Mathematical Statistics Assignment

Added on - 28 May 2020

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Running head: STATISTICS1STATISTICS<Name of Student ><Institutional Affiliation ><Instructor’s Name ><Date of Submission >
STATISTICS2Question 33. Consider the problem; min (x-y) ^2+2(y-2)a) Verify that the problem has multiple optima.By finding the derivative of the equation in which we find x where y is minimum we have;dy/dx = [x^3-y]^2 + 2[y-x]^4dy/dx = [x^6-y^2] = 2[y^4-x^4]dy/dx = x^6-y^2+2[y^4-x^4]6x^5-2yyʹ+2[4y^3yʹ-4x^3]6x^5-2yyʹ+8y^3yʹ-8x^36x^5-8x^3=2yyʹ-8y^3yʹ6x^5-8x^3=yʹ[2y-8y^3]fʹ(x)= (6x^5-8x^3)/(2y-8y^3)= 6x^5-8x^3fʹʹ(x) = 30x^4-24x^2fʹʹʹ (x) =120x^3-48xfʹʹʹ(x) = 360x^2-48For minimum optimality, the value of y=0 give;(6x^5-8x^3)/(2y-8y^3) = 0(6x^5-8x^3)=0Hence X= 2 and 0Therefore from the above calculations, the problem has multiple derivatives.b) Use Newton’s method to solve the problem starting with the initial points Xt=[1.25,0.75],Xt=[0.5,0.5] and Xt=[0.5,-0.5]. In table, print the minimum, the value of the function at theminimum, and the number of iterations needed to reach an approximation of the minimum withan accuracy of atleast 4d.p.Study and explain the pattern of convergence for each initial point.The Newton method for finding a root of an equation is given by;Xk= fʹ(x)/fʹʹ(x)Table 1.Summary of iterationsIterationXifʹ(x)fʹʹ(x)fʹʹ(x)fʹʹʹ(x)
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