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Prove that each of the following sets is convex (a) {(x1, 22, x3) E R3 |...
5.9 Draw an x1,x2 coordinate system and show the convex region which satisfies the constraints that follow: X2 20 and 0 < x < 3 -- X1 + X2 51 X1 + X2 54 Of the points in this region, find the point (or set of points) which yields the maximum of: a. P= 2x1 + x2 b. P = x1 + x2 C. P = xi + 2x2
Algebra Consider the feasible region in R3 defined by the inequalities -X1 + X3 > 4 3x1 + 2x2 – 23 > -3, along with xi > 0, x2 > 0 and x3 > 0. (a) Write down the linear system obtained by introducing slack vari- ables 24 and 25. (b) Write down the basic solution corresponding to the variables xi and X3. (c) Explain whether the solution corresponds to a vertex of the fea- sible region. If it does...
kercise 6. (Rossi 2.6.4, 2.6.29) (a) Let X - (X1, X2) be a random vector with probability density function given by f(x1,x2) = 24x1x2 with support determined by 0 < xit x2 < 1,띠 > 0,x2 > 0 Determine each of the following. (v) Var(Xi/X2) (vi) ElVar(X1|X2)]
Maximize Z = 10x1 + 7x2+ 6x3 Subject to3xi + 2x2 x3 36+C xi x22x33 32 D 2x1 + x2 +x3 <22+F X1 X1, X2, X3
3. Given a second-order ODE, [x1,x2] ER?, study the stability of the equilibrium point at the origin. <=x2–2x17x7 + x3) *2=– *1 –2x2(x} + xž)
2, For each of these sets. A={3n : n E N), B = {r E R : x2 < 7), and C = {x E R : x < 12), (i) Is the set bounded above? Prove your answer.] ( .] ii) Is the set bounded below? Prove your answer answer the following questions:
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Problem 2. (25 points) Consider the following integer nonlinear programming problem. max 2 = x1x2x s.t. X1 + 2x2 + 3x3 < 10, X1 >1, x2 > 1, X3 >1, X1, X2, X3 are integers. Use dynamic programming to solve this problem.
3. Let {X1, X2, X3, X4} be independent, identically distributed random variables with p.d.f. f(0) = 2. o if 0<x< 1 else Find EY] where Y = min{X1, X2, X3, X4}.
3. Consider the following LP. Maximize u = 4x1 + 2x2 subject to X1 + 2x2 < 12, 2x1 + x2 = 12, X1, X2 > 0. (a) Use simplex tableaux to find all maximal solutions. (b) Draw the feasible region and describe the set of all maximal solutions geometrically.
5. Suppose that three random variables Xi, X2, and X3 have a continuous joint distribution with the following p.d.f. (x1+2x2+3z3) and f(1, r2, 3) 0 otherwise. (a) Determine the value of the constant c; (b) Find the marginal joint p.d.f. of Xi and X3; (c) Find P(Xi < 1|X2-2, X3-1)