Search results for query: *

Why is it that, ## \frac{a+\mathcal{O}(h^2)}{b+\mathcal{O}(h^2)} = \frac{a}{b}+\mathcal{O}(h^2) ## as ##h\rightarrow 0##? It seems like the ##\mathcal{O}(h^2)## term should become ##\mathcal{O}(1)##.

I think it is possible since both ##X## and ##Y## are Bernoulli, if their sum is 0, then ##\mathbb{P}[X=0|X+Y=0]=1##. Then, ##\sum_{k=0}^1kf(k)=0*1##.

It's the indicator of the event \{X+Y=0\}. 1_{\{X+Y=0\}}=\begin{cases}1, \qquad \text{if }X+Y=0; \\ 0, \qquad \text{otherwise.}\end{cases}

Homework Statement Let X and Y be independent Bernoulli RV's with parameter p. Find, \mathbb{E}[X\vert 1_{\{X+Y=0\}}] and \mathbb{E}[Y\vert 1_{\{X+Y=0\}}] Homework Equations The Attempt at a Solution I'm trying to show that, \mathbb{E}[X+Y\vert 1_{\{X+Y=0\}}] = 0 by, \begin{align*}...

The wikipedia article on \sgn (x) (http://en.wikipedia.org/wiki/Sign_function) states that, \frac{d}{dx}\vert x\vert = \sgn(x) and \frac{d}{dx}\sgn(x) = 2\delta(x). I'm wondering why the following is not true: \begin{align*} \vert x\vert &= x\sgn(x)\\ \Longrightarrow \frac{d}{dx}\vert x...

Yep, thank you for noticing my error, I meant to say, \mathbf{x}_{n+1} = R\mathbf{x}_n +\mathbf{c} I'm just still unclear why I am allowed to assume \mathbf{x}_0 is a scalar multiple of the eigenvector corresponding to the spectral radius. Doesn't the question read, "If I am provided with some...

Right, that sum diverges, but how do I show that \Vert \mathbf{x}_n\Vert diverges as n\rightarrow\infty? I can only show the norm is not greater than \Vert R^n\mathbf{x}_0\Vert + \infty with the triangle inequality.

Homework Statement Show that if given \mathbf{x}_0, and a matrix R with spectral radius \rho(R)\geq 1, there exist iterations of the form, \mathbf{x}_{n+1}=R\mathbf{x}_0+\mathbf{c} which do not converge. The Attempt at a Solution Let \mathbf{x}_0 be given, and let...

Thanks. How can I go about proving that?

Are you implying that if A\perp B\perp C \perp D, then A+B \perp C+D, where \perp means independent?

Not sure how to proceed.

Hi Ray, yes they are.

Homework Statement Is the process \{X(t)\}_{t\geq 0}, where X(t)=\rho B_1(t)+\sqrt{1-\rho^2}B_2(t) Standard Brownian Motion? Where \rho\in(0,1), \ B_1(t) and B_2(t) are independent standard brownian motions Homework Equations The Attempt at a Solution Obviously X(0)=0. Now let 0\leq...

I computed the distribution of B_s given B_t, where 0\leq s <t and \left\{B_t\right\}_{t\geq 0} is a standard brownian motion. It's normal obviously.. My question is, how do I phrase what I've done exactly? Is it that I computed the distribution of B_s over \sigma(B_t)?

Thank you. I understand it much better now.

Hmm, I'm just trying to make it rigorous... Why do we know that that intersection contains only {0}, when it could conceivably be, * Empty * Non-empty with an infinite amount of points

Don't you need Cantor's intersection theorem for that?

Hmm, I know we need countable additivity. Other than that, not quite sure how to proceed.

Homework Statement Suppose \Omega = \mathbb{R}. Let A_n = [0,\frac{1}{n}]. Let \mathcal{A}=\{A_n:n\in\mathbb{N}\}. Is \{0\} an element of \mathcal{F}=\sigma(\mathcal{A})? Homework Equations The Attempt at a Solution Clearly \lim_{n\rightarrow\infty} A_n = \{0\}. Not sure how to show that...

Thank you. I suspected it was true, but couldn't prove it to myself.

Is it true to say that if X^T X is non-singular, then the column vectors of X must be linearly independent? I know how to prove that if the columns of X are linearly independent, then X^T X is non-singular. Just not sure about the other way around. Thanks!

Thanks.

Homework Statement Suppose A\in \mathbb{R}^{n\times n} is symmetric positive definite, and therefore non-singular. Let M\in\mathbb{R}^{m\times n}. Show that the matrix M^T A M is non-singular if and only if the columns of M are linearly independent. Homework Equations The Attempt...

Homework Statement (i) Show that if A is symmetric positive semi-definite, then there exists a symmetric matrix B such that A=B^2. (ii) Let A be symmetric positive definite. Find a matrix B such that A=B^2. Homework Equations The Attempt at a Solution For part 1, I used: B =...

Ah ok, thanks again!

But since A is symmetric, it must have real eigenvalues, no?

One more question, how do we know that all the eigenvalues of A^2 are positive? Since A is symmetric, if \lambda < 0 were an eigenevalue of A^2 we'd run into a problem...

Thank you.

Oh so all we can say is that either \sqrt{\lambda} or -\sqrt{\lambda} is an eigenvalue of A, but not both?

Then: det(A-\sqrt{\lambda}I)*det(A+\sqrt{\lambda} I) = 0 Not sure how to proceed...

Homework Statement If \lambda_i are the eigenvalues of a matrix A^2, and A is symmetric, then what can you say about the eigenvalues of A? Homework Equations The Attempt at a Solution I know how to prove that if \sqrt(\lambda_i) is an eigenvalue of A, then \lambda_i is an eigenvalue of...

Hmm, somewhat. Could you please provide greater insight?

Homework Statement Show that for any square matrices of the same size, A, B, that AB and BA have the same characteristic polynomial. Homework Equations The Attempt at a Solution I understand how to do this if either A or B is invertible, since they would be similar then. I saw a...

Thank you very much. I knew I wasn't understanding something.

If I know that \sum_{k=1}^n a_{ik} = 1 and \sum_{j=1}^n b_{kj} = 1, why is the following permitted? \sum_{j=1}^n \sum_{k=1}^n a_{ik}b_{kj} = \left(\sum_{j=1}^n b_{kj}\right) \left(\sum_{k=1}^n a_{ik}\right) = 1\cdot 1 = 1 Thanks!

Thank you.

Thank you.

I keep reading that a random vector (X, Y) uniformly distributed over the unit circle has probability density \frac{1}{\pi}. The only proof I've seen is that f_{X,Y}(x,y) = \begin{cases} c, &\text{if }x^2 + y^2 \leq 1 \\ 0 &\text{otherwise}\end{cases} And then you solve for c by integrating...

Why is it that if you have: U=g_1 (x, y), \quad V = g_2 (x,y) X = h_1 (u,v), \quad Y = h_2 (u,v) Then: f_{U,V} (u,v) du dv = f_{X,Y} (h_1(u,v), h_2 (u,v)) \left|J(h_1(u,v),h_2(u,v))\right|^{-1} dxdy While when doing variable transformations in calculus, you have: du dv =...

Thank you very much. I understand the concept of mixed distributions much better now.

I think you can solve this using Bayes' formula.

Homework Statement Jake leaves home at a random time between 7:30 and 7:55 a.m. (assume the uniform distribution) and walks to his office. The walk takes 10 minutes. Let T be the amount of time spends in his office between 7:40 and 8:00 a.m.. Find the distribution function F_T of T and draw...

Homework Statement I need to compute the following using the Jacobian: \int\int_D \frac{x-y}{x+y} dxdy Where D = \left\{(x,y):x\geq 0, y\geq 0, x+y \leq 1\right\} Homework Equations The Attempt at a Solution I've made the transformation: s=x+y \qquad t = x-y My problem is finding...

Thanks for the reply chiro. I was just mainly wondering whether all order derivatives were 0, because we were taking the derivative of 0 as a constant which is uniformly 0, or instead because we are using the exponential and taking a limit. I'm assuming from your response that it is because of...

This might sound like a stupid question. f(x) = \begin{cases} &e^{-\frac{1}{x^2}} &\text{if } x\neq 0 \\ & 0 &\text{if } x = 0 \end{cases} Is the reason f is infinitely differentiable at 0 because we keep differentiating 0 as a constant, or because, \lim_{x\rightarrow 0} f`(x) =...

Homework Statement What is the expected number of flips of a biased coin with probability of heads 'p', until two consecutive flips are heads? Homework Equations The Attempt at a Solution Let T_1 = first flip is tails, H_1 = first flip is heads. and T_2, H_2 for second flip...

Ok so a friend of mine proved it as follows: Assume for the sake of contradiction that x\neq y and: ||x|^\alpha - |y|^\alpha |>|x-y|^\alpha \Leftrightarrow ||x|^\alpha - |y|^\alpha |^{1/ \alpha} >|x-y| Taking limits as alpha -> 0 from above: 0=\lim_{\alpha \rightarrow 0^+}...

Edit: Woops. Derivatives would actually be \ln[a+b](a+b)^\alpha and \ln[a]a^\alpha, \ln[b]b^\alpha, which as you say would not be of much help.

Ah ok, would it be something along the lines of this, then? We wish to show that (|x-y|+|y|)^\alpha \leq |x-y|^\alpha +|y|^\alpha \quad \alpha\in (0,1]. This is equivalent to showing (a+b)^\alpha \leq a^\alpha + b^\alpha \quad a,b >0, \alpha\in (0,1]. We want to show now that the...

Yes, that was rather careless on my part, not your fault at all. I'm not sure what the next step would be, though.