The “random” parts of the algorithm in Self-Test Problem 6.9 &1 can be written in terms of the generated values of a sequence of independent uniform (0, 1) random variables, known as random numbers. With [x] defined as the largest integer less than or equal to x, the first step can be written as follows:
Step 1. Generate a uniform (0, 1) random variable U. Let
a. Explain why the above is equivalent to step I of Problem 6.8.
Hint: What is the
b. Write the remaining steps of the algorithm in a similar style.
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A First Course in Probability (10th Edition)
- 1. Suppose that, in Example 2.27, 400 units of food A, 600 units of B, and 600 units of C are placed in the test tube each day and the data on daily food consumption by the bacteria (in units per day) are as shown in Table 2.6. How many bacteria of each strain can coexist in the test tube and consume all of the food? Table 2.6 Bacteria Strain I Bacteria Strain II Bacteria Strain III Food A 1 2 0 Food B 2 1 1 Food C 1 1 2arrow_forwardSuppose the probability of erroneously transmitting a single digit is P=0.03. Compute the probability of transmitting a 4-bit code word with (a) at most one error, and (b) exactly four errors.arrow_forwardA certain type of digital camera comes in either a 3-megapixel version or a 4-megapixel version. A camera store has received a shipment of 14 of these cameras, of which 6 have 3-megapixel resolution. Suppose that 3 of these cameras are randomly selected to be stored behind the counter; the other 11 are placed in a storeroom. Let X = the number of 3-megapixel cameras among the 3 selected for behind-the-counter storage. (a) What kind of distribution does X have (name and values of all parameters)? Distribution Parameters O binomial O n = 3 O geometric O p = 3/14 hypergeometric O N = 14 O negative binomial O M = 6 O µ = 9/14 O 2 = 9/28 (b) Compute the following. (Enter your answers as fractions.) P(X = 2) P(X S 2) P(X 2 2) (c) Calculate the mean value and standard deviation of X. (Give your answers to three decimal places.) mean value standard deviation O O Oarrow_forward
- . Let a random variable X - U(5,5+0). Based on a single observation X1, find the unbiased estimator of 0.arrow_forwardEach of 12 refrigerators of a certain type has been returned to a distributor because of an audible, high-pitched, oscillating noise when the refrigerator is running. Suppose that 7 of these refrigerators have a defective compressor and the other 5 have less serious problems. If the refrigerators are examined in random order, let X be the number among the first 6 examined that have a defective compressor. Compute the following: a. P(X=5) and P(X≤4) b. The probability that X exceeds its mean value by more than 1 standard deviation. c. Consider a large shipment of 400 refrigerators, of which 40 have defective compressors. If X is the number among 15 randomly selected refrigerator that have defective compressors, describe a less tedious way to calculate (at least approximately) P(X≤5) than to use the hyper-geometric pmf.arrow_forwardThe resumes of two male applicants for a college teaching position in Engineering Data Analysis are placed in the same file as the resumes of two female applicants. Two positions become available, and the first, at the rank of assistant professor, is filled by selecting one of the four applicants at random. The second position, at the rank of instructor, is then filled by selecting at random one of the remaining three applicants. Using the notation M,F,, for example, to denote the simple event that the first position is filled by the second male applicant and the second position is then filled by the first female applicant. Find the probability that the asst. prof position is filled by a male applicant or neither position was filled by a male applicant. O 1/6 O 1/2 O 2/3 O 1/3arrow_forward
- ASK YOUR TEACHER Each of 13 refrigerators of a certain type has been returned to a distributor because of an audible, high-pitched, oscillating noise when the refrigerators are running. Suppose that 8 of these refrigerators have a defective compressor and the other 5 have less serious problems. If the refrigerators are examined in random order, let X be the number among the first 6 examined that have a defective compressor. (a) Calculate P(X= 4) and P(X s 4). (Round your answers to four decimal places.) P(X= 4) = P(X ≤ 4) = (b) Determine the probability that X exceeds its mean value by more than 1 standard deviation. (Round your answer to four decimal places.) (c) Consider a large shipment of 400 refrigerators, of which 40 have defective compressors. If X is the number among 20 randomly selected refrigerators that have defective compressors, describe a less tedious way to calculate (at least approximately) P(X ≤ 3) than to use the hypergeometric pmf. distribution if the population size…arrow_forwardA certain company produces fidget spinners with ball bearings made of either plastic or metal. Under standard testing conditions, fidget spinners from this company with plastic bearings spin for an average of 2.7 minutes, while those from this company with metal bearings spin for an average of 4.2 minutes. A random sample of three fidget spinners with plastic bearings is selected from company stock, and each is spun one time under the same standard conditions; let x¯1x¯1 represent the average spinning time for these three spinners. A random sample of seven fidget spinners with metal bearings is selected from company stock, and each is likewise spun one time under standard conditions; let x¯2x¯2 represent the average spinning time for these seven spinners. What is the mean μ(x¯1−x¯2)μ(x¯1−x¯2) of the sampling distribution of the difference in sample means x¯1−x¯2x¯1−x¯2 ? 3(2.7)−7(4.2)=−21.33(2.7)−7(4.2)=−21.3 A 3−7=−43−7=−4 B 2.7−4.2=−1.52.7−4.2=−1.5 C…arrow_forward1) Consider a memoryless discrete-time source which emits a sequence of random elements X1, X2, · where xk for each integer k can be any of a1, a2, az and a4. Assume ... that Prob(xk a1) = 0.01, Prob(xk a2) 0.1 and Prob(xk аз) = 0.3. Answer the following questions: b) What is the (simplest) relationship between the entropy of xk and the entropy of (Xk, Xk+1)?arrow_forward
- Suppose a man has ordered twelve 1-gallon paint cans of a particular color (lilac) from the local paint store in order to paint his mother’s house. Unknown to the man, three of these can contains an incorrect mix of paint. For this weekend’s big project, the man randomly selects four of these 1-gallon cans to paint his mother’s living room. Let x = the number of the paint cans selected that are defective. Unknown to the man, x follows a hypergeometric distribution. Find the mean of this distribution. A) 12 B) 3 C) 1 D) 4arrow_forwardEach of 15 refrigerators of a certain type has been returned to a distributor because of an audible, high-pitched, oscillating noise when the refrigerators are running. Suppose that 12 of these refrigerators have a defective compressor and the other 3 have less serious problems. If the refrigerators are examined in random order, let X be the number among the first 6 examined that have a defective compressor. (a) Calculate P(X= 4) and P(X 4). (Round your answers to four decimal places.) P(X = 1) = P(X ≤ 4) = (b) Determine the probability that X exceeds its mean value by more than 1 standard deviation. (Round your answer to four decimal places.) (c) Consider a large shipment of 400 refrigerators, of which 40 have defective compressors. If X is the number among 20 randomly selected refrigerators that have defective compressors, describe a less tedious way to calculate (at least approximately) P(X s 6) than to use the hypergeometric pmf. distribution if the population size and the number…arrow_forwardA lumber company has just taken delivery on a shipment of 10,000 2 x 4 boards. Suppose that 40% of these boards (4000) are actually too green to be used in first-quality construction. Two boards are selected at random, one after the other. Let A = {the first board is green} and B = {the second board is green}. (a) Compute P(A), P(B), and P(A ʼn B) (a tree diagram might help). (Round your answer for P(A n B) to five decimal places.) P(A) = P(B) = P(An B) = Are A and B independent? O Yes, the two events are independent. O No, the two events are not independent. (b) With A and B independent and P(A) = P(B) = 0.4, what is P(An B)? How much difference is there between this answer and P(AB) in part (a)? O There is no difference. O There is very little difference. O There is a very large difference. For purposes of calculating P(ANB), can we assume that A and B of part (a) are independent to obtain essentially the correct probability? O Yes O No (c) Suppose the lot consists of ten boards, of…arrow_forward
- Linear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage LearningAlgebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:CengageElements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,