1 0 < x < 1, 2. Consider the Haar scaling function, p(x):= { +, and...
Also, find Cmn. Haar wavelets) Let 6 be a function defined by 0(x) = 1 for x € (0,1), and əla) = 0 otherwise. Let y(x) = º(2x) - (2x - 1). Then the Haar wavelets are the functions Umn(x) = 2m/2v(2" x - n), for m, n = 0, #1, #2, .... Sketch a graph of y(x), and then sketch a graph of Umn(x) for m, n = 0, +1, +2. Generally, what is the graph of Umn(x)? If...
From Arfken 10.3.4 You are given (a) a set of functions un (x)--x", n = 0, 1, 2, (b) an interval (0, oo), (c) a weighting function w(x)-xe. Use the Gram-Schmidt procedure to construct the first three orthonormal functions from the set un(x) for this interval and this weighting function. 10.3.4 You are given (a) a set of functions un (x)--x", n = 0, 1, 2, (b) an interval (0, oo), (c) a weighting function w(x)-xe. Use the Gram-Schmidt procedure...
1. Consider two random variables X and Y with joint density function f(x, y)-(12xy(1-y) 0<x<1,0<p<1 otherwise 0 Find the probability density function for UXY2. (Choose a suitable dummy transformation V) 2. Suppose X and Y are two continuous random variables with joint density 0<x<I, 0 < y < 1 otherwise (a) Find the joint density of U X2 and V XY. Be sure to determine and sketch the support of (U.V). (b) Find the marginal density of U. (c) Find...
3. Which of the following sets spans P2(R)? (a) {1 + x, 2 + 2x 2} (b) {2, 1 + x + x 2 , 3 + 2x + 2x 2} (c) {1 + x, 1 + x 2 , x + x 2 , 1 + x + x 2} 4. Consider the vector space W = {(a, b) ∈ R 2 | b > 0} with defined by (a, b) ⊕ (c, d) = (ad + bc, bd)...
4. Consider the set S = {(x, y) | x ∈ [0, 1], x2 ≤ y ≤ x}. Prove that S is a Jordan region and integrate the function 2x 2 + 3y 2 on S.
Consider the bivariate function f(x.y) = (x + y)/3 for 0< x< 1 and 0<y< 2 and f(x.y) = 0 3. otherwise. (a) Show that f(xy) is a density function. (b) Find the probability that both X and Y are less than one. (c) Find the marginal densities of X and Y and show that they are not independent. (d) Find the conditional density of X given Y when Y = 0.5.
(i) Find an orthonormal basis {~u1, ~u2} for S (ii) Consider the function f : R3 -> R3 that to each vector ~v assigns the vector of S given by f(~v) = <~u1, ~v>~u1 + <~u2; ~v>~u2. Show that it is a linear function. (iii) What is the matrix of f in the standard basis of R3? (iv) What are the null space and the column space of the matrix that you computed in the previous point? Exercise 1. In...
(1 point) Suppose you're given the following Fourier coefficients for a function on the interval [-π, π : ao = 2, ak = 0 for k 2 i, and for k > 1. Find the following Fourier approximations to the Fourier series a0 + 〉 ,(an cos(nz) + bn sin(nx)) bk = F, (z) = F,(z) = Fs(x) (1 point) Suppose you're given the following Fourier coefficients for a function on the interval [-π, π : ao = 2, ak...
1. Consider the joint probability density function 0<x<y, 0<y<1, fx.x(x, y) = 0, otherwise. (a) Find the marginal probability density function of Y and identify its distribution. (5 marks (b) Find the conditional probability density function of X given Y=y and hence find the mean and variance of X conditional on Y=y. [7 marks] (c) Use iterated expectation to find the expected value of X [5 marks (d) Use E(XY) and var(XY) from (b) above to find the variance of...
Consider a signal x(t) which is given as 1 x(t) - 2 <t<2 2 0, otherwise a) Sketch x(t) b) Sketch 3x(t – 1) c) Sketch – 2x(-t - 1) Identify all labels and amplitudes to get the whole score.