peluang, nilai harapan, dan varians variabel acak diskrit dan kontinu.

3 Outline Materi Definisi variabel acak Distribusi probabilitas diskrit Distribusi probabilitas kontinu Nilai harapan dan varians

4 Random Variable –Outcomes of an experiment expressed numerically –E.g., Toss a die twice; count the number of times the number 4 appears (0, 1 or 2 times) –E.g., Toss a coin; assign $10 to head and - $30 to a tail = $10 T = -$30

5 Discrete Random Variable –Obtained by counting (0, 1, 2, 3, etc.) –Usually a finite number of different values –E.g., Toss a coin 5 times; count the number of tails (0, 1, 2, 3, 4, or 5 times)

6 Probability Distribution Values Probability 01/4 =.25 12/4 =.50 21/4 =.25 Discrete Probability Distribution Example Event: Toss 2 Coins Count # Tails T T TT This is using the A Priori Classical Probability approach.

7 Discrete Probability Distribution List of All Possible [X j, P(X j ) ] Pairs –X j = Value of random variable –P(X j ) = Probability associated with value Mutually Exclusive (Nothing in Common) Collective Exhaustive (Nothing Left Out)

8 Summary Measures Expected Value (The Mean) –Weighted average of the probability distribution – –E.g., Toss 2 coins, count the number of tails, compute expected value:

9 Summary Measures Variance –Weighted average squared deviation about the mean – –E.g., Toss 2 coins, count number of tails, compute variance: (continued)

10 Computing the Mean for Investment Returns Return per $1,000 for two types of investments P(X i ) P(Y i ) Economic Condition Dow Jones Fund X Growth Stock Y.2.2 Recession-$100 -$ Stable Economy Expanding Economy Investment

11 Computing the Variance for Investment Returns P(X i ) P(Y i ) Economic Condition Dow Jones Fund X Growth Stock Y.2.2 Recession-$100 -$ Stable Economy Expanding Economy Investment

12 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).

13 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

14 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.

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

16 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.

17 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.

18 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,

19 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

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

21 Short-cut Formula for Variance

22 Selamat Belajar Semoga Sukses.