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1. Weighted Moving Average of Cubic

1. Show that applying a second-order weighted moving average to a cubic polynomial will not change anything. X_t = a_0 + a_1t + a_2t^2 + a_3t^3 is our polynomial Second-order weighted moving average: \sum_{i=-L}^{i=L} B_iX_{t+i} where B_i=(1-i^2I_2/I_4)/(2L+1-I_2^{2}/I_4) where...
2. Estimate p from sample of two Binomially Distributions

1. Suppose X~B(5,p) and Y~(7,p) independent of X. Sampling once from each population gives x=3,y=5. What is the best (minimum-variance unbiased) estimate of p? Homework Equations P(X=x)=\binom{n}{x}p^x(1-p)^{n-x} The Attempt at a Solution My idea is that Maximum Likelihood estimators are...
3. Maximum Likelihood Estimator + Prior

1.Suppose that X~B(1,∏). We sample n times and find n1 ones and n2=n-n1zeros a) What is ML estimator of ∏? b) What is the ML estimator of ∏ given 1/2≤∏≤1? c) What is the probability ∏ is greater than 1/2? d) Find the Bayesian estimator of ∏ under quadratic loss with this prior 2. The attempt...
4. Maximum Likelihood Estimators

Homework Equations L(x,p) = \prod_{i=1}^npdf l= \sum_{i=1}^nlog(pdf) Then solve \frac{dl}{dp}=0 for p (parameter we are seeking to estimate) The Attempt at a Solution I know how to do this when we are given a pdf, but I'm confused how to do this when we have a sample.
5. Find P(X+Y<1) in a different way

1. Let the pdf of X,Y be f(x,y) = x^2 + \frac{xy}{3}, 0<x<1, 0<y<2 Find P(X+Y<1) two ways: a) P(X+Y<1) = P(X<1-Y) b) Let U = X + Y, V=X, and finding the joint distribution of (U,V), then the marginal distribution of U. The Attempt at a Solution a) P(X<1-Y) = ? P(x<1-y) = \int_0^1...
6. Calculating Variances of Functions of Sample Mean

1. Essentially what I'm trying to do is find the asymptotic distributions for a) Y2 b) 1/Y and c) eY where Y = sample mean of a random iid sample of size n. E(X) = u; V(X) = σ2 Homework Equations a) Y^2=Y*Y which converges in probability to u^2, V(Y*Y)=\sigma^4 + 2\sigma^2u^2 So...
7. Conditional Variances

1. Given f(x,y) = 2, 0<x<y<1, show V(Y) = E(V(Y|X)) + V(E(Y|x)) Homework Equations I've found V(Y|X) = \frac{(1-x)^2}{12} and E(Y|X) = \frac{x+1}{2} The Attempt at a Solution So, E(V(Y|X))=E(\frac{(1-x)^2}{12}) = \int_0^y \frac{(1-x)^2}{12}f(x)dx, correct?
8. Covariance - Bernoulli Distribution

1. Consider the random variables X,Y where X~B(1,p) and f(y|x=0) = 1/2 0<y<2 f(y|x=1) = 1 0<y<1 Find cov(x,y) Homework Equations Cov(x,y) = E(XY) - E(X)E(Y) = E[(x-E(x))(y-E(y))] E(XY)=E[XE(Y|X)] The Attempt at a Solution E(X) = p (known since it's Bernoulli, can also...
9. Conditional Expectation

1. Let the joint pdf be f(x,y) = 2 ; 0<x<y<1 ; 0<y<1 Find E(Y|x) and E(X|y) Homework Equations E(Y|x) = \int Y*f(y|x)dy f(y|x) = f(x,y) / f(x) The Attempt at a Solution f(x) = \int 2dy from 0 to y = 2y f(y|x) = f(x,y)/f(x) = 1/2y E(Y|x) = \int Y/2Y dy from x to 1 = \int 1/2 dy from x to 1...
10. Integral + Complex Conjugate

Homework Statement Show that the following = 0: \int_{-\infty}^{+\infty} \! i*(\overline{d/dx(sin(x)du/dx})*u \, \mathrm{d} x + \int_{-\infty}^{+\infty} \! \overline{u}*(d/dx(sin(x)du/dx) \, \mathrm{d} x where \overline{u} = complex conjugate of u and * is the dot product. 2. Work so far...
11. Does diagonalizable imply symmetric?

In order to prove my PDE system is well-posed, I need to show that if a matrix is diagonalizable and has only real eigenvalues, then it's symmetric. Homework Equations I've found theorems that relate orthogonally diagonalizable and symmetric matrices, but is that sufficient? The...

Homework Statement Let A be a real nxn matrix with non-negative elements satisfying \sum_{j=0}^n a_{ij}=1. Determine the spectral radius of A. Homework Equations Denote spectral radius \varsigma(A)=max(\lambda_{i}) We know \varsigma(A) \leq ||A|| for any norm || || 3. Attempt at the solution...
13. Symmetric, irreducible, tridiagonal matrix: Eigenvalues

Homework Statement A) Let A be a symmetric, irreducible, tridiagonal matrix. Show that A cannot have a multiple eigenvalue. B) Let A be an upper Hessenberg matrix with all its subdiagonal elements non-zero. Assume A has a multiple eigenvalue. Show that there can only be one eigenvector...
14. Linear Algebra: Block System

1. See the following picture: http://imageshack.us/photo/my-images/715/math5610.jpg/ Essentially what I'm trying to do is solve a linear block system. I have got to the point where I now need to "add multiples of the top rows to clear out C." Now, I'm sure this is the easy part as I've...
15. Linear Algebra: Symmetric/Positive Definite problem

1. Let A\inRnxn be a symmetric matrix, and assume that there exists a matrix B\inRmxn such that A=BTB. a) Show that A is positive semidefinite B) Show that if B has full rank, then A is positive definite 2. Homework Equations : This is for an operations research class, so most of the...
16. Hermite Polynomial Reccurence Relation

Homework Statement Prove, using Rodrigues form, that Hn+1=2xHn -2nHn-1 Homework Equations The Rodrigues form for Hermite polynomials is the following: Hn = (-1)nex2\frac{d^n}{dx^n}(e-x2) The Attempt at a Solution Hn+1 = (-1)n+1ex2\frac{d^(n+1)}{dx^(n+1)}(e-x2) where...
17. If lcm(a,b) = ab, why is gcd(a,b) = 1?

1. If lcm(a,b)=ab, show that gcd(a,b) = 1 Homework Equations We can't use the fact that lcm(a,b) = ab / gcd(a,b) The Attempt at a Solution I've already shown that gcd(a,b) = 1 → lcm(a,b) = ab but I can't figure out the other direction! Any hints?
18. Fixed Point Iteration Convergence

Homework Statement Consider the system x = \frac{1}{\sqrt{2}} * \sqrt{1+(x+y)^2} - 2/3 y = x = \frac{1}{\sqrt{2}} * \sqrt{1+(x-y)^2} - 2/3 Find a region D in the x,y-plane for which a fixed point iteration xn+1 = \frac{1}{\sqrt{2}} * \sqrt{1+(x_n + y_n)^2} - 2/3 yn+1 =...
19. Proving Quadratic Convergence via Taylor Expansion

Homework Statement The following is a modification of Newton's method: xn+1 = xn - f(xn) / g(xn) where g(xn) = (f(xn + f(xn)) - f(xn)) / f(xn) Homework Equations We are supposed to use the following method: let En = xn + p where p = root → xn = p + En Moreover, f(xn) = f(p + En) = f(p) +...
20. Newton's Method Variation

Homework Statement The most commonly used algorithm for computing \sqrt{a} is the recursion xn+1 = 1/2 (xn + a/xn), easily derived by means of Newton's method. Assume that we have available to us a very simple processor which only supports addition, subtraction, multiplication, and halving (a...
21. Convergence Interval for Newton's Method

1. The problem statement: In what region can we choose x0 and get convergence to the root x = 0 for f(x) = e-1/x^2 Homework Equations xn+1 = xn - f(xn) / f'(xn) The Attempt at a Solution The only thing I've come across is a formula that says |root - initial point| < 1/M where M =...
22. Newton's Method for Root Finding - Infinite Loop

1. Construct a function f (x) so that Newton's method gets 'hanging' in an infinite cycle xn = (-1)n x0 , no matter how the start value x0 is chosen. 2. Homework Equations : xn+1 = xn - f(xn) / f'(xn) The Attempt at a Solution xn+1 = xn - f(xn) / f'(xn) = (-1)n+1x0 = (-1)nx0 -...
23. Fourier Transform (Numerical Analysis)

1. Calculate the finite Fourier transform of order m of the following sequences: a) uk = 1, 0\leqk\leqN-1 b) uk = (-1)k, 0\leqk\leqN-1 N even c) uk = k, 0\leqk\leqN-1 2. Homework Equations Uk = (1/N)\sumuke-2pi*i*k*j/N from j=0 to N-1 ; 0<=k<=N-1 Attempt: a) First thing that I tried is...
24. Solving Dirichlet problem on a Square

1. Problem Statement Solve by inspection the Dirichlet problem, where \Omega is the unit square 0\leqx\leq1, 0\leq y \leq 1, and where the data is: f(x,y) = { x for 0\leqx\leq1, y=0, 1 for x = 1, 0\leq y \leq 1, x for 0\leqx\leq1, y=1, 0 for x=0, 0\leqy\leq1 Homework Equations...
25. Second-Order PDE Help

1. Integrate (by calculus): u''(x) = -4u(x), 0 < x < pi 2. The attempt at a solution I'm not really sure where to start on this one is my problem. I can see that it won't be a e^2x problem because of the negative, which leads me to believe that it will deal with the positive/negative...