1. 1 Pertemuan 7 Variabel Acak-1 Matakuliah: A0064 / Statistik Ekonomi Tahun: 2005 Versi: 1/1

2. 2 Learning Outcomes Pada akhir pertemuan ini, diharapkan mahasiswa akan mampu : Menghitung beberapa persoalan yang berkaitan dengan nilai harapan variabel acak diskrit, bernoulli, dan binomial

3. 3 Outline Materi Nilai Harapan Variabel Acak Diskrit Variabel Acak Bernoulli Variabel Acak Binomial

4. COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 3-4 l Using Statistics l Expected Values of Discrete Random Variables l The Binomial Distribution l Other Discrete Probability Distributions l Continuous Random Variables l Using the Computer l Summary and Review of Terms Random Variables 3

5. COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 3-5 Consider the different possible orderings of boy (B) and girl (G) in four sequential births. There are 2*2*2*2=2 4 = 16 possibilities, so the sample space is: BBBBBGBB GBBB GGBB BBBG BGBG GBBG GGBG BBGB BGGB GBGB GGGB BBGG BGGG GBGG GGGG If girl and boy are each equally likely [P(G)=P(B) = 1/2], and the gender of each child is independent of that of the previous child, then the probability of each of these 16 possibilities is: (1/2)(1/2)(1/2)(1/2) = 1/16. 3-1 Using Statistics

6. COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 3-6 Now count the number of girls in each set of four sequential births: BBBB(0)BGBB(1)GBBB(1)GGBB(2) BBBG(1)BGBG(2)GBBG(2)GGBG(3) BBGB(1)BGGB(2)GBGB(2)GGGB(3) BBGG(2)BGGG(3)GBGG(3)GGGG(4) Notice that: each possible outcome is assigned a single numeric value, all outcomes are assigned a numeric value, and the value assigned varies over the outcomes. The count of the number of girls is a random variable: A random variable, X, is a function that assigns a single, but variable, value to each element of a sample space. Random Variables

7. COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 3-7 Random Variables (Continued) BBBB BGBB GBBB BBBG BBGB GGBB GBBG BGBG BGGB GBGB BBGG BGGG GBGG GGGB GGBG GGGG 0 1 2 3 4 X Sample Space Points on the Real Line

8. COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 3-8 Since the random variable X = 3 when any of the four outcomes BGGG, GBGG, GGBG, or GGGB occurs, P(X = 3) = P(BGGG) + P(GBGG) + P(GGBG) + P(GGGB) = 4/16 The probability distribution of a random variable is a table that lists the possible values of the random variables and their associated probabilities. xP(x) 0 1/16 1 4/16 2 6/16 3 4/16 41/16 16/16=1 Random Variables (Continued)

9. COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 3-9 Consider the experiment of tossing two six-sided dice. There are 36 possible outcomes. Let the random variable X represent the sum of the numbers on the two dice: 234567 1,11,21,31,41,51,68 2,12,22,32,42,52,69 3,13,23,33,43,53,610 4,14,24,34,44,54,611 5,15,25,35,45,55,612 6,16,26,36,46,56,6 xP(x) * 21/36 32/36 43/36 54/36 65/36 76/36 85/36 94/36 103/36 112/36 121/36 1 Example 3-1

10. COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 3-10 Example 3-2

11. COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 3-11 A discrete random variable: l has a countable number of possible values l has discrete jumps (or gaps) between successive values l has measurable probability associated with individual values l counts A discrete random variable: l has a countable number of possible values l has discrete jumps (or gaps) between successive values l has measurable probability associated with individual values l counts A continuous random variable: l has an uncountably infinite number of possible values l moves continuously from value to value l has no measurable probability associated with each value l measures (e.g.: height, weight, speed, value, duration, length) A continuous random variable: l has an uncountably infinite number of possible values l moves continuously from value to value l has no measurable probability associated with each value l measures (e.g.: height, weight, speed, value, duration, length) Discrete and Continuous Random Variables

12. COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 3-12 The probability distribution of a discrete random variable X must satisfy the following two conditions. Rules of Discrete Probability Distributions

13. COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 3-13 The cumulative distribution function, F(x), of a discrete random variable X is: xP(x) F(x) 00.1 0.1 10.2 0.3 20.3 0.6 30.2 0.8 40.1 0.9 50.1 1.0 1 543210 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 x F ( x ) Cumulative Probability Distribution of the Number of Switches Cumulative Distribution Function

14. COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 3-14 xP(x) F(x) 00.1 0.1 10.2 0.3 20.3 0.6 30.2 0.8 40.1 0.9 50.1 1.0 1 The probability that at most three switches will occur: The Probability That at Most Three Switches Will Occur 543210 0.4 0.3 0.2 0.1 0.0 x P ( x ) Cumulative Distribution Function Note: Note: P(X < 3) = F(3) = 0.8 = P(0) + P(1) + P(2) + P(3)

15. COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 3-15 xP(x) F(x) 00.1 0.1 10.2 0.3 20.3 0.6 30.2 0.8 40.1 0.9 50.1 1.0 1 The probability that more than one switch will occur: 543210 0.4 0.3 0.2 0.1 0.0 x P ( x ) The Probability That More than One Switch Will Occur Using Cumulative Probability Distributions (Figure 3-8) Note: Note: P(X > 1) = P(X > 2) = 1 – P(X < 1) = 1 – F(1) = 1 – 0.3 = 0.7

16. COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 3-16 xP(x) F(x) 00.1 0.1 10.2 0.3 20.3 0.6 30.2 0.8 40.1 0.9 50.1 1.0 1 The probability that anywhere from one to three switches will occur: 543210 0.4 0.3 0.2 0.1 0.0 x P ( x ) The Probability That Anywhere from One to Three Switches Will Occur Using Cumulative Probability Distributions (Figure 3-9) Note: Note: P(1 < X < 3) = P(X < 3) – P(X < 0) = F(3) – F(0) = 0.8 – 0.1 = 0.7

17. COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 3-17 543210 The mean of a probability distribution is a measure of its centrality or location, as is the mean or average of a frequency distribution. It is a weighted average, with the values of the random variable weighted by their probabilities. The mean is also known as the expected value (or expectation) of a random variable, because it is the value that is expected to occur, on average. The expected value of a discrete random variable X is equal to the sum of each value of the random variable multiplied by its probability. xP(x)xP(x) 00.1 0.0 10.2 0.2 20.3 0.6 30.2 0.6 40.1 0.4 50.1 0.5 1.0 2.3 = E(X) =  2.3 3-2 Expected Values of Discrete Random Variables

18. COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 3-18 Suppose you are playing a coin toss game in which you are paid $1 if the coin turns up heads and you lose $1 when the coin turns up tails. The expected value of this game is E(X) = 0. A game of chance with an expected payoff of 0 is called a fair game. xP(x)xP(x) -10.5-0.50 10.5 0.50 1.0 0.00 = E(X)=  1 0 A Fair Game

19. COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 3-19 Number of items, xP(x)xP(x) h(x)h(x)P(x) 50000.2 10002000400 60000.3 1800 40001200 70000.2 140060001200 80000.2 160080001600 9000 0.1 900100001000 1.0 67005400 Example 3-3 Example 3-3: Monthly sales of a certain product are believed to follow the given probability distribution. Suppose the company has a fixed monthly production cost of $8000 and that each item brings $2. Find the expected monthly profit h(X), from product sales. The expected value of a function of a discrete random variable X is: The expected value of a linear function of a random variable is: E(aX+b)=aE(X)+b In this case: E(2X-8000)=2E(X)-8000=(2)(6700)-8000=5400 Expected Value of a Function of a Discrete Random Variables Note: h(X) = 2X – 8000 where X = # of items sold

20. COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 3-20 variance The variance of a random variable is the expected squared deviation from the mean: standard deviation The standard deviation of a random variable is the square root of its variance: Variance and Standard Deviation of a Random Variable

21. COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 3-21 Number of Switches, xP(x)xP(x)(x-  )(x-  ) 2 P(x-  ) 2 x 2 P(x) 00.10.0-2.35.290.5290.0 10.20.2-1.31.690.3380.2 20.30.6-0.30.090.0271.2 30.20.60.70.490.0981.8 40.10.41.72.890.2891.6 50.10.52.77.290.729 2.5 2.32.0107.3 Number of Switches, xP(x)xP(x)(x-  )(x-  ) 2 P(x-  ) 2 x 2 P(x) 00.10.0-2.35.290.5290.0 10.20.2-1.31.690.3380.2 20.30.6-0.30.090.0271.2 30.20.60.70.490.0981.8 40.10.41.72.890.2891.6 50.10.52.77.290.729 2.5 2.32.0107.3   22 2 201 22 2 2 7323 2 2                       VXEX x allx Px EXEX x x PxxPx allx ()[()] ()(). ()[()] ()()... Table 3-8 Variance and Standard Deviation of a Random Variable – using Example 3-2 Recall: = 2.3.

22. COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 3-22 The variance of a linear function of a random variable is: Number of items, xP(x)xP(x) x 2 P(x) 50000.2 1000 5000000 60000.3 1800 10800000 70000.2 1400 9800000 80000.2 1600 12800000 9000 0.1 900 8100000 1.0 6700 46500000 Example 3-3: Variance of a Linear Function of a Random Variable

23. COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 3-23 The mean or expected value of the sum of random variables is the sum of their means or expected values: For example: E(X) = $350 and E(Y) = $200 E(X+Y) = $350 + $200 = $550 The variance of the sum of independent random variables is the sum of their variances: For example: V(X) = 84 and V(Y) = 60 V(X+Y) = 144 Some Properties of Means and Variances of Random Variables

24. COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 3-24 Chebyshev’s Theorem applies to probability distributions just as it applies to frequency distributions. For a random variable X with mean  standard deviation , and for any number k > 1: At least Lie within Standard deviations of the mean 234234 Chebyshev’s Theorem Applied to Probability Distributions

25. COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 3-25 Using the Template to Calculate statistics of h(x)

26. COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 3-26 If an experiment consists of a single trial and the outcome of the trial can only be either a success * or a failure, then the trial is called a Bernoulli trial. The number of success X in one Bernoulli trial, which can be 1 or 0, is a Bernoulli random variable. Note: If p is the probability of success in a Bernoulli experiment, the E(X) = p and V(X) = p(1 – p). * The terms success and failure are simply statistical terms, and do not have positive or negative implications. In a production setting, finding a defective product may be termed a “success,” although it is not a positive result. 3-3 Bernoulli Random Variable

27. COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 3-27 Consider a Bernoulli Process in which we have a sequence of n identical trials satisfying the following conditions: 1. Each trial has two possible outcomes, called success *and failure. The two outcomes are mutually exclusive and exhaustive. 2.The probability of success, denoted by p, remains constant from trial to trial. The probability of failure is denoted by q, where q = 1-p. 3. The n trials are independent. That is, the outcome of any trial does not affect the outcomes of the other trials. A random variable, X, that counts the number of successes in n Bernoulli trials, where p is the probability of success* in any given trial, is said to follow the binomial probability distribution with parameters n (number of trials) and p (probability of success). We call X the binomial random variable. * The terms success and failure are simply statistical terms, and do not have positive or negative implications. In a production setting, finding a defective product may be termed a “success,” although it is not a positive result. The Binomial Random Variable

28. COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 3-28 Suppose we toss a single fair and balanced coin five times in succession, and let X represent the number of heads. There are 2 5 = 32 possible sequences of H and T (S and F) in the sample space for this experiment. Of these, there are 10 in which there are exactly 2 heads (X=2): HHTTTHTHTHHTTHTHTTTHTHHTTTHTHTTHTTHTTHHTTTHTHTTTHH The probability of each of these 10 outcomes is p 3 q 3 = (1/2) 3 (1/2) 2 =(1/32), so the probability of 2 heads in 5 tosses of a fair and balanced coin is: P(X = 2) = 10 * (1/32) = (10/32) =.3125 Binomial Probabilities (Introduction)

29. COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 3-29 P(X=2) = 10 * (1/32) = (10/32) =.3125 Notice that this probability has two parts: In general: 1. The probability of a given sequence of x successes out of n trials with probability of success p and probability of failure q is equal to: p x q (n-x) 2. The number of different sequences of n trials that result in exactly x successes is equal to the number of choices of x elements out of a total of n elements. This number is denoted: Binomial Probabilities (continued)

30. COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 3-30 The binomial probability distribution: where : p is the probability of success in a single trial, q = 1-p, n is the number of trials, and x is the number of successes. The Binomial Probability Distribution

31. COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 3-31 Cumulative Binomial Probability Distribution and Binomial Probability Distribution of H,the Number of Heads Appearing in Five Tosses of a Fair Coin Deriving Individual Probabilities from Cumulative Probabilities The Cumulative Binomial Probability Table (Table 1, Appendix C)

32. COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 3-32 60% of Brooke shares are owned by LeBow. A random sample of 15 shares is chosen. What is the probability that at most three of them will be found to be owned by LeBow? Calculating Binomial Probabilities - Example

33. COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 3-33 Mean, Variance, and Standard Deviation of the Binomial Distribution

34. COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 3-34 Calculating Binomial Probabilities using the Template

35. COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 3-35 43210 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 x P ( x ) Binomial Probability: n=4 p=0.5 43210 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 x P ( x ) Binomial Probability: n=4 p=0.1 43210 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 x P ( x ) Binomial Probability: n=4 p=0.3 109876543210 0.5 0.4 0.3 0.2 0.1 0.0 x P ( x ) Binomial Probability: n=10 p=0.1 109876543210 0.5 0.4 0.3 0.2 0.1 0.0 x P ( x ) Binomial Probability: n=10 p=0.3 109876543210 0.5 0.4 0.3 0.2 0.1 0.0 x P ( x ) Binomial Probability: n=10 p=0.5 20191817161514131211109876543210 0.2 0.1 0.0 x P ( x ) Binomial Probability: n=20 p=0.1 20191817161514131211109876543210 0.2 0.1 0.0 x P ( x ) Binomial Probability: n=20 p=0.3 20191817161514131211109876543210 0.2 0.1 0.0 x P ( x ) Binomial Probability: n=20 p=0.5 Binomial distributions become more symmetric as n increases and as p.5. p = 0.1p = 0.3p = 0.5 n = 4 n = 10 n = 20 Shape of the Binomial Distribution

36. COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 3-36 The negative binomial distribution is useful for determining the probability of the number of trials made until the desired number of successes are achieved in a sequence of Bernoulli trials. It counts the number of trials X to achieve the number of successes s with p being the probability of success on each trial. 3-5 Negative Binomial Distribution

37. COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 3-37 Negative Binomial Distribution - Example Example: Suppose that the probability of a manufacturing process producing a defective item is 0.05. Suppose further that the quality of any one item is independent of the quality of any other item produced. If a quality control officer selects items at random from the production line, what is the probability that the first defective item is the eight item selected. Here s = 1, x = 8, and p = 0.05. Thus,

38. COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 3-38 Calculating Negative Binomial Probabilities using the Template

39. 39 Penutup Pembahasan dilanjutkan dengan materi Pokok-8 (Variabel Acak-2)