Instructions: Search the news, or online articles and sites to find the term “probability” (likelihood, odds, and expected outcome are also acceptable terms rather than “probability”). Locate two examples, one where you believe it is used correctly and one that you think seems fishy. Post your sources as a reference. No need to explain the articles, or how the term is used because that will be done in our response later in this module.

Assume you have been recently hired by the Department of Transportation (DoT) to analyze motorized vehicle traffic flows. Your initial goal is to analyze the traffic and traffic delays in a large metropolitan area.

You choose to use a weighted graph to represent this particular scenario. Remember that a graph is a collection of nodes (or vertices) and edges. Each edge will have a corresponding weight.

Describe how you would construct such a weighted graph. What do the nodes represent? What do the edges represent? What values would the weights represent? Is this graph a tree? Justify your design. Feel free to include an example in your description.

Describe an adjacency matrix you might create to represent this graph. What do the rows represent? What do the columns represent? What do the entries in the matrix represent? Feel free to create an example or create an illustration to better explain yourself.

Suppose you wish to identify the shortest travel time between nodes on the graph. What type of path would represent such a route? Describe this path in detail.

Suppose that the DoT is considering repairing some of the roads in the metropolitan area. But first, the DoT wishes to assess each and every road to determine which of the roads require repair. You have reported to the manager of the repair project and noted that you can use your graphical representation and graph-based methods to suggest the most efficient route the inspection crew can take to inspect all the roads. Describe in detail and name the type of the particular path on the graph that would represent this most efficient route. Consider all aspects of travel time undertaken by the inspection crew. Justify your answer.

Suppose that the DoT is planning to widen roads that create a bottleneck in the flow of traffic. As such, you pose this problem as a max flow/min cut problem. You have determined that roads are in need of widening if the traffic flow on the roads is at maximum capacity; these roads are essentially the bottlenecks of the road system. These bottleneck roads can be identified using the Max Flow/Min Cut theorem.

Using the graph representation of the road system, explain how the Max Flow/Min Cut theorem can be used to identify the bottleneck roads.

Create an example graph one might use to analyze this network as a Max Flow problem. Include at least 8 nodes and 16 edges. Include capacities and flows and identify the max flow and/or min cut.

Write a brief report on a real-life project that can be organized using PERT. Books on operations research and project management often have sections on PERT which is sometimes called the critical path method or CPM. Find references on the internet for commercial software that draws PERT diagrams and describe sine of these products and their applications.

compare and contrast slope and average rate of change

identify the characteristics of the graph of a quadratic equation

give a real life application of a quadratic function

1a)

An invertible function f(x) is given along with a point that lies on its graph. Using Theorem 2.7.1, evaluate (f−1)′(x) at the indicated value.

f(x)=11e^2x, Point=(0,11)

(f^−1)′(11)=

1b)

An invertible function f(x) is given along with a point that lies on its graph. Using Theorem 2.7.1, evaluate (f−1)′(x) at the indicated value.

f(x)=2x+sin3x.   −π/6 ≤ x ≤ π/6 Point=(π/18, 2π/18+0.5)

(f^−1)′(2π/18+0.5)=

1c)

An invertible function f(x) is given along with a point that lies on its graph. Using Theorem 2.7.1, evaluate (f−1)′(x)at the indicated value.

f(x)=x^2−11x+36, x≥5.5   Point=(8,12)

(f^−1)′(12)=

2- Ordinary Differential Equations

a) y'+y = sen(x)

b)By what technique do you solve an ODE below: (x + yˆ2) dy + (y-xˆ2) dx = 0?

c) Solve the following ODE by Exact Equation: y '= 2x

d) Resolution y '= (2 + exp (xy)) / (2y-xexp (xy))

Consider Matrix A = ([5, 0, 4],[1, -1, 0],[1, 1, 0]). Note that [5, 0, 4] is row 1. [1, -1, 0] is row 2. [1, 1, 0] is row 3.

a) Find all Eigenvalues and Eigenvectors.

Let R be a commutative ring with unity. If I is a prime ideal of R, prove that I[x] is a prime ideal of R[x].

Find the standard normal area for each of the following (Round your answers to 4 decimal places.): Standard normal area a. P( Z ≤ 2.16) b. P(Z ≤ 3.05) c. P(Z ≤ 2.05) d. P(Z > 0.55)

For this Discussion, think of a specific testing scenario. Then consider a reliable test item for that testing scenario and an unreliable item for that same testing scenario. Consider how you might know if these items are reliable or unreliable.

With these thoughts in mind:

Post a brief description of a specific testing scenario. Then describe one reliable test item and one unreliable test item for that testing scenario. Finally, explain what determines whether an item is reliable or unreliable within the scenario you presented.

Identify at least three cash flow model inputs that are significantly stochastic in your opinion. Define appropriate probability distribution to covert these input data to random variables. Estimate the host governments take (HGT) statistic and the earning power of the investment on this project with 50 percent degree of certainty using a stochastic spreadsheet modeling tool of your choice

Problem 3. Let C_ξ and C_ν be two Cantor sets (constructed in previous HW ). Show that there exist a function F : [0, 1] → [0, 1] with the following properties

(a) F is continuous and bijective.

(b)F is monotonically increasing.

(c) F maps C_ξ surjectively onto Cν.

(d) Now give an example of a measurable function f and a continuous function Φ so that f ◦ Φ is non-measurable. One may use function F constructed above (BUT YOU NEED TO PLAY WITH IT). One of the ideas is to take two measurable sets C₁ and C₂ such that m(C₁) > 0 but m(C₂) = 0 and function Φ : C₁ → C₂, continuous. Also take N ⊂ C₁ - non-measurable set and define f = χΦ(N) .

(d) Use the above construction to show that there exists a Lebesque measurable set that is not a Borel set.

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THIS WAS THE PREVIUS HW, WHERE WE DEFINE C_ξ and C_ν

Problem 3. Consider the unit interval [0, 1], and let ξ be fixed real number with ξ ∈ (0, 1) (note that the case ξ = 1/3 corresponds to the regular Cantor set we learned in our lectures). In stage 1 of the construction, remove the centrally situated open interval in [0, 1] of length ξ. In stage 2 remove the centrally situated open intervals each of relative length ξ (i.e. if the interval has length a you remove an interval of length ξ × a), one in each of the remaining intervals after stage 1, and so on. Let C_ξ denote the set which remains after applying the above procedure indefinitely

(a) Prove that C_ξ is compact.

(b) Prove that C_ξ is totally disconnected and perfect.

(c) Atually, prove that the complement of C_ξ in [0, 1] is the union of open intervals of total length equal to 1.

Problem 2. Let N denote the non-measurable subset of [0, 1], constructed in class and in the book "Real Analysis: Measure Theory, Integration, and Hilbert Spaces" by E. M. Stein, R. Shakarchi.

(a) Prove that if E is a measurable subset of N , then m(E) = 0.

(b) Assume that G is a subset of R with m_∗(G) > 0, prove that there is a subset of G such that it is non-measurable.

(c) Prove that if N^c = [0, 1] \ N , then m_∗(N^c) = 1.

(d) Now, conclude that

m_∗(N ) + m∗(N^c ) ≠ m_∗(N ∪ N^c ).

For each relation below, determine the following.(i) Is it a function? If not, explain why not and stop. Otherwise, answer part (ii).(ii) What are its domain and image?

(a){(x, y) :x, y∈Z, y- 2x}.

(b){(x, y) :x, y∈Z, xy- 0}.

(c){(x, y) :x, y∈Z, y-x2}.

(d){(x, y) :x, y∈Z, x|y}.

(e){(x, y) :x, y∈Z, x+y= 0}.

(f){(x, y) :x, y∈R, x2+y2= 1}.