Get questions and answers for Advanced Engineering Mathematics

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If the mathematics scores of the SAT college entrance exams are normal with mean 480 and standard deviation 100 (these are about the actual values"

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State the main theorems on probability. Illustrate them by simple examples."

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Determine the number of different bridge hands. (A bridge hand consists of 13 cards selected from a full deck of 52 cards.)"

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Let X be the number of years before a certain kind of pump needs replacement. Let X have the probability function f(x) = kx3, x = 0, 1, 2, ,3, 4."

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Graph a sample space for the experiments:Tossing a coin until the first Head appears"

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Graph a sample space for the experiments:Rolling 2 dice"

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Graph f and F when the density of X is f(x) = k = const if -2 ≤ x ≤ 2 and 0 elsewhere. Find P(0 ≤ X ≤ 2)."

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Let f(x, y) = k if x > 0, y > 0, x + y < 3 and 0 otherwise. Find k. Sketch f(x, y). Find P(X + Y ≤ 1), P(Y > X)."

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Find the adjacency matrix of the given graph or digraph."

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State the most important facts about distributions of two random variables and their marginal distributions."

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Graph a sample space for the experiments:Drawing gaskets from a lot of 10, containing one defective D, unitil D is drawn, one at a time and assuming"

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Let X [millimeters] be the thickness of washers. Assume that X has the density f(x) = kx if 0.9 < x < 1.1 and 0 otherwise. Find k. What is the"

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Represent the data by a stem-and-leaf plot, a histogram, and a boxplot:Efficiency [%] of seven Voith Francis turbines of runner diameter 2.3 m under"

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In 1910, E. Rutherford and H. Geiger showed experimentally that the number of alpha particles emitted per second in a radioactive process is a random"

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If the diameter X [cm] of certain bolts has the density f(x) = k(x - 0.9)(1.1 - x) for 0.9 < x < 1.1 and 0 for other x, what are k, μ, and"

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In how many different ways can 6 people be seated at a round table?"

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If we inspect photocopy paper by randomly drawing 5 sheets without replacement from every pack of 500, what is the probability of getting 5 clean"

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A manufacturer knows from experience that the resistance of resistors he produces is normal with mean μ = 150 Ω and standard deviation σ = 5 Ω."

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Graph a sample space for the experiments:Recording the daily maximum temperature X and the daily maximum air pressure Y at Times Square in New York"

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Let X be the number of cars per minute passing a certain point of some road between 8 A.M. and 10 A.M. on a Sunday. Assume that X has a Poisson"

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Represent the data by a stem-and-leaf plot, a histogram, and a boxplot:Release time [sec] of a relay1.3 1.2 1.4 "

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When is the Poisson distribution a good approximation of the binomial distribution? The normal distribution?"

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What are the mean thickness and the standard deviation of transformer cores each consisting of 50 layers of sheet metal and 49 insulating paper"

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Of a lot of 10 items, 2 are defective.(a) Find the number of different samples of 4. Find the number of samples of 4 containing(b) No defectives(c) 1"

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What is sampling with and without replacement? What distributions are involved?"

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If the lifetime X of a certain kind of automobile battery is normally distributed with a mean of 5 years and a standard deviation of 1 year, and the"

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Five fair coins are tossed simultaneously. Find the probability function of the random variable X = Number of heads and compute the probabilities of"

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Graph f and F when f(-2) = f(2) = 1/8, f(-1) = f(1) = 3/8. Can f have further positive values?"

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Represent the data by a stem-and-leaf plot, a histogram, and a boxplot:Weight of filled bags [g] in an automatic filling203 199 "

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State the definition of probability from memory. Give simple examples."

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Find the mean and variance of the random variable X with probability function or density f(x).f(x) = 4e4x (x ≥ 0)"

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Find the density of the marginal distribution of Y in Fig. 524."

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In how many different ways can we select a committee consisting of 3 engineers, 2 physicists, and 2 computer scientists from 10 engineers, 5"

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If a box contains 10 left-handed and 20 right-handed screws, what is the probability of obtaining at least one right-handed screw in drawing 2 screws"

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What is a random variable? Its distribution function? Its probability function or density?"

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Let X be normal with mean 50 and variance 9. Determine c such that P(X < c) = 5%, P(X > c) = 1%, P(50 - c X < 50 + c) = 50%"

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Represent the data by a stem-and-leaf plot, a histogram, and a boxplot:Systolic blood pressure of 15 female patients of ages 20–22156 "

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Find the mean and variance of the random variable X with probability function or density f(x).Uniform distribution on [0, 2π]"

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What do we mean by an experiment? An outcome? An event? Give examples."

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If a box contains 4 rubber gaskets and 2 plastic gaskets, what is the probability of drawing(a) First the plastic and then the rubber gaskets(b)"

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Three screws are drawn at random from a lot of 100 screws, 10 of which are defective. Find the probability of the event that all 3 screws drawn are"

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What properties of data are measured by the mean? The median? The standard deviation? The variance?"

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In rolling 2 fair dice, what is the probability of a sum greater than 3 but not exceeding 6?"

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Let X be normal with mean 10 and variance 4. Find P(X > 12), P(X < 10), P(X < 11), P(9 < X < 13)."

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Graph a sample space for the experiments:Drawing 3 screws from a lot of right-handed and left handed screws"

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Graph the probability function f(x) = kx2 = (x = 1, 2, 3, 4, 5; k suitable) and the distribution function."

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Mark the positions of μ in Fig. 517. Comment."

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Represent the data by a stem-and-leaf plot, a histogram, and a boxplot:Length of nails [mm]19 21 19 20 "

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What are stem-and-leaf plots? Boxplots? Histograms? Compare their advantages."

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Find the mean and variance of the random variable X with probability function or density f(x).f(x) = kx (0 ≤ x ≤ 2, k suitable)"

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Let f(x, y) = k when 8 ≤ x ≤ 12 and 0 ≤ y ≤ 2 and zero elsewhere. Find k. Find P(X ≤ 11, 1 ≤ Y ≤ 1.5) and P(9 ≤ X ≤ 13, Y ≤ 1)."

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In how many ways can a company assign 10 rivers to n buses, one driver to each bus and conversely?"

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In rolling 3 fair dice, what is the probability of obtaining a sum not greater than 16?"

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Find a graph, as simple as possible, that cannot be vertex colored with three colors. Why is this of interest in connection with Prob. 24?Data from"

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Using augmenting paths, find a maximum cardinality matching."

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What would be the answer to Prob. 22 if only the five ships S1, · · ·, S5 had to be accommodated?Data from Prob. 22How many piers does a harbor"

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How many piers does a harbor master need for accommodating six cruise ships S1, · · ·, S6 with expected dates of arrival A and departure D in"

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Company A has offices in Chicago, Los Angeles, and New York; Company B in Boston and New York; Company C in Chicago, Dallas, and Los Angeles."

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How many colors do you need for vertex coloring any tree?"

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Find a shortest spanning tree."

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Three teachers x1, x2, x3 teach four classes y1, y2, y3, y4 for these numbers of periods:Show that this arrangement can be represented by a bipartite"

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If a network has several sources s1, · · ·, sk, show that it can be reduced to the case of a single-source network by introducing a new vertex s"

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The definition is B∼ = [bjk], whereFind the incidence matrix of the digraph in Prob. 11.Data from Prob. 11Find the adjacency matrix of the given"

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Find the maximum flow by inspection:"

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If two vertices in a tree are joined by a new edge, a cycle is formed."

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Find shortest paths by Dijkstra’s algorithm."

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What is the smallest number of exam periods for six subjects α, b, c, d, e, f if some of the students simultaneously take α, b, f, some c, d, e,"

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The definition is B = [bjk], whereFind the incidence matrix of the graph in Prob. 8.Data from Prob. 8Find the adjacency matrix of the given graph or"

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Find a shortest path and its length by Moore’s BFS algorithm, assuming that all the edges have length 1."

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In Prob. 15, the cut set contains precisely all forward edges used to capacity by the maximum flow (Fig. 501). Is this just by chance?Data from Prob."

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In what case are all the off-diagonal entries of the adjacency matrix of a graph G equal to one?"

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Find the maximum flow by inspection:"

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A tree with exactly two vertices of degree 1 must be a path."

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Make it for the graph in Prob. 15.Data from Prob. 15Sketch the graph whose adjacency matrix is:"

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A planar graph is a graph that can be drawn on a sheet of paper so that no two edges cross. Show that the complete graph K4 with four vertices is"

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Is the graph in Fig. 484 an Euler graph. Give reason."

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Fi Sketch the graph whose adjacency matrix is:"

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Find a minimum cut set in Fig. 500 and its capacity."

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Sketch the graph for the given adjacency matrix."

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An Euler graph G is a graph that has a closed Euler trail. An Euler trail is a trail that contains every edge of G exactly once. Which subgraph with"

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Using augmenting paths, find a maximum cardinality matching:In Prob. 12Data from Prob. 12Find an augmenting path:"

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Find flow augmenting paths:"

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Prove the following.If in a graph any two vertices are connected by a unique path, the graph is a tree.Data from Prob. 14Prove the following.The path"

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Sketch the graph for the given adjacency matrix."

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Sketch the graph whose adjacency matrix is:"

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Sketch the graph whose adjacency matrix is:"

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Find a shortest spanning tree by Prim’s algorithm.Write a program and apply it to Probs. 6.Data from Prob. 6Find a shortest spanning tree by"

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Are the consecutive flow augmenting paths produced by Ford–Fulkerson unique?"

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Find the adjacency matrix of:"

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Find flow augmenting paths:"

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Using augmenting paths, find a maximum cardinality matching:Data from Prob. 11Find an augmenting path:"

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The postman problem is the problem of finding a closed walk W: s→s (s the post office) in a graph G with edges (i, j) of length lij > 0"

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How can you see that Ford–Fulkerson follows a BFS technique?"

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Find the adjacency matrix of the given graph or digraph."

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A (not necessarily connected) graph without cycles is called a forest. Give typical examples of applications in which graphs occur that are forests"

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Write a program and apply it to Probs. 6–9.Data from Prob. 6Find the maximum flow by Ford-Fulkerson:"

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Find a shortest spanning tree by Prim’s algorithm.For the graph in Prob. 4Data from Prob. 4Find a shortest spanning tree by Kruskal’s algorithm."

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Find the adjacency matrix of the given graph or digraph."

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Which edges could be omitted from the network in Fig. 499 without decreasing the maximum flow?"

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Apply the algorithm in Prob. 10 to the graph in Example 1. Compare with the result in Example 1.Data from Prob. 10Design an algorithm for obtaining"

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