4. Suppose Σί, 1 Xi-1, ΣΙ-, x-2, and n-5. Evaluate the followings: a) Σί-1[Xi + X) b) 5. Let X-1 Σ., x1-1 with n-100. (i) Obtain ΣΙ.i Xu (ii) Evaluate Σ.1x,-X) and X +1
1. Suppose E[X]-2 and E Y]-1. Evaluate the followings: a) E3X ] b) EX 2Y] c)
Question 3 By hand get, for x(n) = {1, 2, 3, 4, 5), the followings: A. Circular Folding: x(-n) B. Circular shift x(n-2) C. Circular convolution for N=6 of x(n) and x(n-2) Then get the same result using Matlab for proof.
7. Suppose X ~N(3, 22) (1) Evaluate P (2 Xs5), P-4<Xs10), P>2) (2) Decide C so that P (X> C) P (sc) Suppose the density function of X is 04 x)8 0, else Find the density function of Y-2X+8.
Book: A Course in Enumeration. Author: Martin Aigner
Chapter 1 Page:9
1.10 Evaluate ΣΙ.1 12 and Σ.1 13 by counting configurations of dots as in the proof of Σ-is n(n+1)
1.10 Evaluate ΣΙ.1 12 and Σ.1 13 by counting configurations of dots as in the proof of Σ-is n(n+1)
10) Use Theorem 4 to evaluate. n(n+1) 2 (=1 i = 10 pts 21-11 = n) 2,12 = n(n+1)(2n+1) E=1 43 = {*+1), 2 4 Theorem Iff is integrable on (a, b), then (x) dx = lim 8(xAx where and X; - a +i Ax (2x + 3 (2x + 5)dx
5) Suppose 2, x are an eigenpair for an n by n nonsingular matrix A. a) Show that 1k , x are an eigenpair for Ak. (10 points) b) Show that 2-1 , x are an eigenpair for A-?. (10 points)
c. Evaluate the following determinants: (5 pts) 2 x 2 1. 1 -4 91 | 5 - 121 3x3 (5 pts) 2. 1-4 2 3 14 -5 41= 1-12 8 91
n=0 (5 points each) Suppose the power series an(x - 1)" converges for 1 = 4 and diverges for x = -4. Answer each question below as yes, no, or can not be determined. (a) Does the power series converge for x = -1? (b) Does the power series converge for x = -2? (c) Does the power series converge for x = 5? (d) Does the power series converge for x = 7?
4. For: 1 + x3 dx a) Evaluate I using the trapezoidal rule with n= 4. (15 pts) b) Evaluate I using the 1/3 Simpson's rule with n=2. (10 pts) Trapezoidal Rule Single Application 1 = (6-a) f(b) + f(a) Composite (b-a) 2n I= i=1 Simpson's 1/3 Rule Single Application Composite b) Evaluate I using the 1/3 Simpson's rule with n=2. (10 pts) Trapezoidal Rule Single Application f(b) + f(a) I = (b-a) 2 Composite I = (b − a)...