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    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...
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    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...
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    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...
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    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.
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    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...
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    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...
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    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?
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    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...
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    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...
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    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...
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    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...
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    Spectral Radius: Matrix

    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...
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    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...
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    Linear Algebra: Block System

    1. See the following picture: 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...
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    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...
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    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...
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    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?
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    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 =...
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    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) +...
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    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...
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    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 =...
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    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 -...
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    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...
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    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...
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    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...