Concept explainers
Suppose A[1], A[2],…A[n] is a one-dimensional array and
- How many elements are in the subarray (i) if n is even? and (ii) if n is odd?
- What is the probability that a randomaly chosen array element is the subarray (i) if n is even? and (ii) if n is odd?
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Chapter 9 Solutions
Discrete Mathematics With Applications
- 2. Each of 12 refrigerators of a certain type has been returned to a distributor because of the presence of a high-pitched oscillating noise when the refrigerator is running. Suppose that four of these 12 have defective compressors and the other eight have less serious problems. If they are examined in random order, let ? be the number among the first six examined that have a defective compressor. Compute (a) ?(? = 1), (b) ?(? ≥ 4), and (c) ?(1 ≤ ? ≤ 3).arrow_forwardEach of 14 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 9 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. (I have figured out part "a" but need help with "b" and P(X ≤ 3) in "c") (a) Calculate P(X = 4) and P(X ≤ 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 25 randomly selected refrigerators that have defective compressors, describe a less tedious way to calculate (at least approximately)…arrow_forwardAs a generalization of Example 5.3(figure), consider a test of n circuits such that each circuit is acceptable with probability p, independent of the outcome of any other test. Show that the joint PMF of X, the number of acceptable circuits, and Y, the number of acceptable circuits found before observing the first reject, is PX,Y(x,y) = ((n-y-1)C(x-y))*p^(x)*(1-p)^(n-x) For 0 ≤ y ≤ x < n p^(n) For x=y=n 0 otherwise Hint: For 0 ≤ y ≤ x < n, show that {X = x, Y = y} = A ∩ B ∩ C, where A: The first y tests are acceptable. B: Test y + 1 is a rejection. C: The remaining n − y − 1 tests yield x − y acceptable circuitsarrow_forward
- Generate the first five random numbers (0 to 1) using Linear Congruential method. Given: a = 7, c = 6, m = 17, X0 = 110arrow_forwardA manufacturer of “Keep it Warm” bags is interested in comparing the heat retention of bagswhen used at five different temperatures (100 oF, 125 oF, 150 oF, 175 oF, and 200 oF). Thirty bagsare selected randomly from last week’s production and randomly assigned, six each, to fivedifferent groups. Items from group 1 at beginning temperature 100 oF were kept in bags for anhour, and the temperatures of those items were recorded after an hour. Similarly, groups 2 to 5were assigned items at 125 oF, 150 oF, 175 oF, and 200 oF, respectively.a. Identify the type of study used here.b. What type of inference is possible from this study?arrow_forwardHow many elements of order 4 does Z4 ⊕ Z4 have? (Do not do this by examining each element.) Explain why Z4 ⊕ Z4 has the same number of elements of order 4 as does Z8000000 ⊕ Z400000. Generalize to the case Zm ⊕ Zn.arrow_forward
- A purchaser of transistors buys them in lots of 20. It is his policy to randomly inspect 4components from a lot and to accept the lot if at least 3 are nondefective. Suppose eachlot contains exactly five defective transisters. What proportion of lots are rejected?arrow_forwardConsider a biased random walk on the set { 1, 2, 3, 4} withprobability p = .2 of moving to the left. What is the probability of moving from 2 to 3 in exactly 3 steps if the walk hasa. reflecting boundaries? b. absorbing boundaries?arrow_forwardA consumer products testing group is evaluating two competing brands of tires, Brand 1 and Brand 2. Tread wear can vary considerably depending on the type of car, and the group is trying to eliminate this effect by installing the two brands on the same 12 cars, chosen at random. In particular, each car has one tire of each brand on its front wheels, with half of the cars chosen at random to have Brand 1 on the left front wheel, and the rest to have Brand 2 there. After all of the cars are driven over the standard test course for 20,000 miles, the amount of tread wear (in inches) is recorded, as shown in the table below. Car 1 2 3 4 5 6 7 8 9 10 11 12 Brand 1 0.64 0.53 0.32 0.61 0.59 0.64 0.34 0.58 0.53 0.43 0.38 0.34 Brand 2 0.47 0.48 0.31 0.37 0.56 0.37 0.30 0.57 0.49 0.41 0.50 0.17 Difference(Brand 1 - Brand 2) 0.17…arrow_forward
- Assume that a simple random Salomearrow_forwardGenerate a sequence U1, U2,..., U1000 of independent uniform random variables on a computer. Let Sn = ni=1 Ui for n = 1, 2,..., 1000. Plot each ofthe following versus n:a. Snb. Sn/nc. Sn − n/2d. (Sn − n/2)/ne. (Sn − n/2)/√nExplain the shapes of the resulting graphs using the concepts of this chapter.arrow_forwardConsider the following parameters:a = 1,298, Xo = 6,286, C = 9,807 and M = 341.i). Using the linear congruential method, determine the first 10 random numbers.ii). How robust is this algorithm?arrow_forward
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