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study help
mathematics
advanced engineering mathematics
Advanced Engineering Mathematics 10th edition Erwin Kreyszig - Solutions
Do steepest descent steps when:f(x) = 0.1x21 + x22 - 0.02x1, x0 = (3, 3), 5 steps
From memory: Make a list of the three types of alternatives, each with a typical example of your own.
What is a sample? A population? Why do we sample in statistics?
Find and graph the sample regression line of y on x and the given data as points on the same axes. Show the details of your work.(0, 1.0), (2, 2.1), (4, 2.9), (6, 3.6), (8, 5.2)
Let X be normal with mean 80 and variance 9. Find P(X > 83), P(X < 81), P(X < 80), and P(78 < X < 82).
Let X be normal with mean 14 and variance 4. Determine c such that P(X ≤ C) = 95%, P(X ≤ C) = 5%, P(X ≤ C) = 99.5%
Find the mean and variance of a discrete random variable X having the probability function f(0) = 1/4, f(1) =1/2, f(2) = 1/4.
Find the probability function of X = Number of times of tossing a fair coin until the first head appears.
Using Venn diagrams, graph and check De Morgan’s laws
Construct the simplest possible data withx̅ = 100 but qM = 0. What is the point of this problem?
Suppose that 3% of bolts made by a machine are defective, the defectives occurring at random during production. If the bolts are packaged 50 per box, what is the binomial approximation of the probability that a given box will contain x = 0, 1, · · ·, 5 defectives?
Let X be discrete with probability function f(0) = f(3) = 1/8, f(1) = f(2) = 3/8. Find the expectation of X3.
Show that the random variables with the densitiesf(x, y) = x + yandg(x, y) = (x + 1/2)(y + 1/2)if 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 and f(x, y) = 0 and g((x, y) = 0 elsewhere, have the same marginal distribution.
Make up an example similar to Prob. 16, for instance, in terms of divisibility of numbers.Data from Prob. 16You may wonder whether in (16) the last relation follows from the others, but the answer is no. To see this, imagine that a chip is drawn from a box containing 4 chips numbered 000, 011, 101,
Using a Venn diagram, show that if and only if A ⊆ B if and only if A ∪ B = B.
Calculate s for the data 4 1 3 10 2. Then reduce the data by deleting the outlier and calculate s. Comment.
Suppose a trial can result in precisely one of k mutually exclusive events A1, · · ·,Ak with probabilities p1, · · ·, pk, respectively, where P1 + · · · + Pk = 1. Suppose that n independent trials are performed. Show that the probability of getting x1A1’s, · · ·, xkAk's is where 0
Plot a histogram of the data 8, 2, 4, 10 and guess x̅ and s by inspecting the histogram. Then calculate x̅, s2, and s.
James rolls 2 fair dice, and Harry pays k cents to James, where k is the product of the two faces that show on the dice. How much should James pay to Harry for each game to make the game fair?
Let (X, Y) have the probability functionf(0, 0) = f(1, 1) = 1/8,f(0, 1) = f(1, 0) = 3/8,Are X and Y independent?
Show that, by the definition of complement, for any subset A of a sample space S.
Give a systematic discussion of the use of Tables A7 and A8 for obtaining P(X α), P(α b), P(X c) = k, as well as P(μ - c Table A7 Table A8
Let X be a random variable that can assume every real value. What are the complements of the events X ≤ b, X < b, X ≥ c, X ≥ c, b ≤ X ≤ c, b < X ≤ c?
In connection with a trip to Europe by some students, consider the events P that they see Paris, G that they have a good time, and M that they run out of money, and describe in words the events 1, · · ·, 7 in the diagram.
Find the mean and compare it with the median. Find the standard deviation and compare it with the interquartile range.For the release times in Prob. 7Data from Prob. 7Represent the data by a stem-and-leaf plot, a histogram, and a boxplot:Release time [sec] of a relay1.3 1.2
Suppose that in the production of 60-ohm radio resistors, nondefective items are those that have a resistance between 58 and 62 ohms and the probability of a resistor’s being defective is 0.1%. The resistors are sold in lots of 200, with the guarantee that all resistors are nondefective. What is
Show that the mean always lies between the smallest and the largest data value.
A small filling station is supplied with gasoline every Saturday afternoon. Assume that its volume X of sales in ten thousands of gallons has the probability density f(x) = 6x(1 - x) if 0 ≤ x ≤ 1 and 0 otherwise. Determine the mean, the variance, and the standardized variable.
Give an example of two different discrete distributions that have the same marginal distributions.
What is the probability that in a group of 20 people (that includes no twins) at least two have the same birthday, if we assume that the probability of having birthday on a given day is 1/365 for every day. First guess. Consider the complementary event.
What gives the greater probability of hitting at least once:(a) Hitting with probability and firing 1 shot(b) Hitting with probability and firing 2 shots(c) Hitting with probability and firing 4 shots? First guess.
Find the expectation of g(X) = X2, where X is uniformly distributed on the interval -1 ≤ x ≤ 1.
If the resistance X of certain wires in an electrical network is normal with mean 0.01 Ω and standard deviation 0.001 Ω, how many of 1000 wires will meet the specification that they have resistance between 0.009 and 0.011 Ω?
Suppose that in an automatic process of filling oil cans, the content of a can (in gallons) is Y = 100 + X, where X is a random variable with density f(x) = 1 - |x| when |x| ≤ 1 and 0 when |x| > 1. Sketch f(x) and F(x). In a lot of 1000 cans, about how many will contain 100 gallons or more?
Find the mean and compare it with the median. Find the standard deviation and compare it with the interquartile range.For the medical data in Prob. 3Data from Prob. 3Represent the data by a stem-and-leaf plot, a histogram, and a boxplot:Systolic blood pressure of 15 female patients of ages
Suppose that a test for extrasensory perception consists of naming (in any order) 3 cards randomly drawn from a deck of 13 cards. Find the probability that by chance alone, the person will correctly name(a) No cards(b) 1 card(c) 2 cards(d) 3 cards.
Find the mean, standard deviation, and variance in Prob. 11.Data from Prob. 11Make a stem-and-leaf plot, histogram, and boxplot of the data 110, 113, 109, 118, 110, 115, 104, 111, 116, 113.
What is the expected daily profit if a store sells X air conditioners per day with probability f(10) = 0.1, f(11) = 0.3, f(12) = 0.4, f(13) = 0.2 and the profit per conditioner is $55?
Find P(X > Y) when (X, Y) has the densityf(x, y) = 0.25e0.5(x+y) if x ≥ 0, y ≥ 0and 0 otherwise.
(a) Using (7), compute approximate values of for n! for n = 1, · · ·, 20.(b) Determine the relative error in (a). Find an empirical formula for that relative error.(c) An upper bound for that relative error is e1/12n - 1. Try to relate your empirical formula to this.(d) Search through the
A pressure control apparatus contains 3 electronic tubes. The apparatus will not work unless all tubes are operative. If the probability of failure of each tube during some interval of time is 0.04, what is the corresponding probability of failure of the apparatus?
If sick-leave time X used by employees of a company in one month is (very roughly) normal with mean 1000 hours and standard deviation 100 hours, how much time t should be budgeted for sick leave during the next month if t is to be exceeded with probability of only 20%?
Find the probability that none of three bulbs in a traffic signal will have to be replaced during the first 1500 hours of operation if the lifetime X of a bulb is a random variable with the density f(x) = 6[0.25 - (x - 1.5)2] when 1 ≤ x ≤ 2 and f(x) = 0 otherwise, where x is measured in
Find the mean and compare it with the median. Find the standard deviation and compare it with the interquartile range.For the data in Prob. 1Data from Prob. 1Represent the data by a stem-and-leaf plot, a histogram, and a boxplot:Length of nails [mm]19 21 19 20
Make a stem-and-leaf plot, histogram, and boxplot of the data 110, 113, 109, 118, 110, 115, 104, 111, 116, 113.
For what choice of the maximum possible deviation from 1.00 cm shall we obtain 10% defectives in Probs. 9 and 10?
A 5-gear assembly is put together with spacers between the gears. The mean thickness of the gears is 5.020 cm with a standard deviation of 0.003 cm. The mean thickness of the spacers is 0.040 cm with a standard deviation of 0.002 cm. Find the mean and standard deviation of the assembled units
How many automobile registrations may the police have to check in a hit-and-run accident if a witness reports KDP7 and cannot remember the last two digits on the license plate but is certain that all three digits were different?
A batch of 200 iron rods consists of 50 over sized rods, 50 undersized rods, and 100 rods of the desired length. If two rods are drawn at random without replacement, what is the probability of obtaining(a) Two rods of the desired length(b) Exactly one of the desired length(c) None of the desired
Find the mean and variance of the random variable X with probability function or density f(x).f(x) = Cex/2 (x = 0)
Find a shortest spanning tree by Prim’s algorithm. 10/ (5) 3 14 2 8 6 (4) 1 4 4 16 (2
Find the adjacency matrix of the given graph or digraph.
Why are backward edges not considered in the definition of the capacity of a cut set?
Apply the method suggested in Prob. 8 to the graph in Example 1. Do you get the same tree?Data from Prob. 8To get a minimum spanning tree, instead of adding shortest edges, one could think of deleting longest edges. For what graphs would this be feasible? Describe an algorithm for this.
Define bipartite graphs and describe some typical applications of them.
For each graph find the shortest paths.
Can you obtain a bipartite subgraph in Prob. 4 by omitting two edges? Any two edges? Any two edges without a common vertex? Data from Prob. 4 If you answer is yes, find S and T:
Write a computer program for the algorithm in Table 23.1. Test the program with the graph in Example 1. Apply it to Probs. 1–3 and to some graphs of your own choice. Table 23.1
Find the adjacency matrix of the given graph or digraph.
What is combinatorial optimization? Which sections of this chapter involved it? Explain details.
Find the maximum flow by Ford-Fulkerson:In Prob. 15Data from Prob. 15 (1) S - (2) 10,3 5,3 6,0 4, 2 3, 1 (3) 8,5 (4) 1, 1 (5) t
Find a shortest spanning tree by Prim’s algorithm.
When will the adjacency matrix of a digraph be symmetric?
Find a minimum cut set and its capacity for the network:In Fig. 499 s (1) 8,4 7,5 4 2 6, 1 6,5 8,5 4, 2 5 (3 2, 1 4,2 10, 8 (6) 6, 1 7
Write a corresponding program.
What are the basic ideas and concepts in handling flows?
For each graph find the shortest paths. (1) 17 (3) 10 2 6 (2) 3 4
If you answer is yes, find S and T: (1) (6) (2) (G) 5 (3) (4)
Show that the adjacency matrix of a graph is symmetric.
Find a shortest spanning tree by Kruskal’s algorithm. Sketch it.
Give typical applications involving spanning trees.
How does Prim’s algorithm prevent the generation of cycles as you grow T?
How does Ford–Fulkerson prevent the formation of cycles?
Find further situations that can be modeled by a graph or diagraph.
Find T and cap (S, T ) for:Fig 499, S = {1, 2, 4, 5} 8,4 7,5 4 6, 1 6,5 8,5 4, 2 5 3 2, 1 4, 2 10, 8 6 6, 1 7) t
Find a shortest spanning tree by Kruskal’s algorithm. Sketch it.
What situations can be handled in terms of the traveling salesman problem?
For each graph find the shortest paths. (5) 2 4 3 1 (4) 3 LO 5 1 (2) 2 (1)
If you answer is yes, find S and T: (1 7 (3) 5 4 6 2 (8
Show that if vertex v has label λ(v) = k, then there is a path s→v of length k.
Worker can do jobs J1, J3, J4, worker W2 job J3, and worker W3 jobs J2, J3, J4. Represent this by a graph.
Find a shortest spanning tree by Kruskal’s algorithm. Sketch it.
What is a shortest path problem? Give applications.
What is the result of applying Prim’s algorithm to a graph that is not connected?
How would you represent a net of two-way and one way streets by a digraph?
Which are the “bottleneck” edges by which the flow in Example 1 is actually limited? Hence which capacities could be decreased without decreasing the maximum flow?
Find T and cap (S, T ) for:Fig 498, S = {1, 2, 3} (1) 20, 8 10, 6 2 4 Cut 11, 11 4,3 7,6 3 5,0 (5) 13, 11 3,3 (6) t
Find a shortest spanning tree by Kruskal’s algorithm. Sketch it.
State from memory how graphs can be handled on computers.
Show that in Dijkstra’s algorithm, at each instant the demand on storage is light (data for fewer than n edges).
Find a shortest path P: s→t and its length by Moore’s algorithm. Sketch the graph with the labels and indicate P by heavier lines as in Fig. 482. S
Sketch the graph consisting of the vertices and edges of a triangle. Of a pentagon. Of a tetrahedron.
State some typical problems that can be modeled and solved by graphs or digraphs.
When will S = E at the end in Prim’s algorithm?
Find T and cap (S, T ) for:Fig 498, S = {1, 2, 4, 5} 8 20,8 10, 6 2 4 Cut 11, 11 4,3 7,6 3 5,0 5 13, 11 3,3 6) t
Find a shortest spanning tree by Kruskal’s algorithm. Sketch it.
What is a graph, a digraph, a cycle, a tree?
The net of roads in Fig. 488 connecting four villages is to be reduced to minimum length, but so that one can still reach every village from every other village. Which of the roads should be retained? Find the solution (a) By inspection (b) By Dijkstra’s algorithm.
If you answer is yes, find S and T: (1) 3 (2)
Find a shortest path P: s→t and its length by Moore’s algorithm. Sketch the graph with the labels and indicate P by heavier lines as in Fig. 482. S
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