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 EquationsThe 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 motionsHomework 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)?