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We have considered, in the context of a randomized
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- please try to simulate the probability of rolling a Die with Sample Space* S={1,2,3,4,5,6} and the probability of each sample point has a 1/6 chance of occurring, i.e., you need to verify that your simulation converges to 1/6 when you select one point of sample space. When X is a random variable for sample point of rolling a Die, Pr(X<=4)=2/3. Please verify this result by simulation. Please let me know how to make an Excel file as stated above.Simulated annealing is an extension of hill climbing, which uses randomness to avoid getting stuck in local maxima and plateaux. a) As defined in your textbook, simulated annealing returns the current state when the end of the annealing schedule is reached and if the annealing schedule is slow enough. Given that we know the value (measure of goodness) of each state we visit, is there anything smarter we could do? (b) Simulated annealing requires a very small amount of memory, just enough to store two states: the current state and the proposed next state. Suppose we had enough memory to hold two million states. Propose a modification to simulated annealing that makes productive use of the additional memory. In particular, suggest something that will likely perform better than just running simulated annealing a million times consecutively with random restarts. [Note: There are multiple correct answers here.] (c) Gradient ascent search is prone to local optima just like hill climbing.…Recall the Monte Carlo method, from week 6 (section 6.2.2), for approximating . Suppose we choose a point (x, y) randomly (with uniform distribution) in the unit square. The probability that it lies inside a circle of diameter 1 contained in the unit square is equal to the area of that circle, or π/4. So this Monte Carlo method works as follows: Write a function montecarlo (M) which takes an integer M and returns an approximation to π. (I can't give you an example output, as the random nature of the procedure means approximations will differ!)
- SIMULATION AND MODELING Using the mid- square method obtain the random variables using Z0= 1009 until the cycle degenerates to zero.Pick one million sets of 12 uniform random numbers between 0 and 1. Sum up the 12 numbers in each set. Make a histogram with these one million sums, picking some reasonable binning. You will find that the mean is (obviously?) 12 times 0.5 = 6. Perhaps more surprising, you will find that the distribution of these sums looks very much Gaussian (a "Bell Curve"). This is an example of the "Central Limit Theorem", which says that the distribution of the sum of many random variables approaches the Gaussian distribution even when the individual variables are not gaussianly distributed. mean Superimpose on the histogram an appropriately normalized Gaussian distribution of 6 and standard deviation o = 1. (Look at the solutions from the week 5 discussion session for some help, if you need it). You will find that this Gaussian works pretty well. Not for credit but for thinking: why o = 1 in this case? (An explanation will come once the solutions are posted).Let pn(x) be the probability of selling the house to the highest bidder when there are n people, and you adopt the Look-Then-Leap algorithm by rejecting the first x people. For all positive integers x and n with x < n, the probability is equal to p(n(x))= x/n (1/x + 1/(x+1) + 1/(x+2) + … + 1/(n-1)) If n = 100, use the formula above to determine the integer x that maximizes the probability n = 100 that p100(x). For this optimal value of x, calculate the probability p100(x). Briefly discuss the significance of this result, explaining why the Optimal Stopping algorithm produces a result whose probability is far more than 1/n = 1/100 = 1%.
- 4. Consider a random walk on the infinite line. At each step, the position of the particle is one of the integer points. At the next step, it moves to one of the two neighboring points equiprobably. Show that the expected distance of the particle from the origin after n steps is O(n/2).Write a program to find the solution to Maxone problem (You want to maximize the number of ones) using agenetic algorithm where the population size is 2000 and each chromosome is 20 genes long. Use a mutationprobability of 0.02 and a cross-over probability of 0.5. Make separate functions for different components of thegenetic algorithm.Consider the following procedure for initializing the parameters of a neural network: 1. Pick a random number r r = rand(1,1) * (2 + INIT_EPSILON ) – INIT_EPSILON 2. Set e =r for all i, j,l Does this work? No, because the procedure fails to break symmetry. O b. Yes, unless we are unlucky and get r = 0 (up to numerical precision). O. Yes or no, depending on the training set inputs x(i). d. Yes, because the parameters are chosen randomly.
- an iterative algorithm for separating n VLSI chips into those that are good and those that are bad by testing two chips at a time and learning either that they are the same or that they are different. To help, at least half of the chips are promised to be good. Now design (much easier) a randomized algorithm for this problem. Here are some hints.Randomly select one of the chips. What is the probability that the chip is good? How can you learn whether or not the selected chip is good? If it is good, how can you easily partition the chips into good and bad chips? If the chip is not good, what should your algorithm do? When should the algorithm stop? What is the expected running time of this algorithm?Q1/ The ideal gas equation of state is given by: PV = nRT Where: P is the pressure (atm), V is the volume (L), T is the temperature (K), R=0.08206 (L atm)/(mol K) is the gas constant, and n is the number of moles. Real gases, especially at high pressures, deviate from this behavior. Their responses can be modeled with the van der Waals equation: P-- nRT n² a + = 0 V-nb V2 Where a and b are material constants. For CO₂ a 3.5924 L'atm/mol², and b=0.04267 L/mol. Calculate P from both equations for CO₂ gas with 40 values of V between 0.01 and 1.5 and display the results in: 1- Three-column table where the values of Vand both P are displayed in the first, second, and third columns, respectively. 2-Plot V versus both P in two different plots in the same figure with a solid line, black color, with circle marker. Add a title, labels, and the grid to the plot. Make all texts bold with font size of 13. Take T-298K and n-3 moles.Let l be a line in the x-yplane. If l is a vertical line, its equation is x = a for some real number a. Suppose l is not a vertical line and its slope is m. Then the equation of l is y = mx + b, where b is the y-intercept. If l passes through the point (x₀, y₀), the equation of l can be written as y - y₀ = m(x - x₀). If (x₁, y₁) and (x₂, y₂) are two points in the x-y plane and x₁ ≠ x₂, the slope of line passing through these points is m = (y₂ - y₁)/(x₂ - x₁). Instructions Write a program that prompts the user for two points in the x-y plane. Input should be entered in the following order: Input x₁ Input y₁ Input x₂