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...
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...
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...
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.
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...
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...
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?
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...
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...
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...
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...
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...
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...
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...
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...
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?
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 =...
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) +...
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...
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 =...
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 -...
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...
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...
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...