# Search results for query: *

1. ### Landau notation division

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)##.
2. ### Conditional expectation on an indicator

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##.
3. ### Conditional expectation on an indicator

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}
4. ### Conditional expectation on an indicator

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*}...
5. ### Exploring Derivatives of |x| and \sgn(x)

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...
6. ### Convergence of iterative method and spectral radius

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...
7. ### Convergence of iterative method and spectral radius

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.
8. ### Convergence of iterative method and spectral radius

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...
9. ### Proving a process is Brownian Motion

Thanks. How can I go about proving that?
10. ### Proving a process is Brownian Motion

Are you implying that if A\perp B\perp C \perp D, then A+B \perp C+D, where \perp means independent?
11. ### Proving a process is Brownian Motion

Not sure how to proceed.
12. ### Proving a process is Brownian Motion

Hi Ray, yes they are.
13. ### Proving a process is Brownian Motion

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...
14. ### Conditional Brownian motion

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)?
15. ### Sigma Algebra elements

Thank you. I understand it much better now.
16. ### Sigma Algebra elements

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
17. ### Sigma Algebra elements

Don't you need Cantor's intersection theorem for that?
18. ### Sigma Algebra elements

Hmm, I know we need countable additivity. Other than that, not quite sure how to proceed.
19. ### Sigma Algebra elements

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...
20. ### Non-singularity of A^T*A

Thank you. I suspected it was true, but couldn't prove it to myself.
21. ### Non-singularity of A^T*A

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.
23. ### Proof of non-singularity of triple matrix product

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...
24. ### Symmetric positive definite matrix

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 =...
25. ### Eigenvalues of the power of a matrix

Ah ok, thanks again!
26. ### Eigenvalues of the power of a matrix

But since A is symmetric, it must have real eigenvalues, no?
27. ### Eigenvalues of the power of a matrix

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.
29. ### Eigenvalues of the power of a matrix

Oh so all we can say is that either \sqrt{\lambda} or -\sqrt{\lambda} is an eigenvalue of A, but not both?
30. ### Eigenvalues of the power of a matrix

Then: det(A-\sqrt{\lambda}I)*det(A+\sqrt{\lambda} I) = 0 Not sure how to proceed...
31. ### Eigenvalues of the power of a matrix

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...
32. ### Prove AB and BA have the same characteristic polynomial

Hmm, somewhat. Could you please provide greater insight?
33. ### Prove AB and BA have the same characteristic polynomial

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...
34. ### Double summation

Thank you very much. I knew I wasn't understanding something.
35. ### Double summation

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.
38. ### Uniform Distribution on unit Circle

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...
39. ### Jacobian Transformations

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 =...
40. ### Probability - Uniform distribution word problem.

Thank you very much. I understand the concept of mixed distributions much better now.
41. ### Poisson distribution help

I think you can solve this using Bayes' formula.
42. ### Probability - Uniform distribution word problem.

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...
43. ### Jacobian Transformation - new domain of integration

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...
44. ### Infinitely differentiable function

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...
45. ### Infinitely differentiable function

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) =...
46. ### Probability: Conditional expectation

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...
47. ### Prove Inequality: ||x|^α - |y|^α| ≤ |x-y|^α

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^+}...
48. ### Prove Inequality: ||x|^α - |y|^α| ≤ |x-y|^α

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.
49. ### Prove Inequality: ||x|^α - |y|^α| ≤ |x-y|^α

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...
50. ### Prove Inequality: ||x|^α - |y|^α| ≤ |x-y|^α

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