How to Obtain parameters δ Appearing in The Weibull Probability
Question Description
Write well-organized, detailed solutions. You must define any new variables you introduce. Please submit a hard copy of your code and the results published using MATLAB publish command and submit also a soft copy of your original M-file code. Question: 1 2 3 4 5 6 7 8 Total Points: 5 10 10 15 15 15 15 15 100 Score: 1. (5 points) One means of obtaining of a parameter δ appearing in the Weibull probability density function is obtained from ” n #1/β 1X β δ= x n i=1 i where xi are a random sample of size n and β is another (known) parameter. If x = [72 82 97 103 113 117 126 127 127 139 154 159 199 207] and β = 3.644, then determine the value of δ. [Ans: δ = 144.2741.] 2. (10 points) The Fibonacci numbers can be generated from the following relation ” √ !n √ !n # 1+ 5 1− 5 1 − Fn = √ 2 2 5 Generate the first sixteen numbers using both fprintf and disp and present them to the MATLAB command window as follows F 0 F 1 F 2 F 3 : F15 = = = = 0 1 1 2 =610 3. (10 points) The velocity of a rocket might be described by 2 10t − 5t 624 − 3t v(t) = 36t + 12(t − 16)2 2136e−0.1(t−26) 0 piecewise functions 0≤t≤8 8 ≤ t ≤ 16 16 ≤ t ≤ 26 t > 26 otherwise Develop a MATLAB function to compute v as a function of t. Then, develop a script that uses this function to generate a plot of v versus t for t = −5 to 50. 4. (15 points) Consider the following product SN N Y = 1− n=1 Mechanical Engineering Department x2 n2 − a 2 Kuwait University ENG 307 Assignment #2 Fall 2018 where when N goes to infinity, S∞ = √ p a sin π a2 + x2 a2 + x2 sin πa A percentage error is defined as N = SN − S∞ × 100 % S∞ √ If x varies from 1 to 5 by 0.5 and a = 2.8, then what is the percentage error at these nine values of x when N = 100. Use the function prod. [Ans: 100 = [1.0001 2.2643 4.0610 6.4176 9.3707 12.9670 17.2642 22.3330 28.2588].] 5. (15 points) Develop a MATLAB function to determine whether a year is a leap year by following these steps: 1. If the year is evenly divisible by 4, go to step 2. Otherwise, go to step 5. 2. If the year is evenly divisible by 100, go to step 3. Otherwise, go to step 4. 3. If the year is evenly divisible by 400, go to step 4. Otherwise, go to step 5. 4. The year is a leap year (it has 366 days, i.e., February month has 29 days). 5. The year is not a leap year (it has 365 days, i.e., February month has 28 days) Test it 1996, 2000. 2002, 2004, 2008, 2018. Hint: Use mod(n,m) command to determine whether the number n is divisible by m or not. 6. (15 points) Consider the following relation that determine the positive square value of a, i.e., xn+1 = x2n + a √ a, n = 0, 1, 2, . . . √ where xn+1 approaches a as n approaches infinity. Write a script using for structure to determine √ 0.25, i.e., a = 0.25, with x0 = 0 and for n = 0, 1, 2, . . . , 200. Plot the values of xn for n = 0, 5, 10, . . . , 200 using black x-mark. The x-axis values are the values of n and the y-axis values are the xn . To what value does x200 appear to converge?. 7. (15 points) For a given a > 0, the following relationship will determine the positive square value of a, √ i.e., a, to within a tolerance s for any starting value (guess) x0 > 0 1 a xn+1 = xn + n = 0, 1, 2, . . . 2 xn √ where xn+1 approaches a as n approaches√infinity. When x0 < 0, the negative root is found. Write a script using while structure to determine a to within a < 0.0001% for a = 7. How many iterations does it take if (a) x0 = 3; and (b) x0 = 100. The first iteration is the determination of x1 . 8. (15 points) The cosine function can be evaluated using Maclaurin infinite series expansion cos x = ∞ X (−1)k k=0 (2k)! x2k = 1 − x2 x4 x6 + − + ··· 2! 4! 6! ∀x ∈ R Create a MATLAB function that implement this formula to compute and display the values of cos x as each term in the series is added. The input of the function is x and the order term k. For each adding term, compute and display the true percent relative error as t = True Value − Approximation Value × 100% True Value Test your function by computing cos 0.9 for up to and including the eighth term. Mechanical Engineering Department Kuwait University …
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