3. In your local nuclear power station, there is an alarm that senses when a temperature gauge exceeds a given threshold. The gauge measures the temperature of the core. Consider the Boolean variables A (alarm sounds), FA (alarm is faulty), and FG (gauge is faulty) and the multivalued nodes G (gauge reading) and T (actual core temperature). a. Draw a Bayesian network for this domain, given that the gauge is more likely to fail when the core temperature gets too high. b. Is your network a polytree? Why or why not?

3. In your local nuclear power station, there is an alarm that senses when a temperature gauge exceeds a given threshold. The gauge measures the temperature of the core. Consider the Boolean variables A (alarm sounds), FA (alarm is faulty), and FG (gauge is faulty) and the multivalued nodes G (gauge reading) and T (actual core temperature). a. Draw a Bayesian network for this domain, given that the gauge is more likely to fail when the core temperature gets too high. b. Is your network a polytree? Why or why not?

Operations Research : Applications and Algorithms

4th Edition

ISBN:9780534380588

Author:Wayne L. Winston

Publisher:Wayne L. Winston

Chapter12: Review Of Calculus And Probability

Section12.5: Random Variables, Mean, Variance, And Covariance

Problem 5P

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When a temperature gauge surpasses a threshold, your local nuclear power station sounds an alarm. Core temperature is gauged. Consider the Boolean variables A (alarm sounds), FA (faulty alarm), and FG (faulty gauge) and the multivalued nodes G (gauge reading) and T (real core temperature). Since the gauge is more likely to fail at high core temperatures, draw a Bayesian network for this domain.
Suppose a biking environment consists of n ≥ 3 landmarks,which are linked by bike route in a cyclical manner. That is, thereis a bike route between landmark 1 and 2, between landmark 2 and 3,and so on until we link landmark n back to landmark 1. In the centerof these is a mountain which has a bike route to every single landmark.Besides these, there are no other bike routes in the biking environment.You can think of the landmarks and the single mountain as nodes, andthe bike routes as edges, which altogether form a graph G. A path is asequence of bike routes.What is the number of paths of length 2 in the graph in termsof n?What is the number of cycles of length 3 in the graph in termsof n?What is the number of cycles in the graph in terms of n?
B. Suppose we wish to animate a bouncing ball. We are given keyframes for the ball's motion. The key frames are at times T = {0, 3, 9, 15} with associated height values of H = {1, 6, 8, 3}. 1. Plot the graph of height versus time using linear interpolation between key frames. 2. Since the ball's speed changes as it falls and as it bounces, linear interpolation will not give a realistic result. Which method would create the most realistic bounce? And how would the graph look like? H 16- 14- 12- 10- 8- 6- 4- 2- 0- →T 4 6 8. 10 12 14 16
1. The impulse response of a causal system is: h(t) = A cos(wt) e¯¹/¹u(t) where u(t) is the Heaviside step function. The response is measured experimentally with a sampling interval of T. a. Write an expression for the sampled impulse response h[n]. b. Calculate the z transform of h[n] and write an expression for H[z]. Use the tables provided below as necessary. c. Does the system have an infinite impulse response (IIR) or finite impulse response (FIR)? Justify your answer. d. What is the DC gain of H[z]? e. Write a difference equation that describes the output y[n] in terms of input x[n].
We now consider two sound waves with different frequencies which have to the same amplitude. The wave functions of these waves are as follows: y1 (t) = A sin (2πf1t) y2 (t) = A sin (2πf2t) 1) Using any computer program, construct the wave dependency graph resultant y (t) from time t in the case when the frequencies of the two sound waves are many next to each other if the values are given: A = 1 m, f1 = 1000 Hz and f2 = 1050 Hz. Doing the corresponding numerical simulations show what happens with the increase of the difference between the frequencies of the two waves and vice versa.
Give the truth value of each of the following wffs in the interpretation where the domain consists of the states of the United States, Q(x, y) is “x is north of y," P(x) is "x starts with the letter M," and a is "Massachusetts." a. (Vx)P(x) b. (Vx)(Vy)(Vz)[Q(x, y) ^ Q(y, z) → Qx, z)] c. Ay)Ex)Q(y, x) d. (Vx)Ay)[P(y) Qx, y)] e. (dy)Q(a, y) f. Ax)[P(x) ^ Q(x, a)]
The following Boolean function: f(x1, x2, x3) = 1 if and only if exactly one of x1, x2 and x3 is 1. Construct by hand a neural network that computes f, using the minimal number of units. Make sure to specify what sort of units you are using. Prove that the function f defined in (i) is not implementable with a single perceptron.
A wave is modeled by the wave function: y (x, t) = A sin [ 2π/0.1 m (x - 12 m/s*t)] y1 (t) = A sin (2πf1t) y2 (t) = A sin (2πf2t) Using any computer program, construct the wave dependency graph resultant y (t) from time t in the case when the frequencies of the two sound waves are many next to each other if the values are given: A = 1 m, f1 = 1000 Hz and f2 = 1050 Hz. Comment on the results from the graph and determine the value of the time when the waves are with the same phase and assemble constructively and the time when they are with phase of opposite and interfere destructively. Doing the corresponding numerical simulations show what happens with the increase of the difference between the frequencies of the two waves and vice versa.
Choose an elliptic curve and three points P,Q,R and construct via acomputer program (e.g. Geogebra) the two points (P ⊕ Q) ⊕ R andP ⊕ (Q ⊕ R). Drag the dots and explore what that means for associativity.
6. Give the truth value of each of the following wffs in the interpretation where the domain consists of people, M(x, y) is "x is the mother of y", F(x) is "x is female", M(x) is "x is male." a. (Vx)(y)M(y,x) b. (x)(y)(M(y,x) → M(y)) 7. Which of the following sentences are equivalent to the statement Cats are smarter than dogs. a. Some cats are smarter than some dogs. b. There is a cat that is smarter than all dogs. c. All cats are smarter than all dogs. d. Only cats are smarter than dogs. e. All cats are smarter than any dog. 8. Using the predicate symbols shown and appropriate quantifiers, write each English language statement as a predicate wff. (The domain is the whole world.) B(x): x is a ball R(x): x is round S(x): x is a soccer ball All soccer balls are round. a. b. Some balls are round but soccer balls are not.
2. Consider the network for car diagnosis shown in the figure above. a. Extend the network with the Boolean variables IcyWeather and StarterMotor.
Consider a robot that is capable of picking objects from a shelf and placing them on the table. Suppose that the robot’s arm works perfectly, and the environment is fully-observable. (i) Define a STRIPS operator that could be used for planning the actions of this robot. (ii) Give an example of a state S of the world at which this STRIPS operator is applicable. (iii) Describe the operation of this STRIPS operator at this state S, to show the next state S0 of the world.