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)##.
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...
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)?
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
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...
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!
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 =...
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...
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...
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...
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!
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 =...
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^+}...
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...