MATH 589B due May 11, 2023 Algorithms of Applied Mathematics II Section 001, Spring 2023 (instructor: Misha Stepanov) Final Exam 1. For a function ��� : D → R, its Legendre transform is defined by ���(���) = sup���∈D ?������ − ��� (���)?. Plot the Legendre transform of ���1(���) = 12 ln?exp(���2) + exp(5 + ���)? over the interval −4 ≤ ��� ≤ 4. 2. Inside a [two-dimensional] circular chamber ���2 + ���2 ≤ 1 there are three disks of radii ���1 = 12, ���2 = 13, and ���3 = 41, with the same density. Minimize the gravitational potential energy ���2(���1,���1,���2,���2,���3,���3) = ���12���1 +���2���2 +���32���3 subject to the inequality con- straints������2+������2 ≤(1−������)2forall���=1,2,3;and(������−������)2+(������−������)2 ≥ (������ + ������)2 for all ��� ≠ ���. 3. Alice and Bob play the “game of the goose” or “snakes and ladders” with the following board: Alice/Bob are at the spaces 16/3, respectively. The players are making moves alternately, a player tosses a coin and with equal probabilities moves forward by 1 or 2 spaces. It is Bob’s turn. From the spaces 4/12/15/19 a player is automatically moved to 10/16/9/6, respectively. Estimate the probabilities of Alice or Bob winning (i.e., reaching the space 20 first). 4. Each cookie box independently of others contains one of 100 trading cards with equal prob- ability. Let ��� be a random variable that is the number of boxes needed to obtain a full collection. Estimate the probability P(��� ≤ 200) that a full collection is obtained from 200 boxes or less. 5. Random variables ���1, ���2, …, ���10 are having the joint probability density function 1?10 910 ? ���5(���1,���2,…,���10) = ��� exp −Õ������2 −Õ Õ ������2 ������2 ���=1 ���=1 ���=���+1 where ��� is an unknown normalization constant. Depict the distribution of ��� = ���1 + ���2 + … + ���10. Minimizing means finding the position where the minimum is achieved, also report the optimal value. You can use built-in elementary functions, linear algebra and ODE functions/solvers. Don’t use built-in functions for optimization in problems 1 and 2. You can use built-in random number generator for Unif (0, 1) only. 1 2