3. (a) Let z1,z2, z3 € C, prove the following identity: (21 - 22)(22 – 23)(23...
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 U ⊂ C4 be the subspace U = {(z1, z2, z3, z4) ∈ C4: z1 + z2 + z3 + z4 = 0 and z1 = iz2}. (a) Find a basis for U. What is dim U? (b) Extend the basis from part (a) to a basis for C4. (c) Find a subspace W ⊂ C4 such that C4 = U ⊕ W. What is dim W?
Suppose z1, z2 and z3 are distinct points in the extended complex plane C ∗ . Show that the unique M¨obius transform taking these points to 1, 0,∞ in order is z → (z, z1, z2, z3), where (z, z1, z2, z3) is the cross ratio
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
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).
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...
9. (a) Let P1(21, 91, zı), P2 (22, 42, z2),P3 (13, 93, 23) be three non-collinear points in R, that is, three points which do not all lie on a straight line. Then the equations of the plane through these three points is: 2 y 21 1 = 0 22 Y2 22 1 2 1 2141 I3 Y3 23 1 Page 1 (b) Find the equation of the plane through P1(1,2,2), P2(1, 2, -1), P3(0,1,2)
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?
4 points) Let Z1,Z2,...,Z1 be 11 independent N(O, 1) variables, and let Provide answers to the following to two decimal places Part a) Evaluate the moment generating function Mz2 (t) of Z2 at the point 0.23 Part b) Evaluate the moment generating function My(t) of Y at the point t = 0.31 . Part c) Find the mean of Y. Part d) Find the variance of Y.