Let Z! = 3H4, Z2-5-2, Z,--3-12, Z4--10-j6, and Z5--6-3. 1. Calculate Z1 + Z2 in rectangular form. 2. Calculate Z1 - Z2 in rectangular form. 3. Calculate Z3 + Z4 in polar form. 4. Calculate Za - Z5 in polar form. 5. Calculate Z1Z2-Z3 in rectangular form. 6. Find ZsZ7 in polar form. 7. Find Z7Zs in rectangular form. 8. Find ZsZs+Z7 in rectangular form Reduce the following to rectangular form. 10. Z1/Z2
1. Let Z = (Z1, Z2, Z3) be a vector with i.i.d. N(0, 1) components. Let r be a constant with 0 < r < 1. Define X1 = √ rZ1 + √ 1 − rZ2 and X2 = √ rZ1 + √ 1 − rZ3. (a) Give the distribution of X1 and the distribution of X2. Find Cov(X1, X2). (b) Give the matrix A so that the vector X = (X1, X2) is a transform X = AZ. Give...
Note: if z = (z1, z2, z3), then the vectors x = (−z2, z1, 0) and y = (−z3, 0, z1) are both orthogonal to z. Consider the plane P = H4 (1,−1,3) in R 3 . Find vectors w, x, y so that P = w + Span(x, y). Note: if z = (2,22,23), then the vectors x = (-22,21,0) and y = (-23,0,2) are both orthogonal to z. Consider the plane P = H(1,-1,3) in R3. Find vectors...
Let X1, X2, X3 be independent Binomial(3,p) random variables. Define Y1 = X1 + X3 and Y2 = X2 + X3. Define Z1 = 1 if Y1 = 0; and 0 otherwise. Define Z2 = 1 if Y2 = 0; and 0 otherwise. As Z1 and Z3 both contain X3, are Z1 and Z3 independent? What is the marginal PMF of Z1 and Z2 and joint PMF of (Z1, Z2) and what is the correlation coefficient between Z1 and Z2?
2. Let U be a set, and let A CU. Recall the indicator function XA: U → Z2 defined by XA() : S 1, XEA 10, x¢ A. Now, let A, B CU and consider the symmetric difference of A and B defined by A A B = (A – B) U (B – A). (a) Show that A AB CU, and compute Ø A A. (b) Prove that Vx € U, XAAB(x) = x1(x) + XB(x), where addition is...
Ifz-I+),22-1-j, and 3=-2, calculate the magnitude and phase (in radians) of (a) zi (b) z2 (c) z3 (d) z1 +z (e) z z3 (f) z1z2 (g) t22 (h) 을 2. 21 23 21-23 Z1
Exercise 9.2. Let Z~ N(0, 1). Find the variance of Z2 and Z3 N0 1) Find the density of . Is the density bounded?
Problem 2. (Conditional Distribution of MVN) Let Z1, Z2, Z3 be i.i.d. N(0,1) dis- tributed random variables, and set X1 = 21 – Z3 X2 = 2Z1 + Z2 – 223 X3 = -221 +3Z3 1) What distribution does X = (X1, X2, X3)T follow? Specific the parameters. 2) Find out P(X2 > 0|X1 + X3 = 0).
Describe by words and/or pictures, z, z1, z2, such that: 1) |z+5| = 3 2) -3 < Re(z) < 5 3) Arg(z1) = Arg(z2) 4) |z1| = |z2| 5) Im(z1 z*2) = 0 6) |z| = |z*| ** z* = "complex conjugate of z"
3 Cual es magnitud de E debido a Q1-38 nC r1( x1-1, y1-3, z1-3) y Q2-47_nC r1( x2-3, y2-5, z2-1 ) en el origen (0,0,0) A) 45.81 N/C B) 24.98 NIC C) 35.18 NIC D) 51.02 NIC r (x1,y1,z1) Q2 r2 (x2,y2,z2) 0 (0,0,0)