For a given k, 1 ≤ k ≤ n, how many vectors (x1, x2, . . . , xk) are there for which each xi is a positive integer such that 1 ≤ x1 < x2 < · · · < xk ≤ n?
For a given k, 1 ≤ k ≤ n, how many vectors (x1, x2, . . . , xk) are there for which each xi is a positive integer such that 1 ≤ x1 < x2 < · · · < xk ≤ n?
1. Fix n and k. How many positive integer solutions are there to x1 + + xk = n where xi i for all i?
(a) How many vectors (x1, x2, x3, . . . , xn) are there for which each xi is either 0 or 1 and x1 + x2 + · · · + xn = k. (b) Do the same problem as before but under the condition that x1 + x2 + · · · + xn ≥ k.
Problem 3 Let n and k > l be positive integers. How many different integer solutions are there to x1 +...+ In = k, with all xi <l?
How many integer solutions are there for the inequality : x1 + x2 + x3 + x4 ≤ 15 (a) if xi ≥ 0 (b) if 6 ≥ x1 ≥ 1, 6 ≥ x2 ≥ 1, x3 ≥ 0, x4 > 0 How many integer solutions are there for the inequality : x++ (a) if z 20 How many integer solutions are there for the inequality : x++ (a) if z 20
1. Let X1 ~N(1,2) and X2 ~N(-1,2) be two Gaussian variables, and let Z = X1 +X2. (a) Express FX1 and FX2 in terms of Ф. b) Find Fz given that Xi, X2 are independent. (c)Find Fz given that it is Gaussian, and that E(X2-3 1. Let X1 ~N(1,2) and X2 ~N(-1,2) be two Gaussian variables, and let Z = X1 +X2. (a) Express FX1 and FX2 in terms of Ф. b) Find Fz given that Xi, X2 are independent....
How to solve these problem, I need detailed answer process. 7. a) What is the multinomial coefficient of x1x2'- ...xx*, ii + /2ikn, in the expansion of (x1+ x2 + .. + xk)"? b) How many terms are there of the form the multinomial (xi + x2 xx)? i. i.i.x1'x212 ...xxk in the expansion of 1 L2.k 7. a) What is the multinomial coefficient of x1x2'- ...xx*, ii + /2ikn, in the expansion of (x1+ x2 + .. + xk)"?...
Check if the given vectors are optimal solutions of the corresponding problems. 1. XI + 4X2 + X3 → max. 4X1 + 11X2 + 3X3 7, rI X2-X3=0, X1+ 13, .y20, j = 1.2.3. Check if the given vectors are optimal solutions of the corresponding problems. 1. XI + 4X2 + X3 → max. 4X1 + 11X2 + 3X3 7, rI X2-X3=0, X1+ 13, .y20, j = 1.2.3.
how to calculate cov(x1,x2), cov(x2,x3),cov(x3,x1)? and how to calculate var(x1),var(x2),var(x3)? Given three random variables Xi, X2, and X such that X[Xi X2 X 20 -1 E [X] ,1-10 | and var(X)=Σ-| 0 3 0. 1 0.5 1 compuite: 2
1. Consider a set of vectors S = {X1, X2, X3 } in which X1 = (1,0,0), X2 = (a, 1, -a), X3 = (1, 2, 3a +1) Determine the value (or values) of a for which the set Sabove is linearly independent (LI). 2. Consider a set of vectors T = {y1, y2.ya} in which yı = (1,2,0), y2 = (1, m,5), and y3 = (0,4, n) Determine a condition on m and n such that the set T...