Recent content by IniquiTrance

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    Graduate How about Why does the Landau notation for division work when h approaches 0?

    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)##.
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    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##.
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    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}
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    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 EquationsThe Attempt at a Solution I'm trying to show that, \mathbb{E}[X+Y\vert 1_{\{X+Y=0\}}] = 0 by, \begin{align*}...
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    Graduate Why is the derivative of |x| not equal to sgn(x) + 2xδ(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...
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    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...
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    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.
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    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...
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    Proving a process is Brownian Motion

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

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

    Hi Ray, yes they are.
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    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 motionsHomework Equations The Attempt at a Solution Obviously X(0)=0. Now let 0\leq...
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    Graduate How Is the Distribution of B_s Given B_t Computed in 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)?
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    Proving \{0\} is an Element of Sigma Algebra Generated by A_n on \mathbb{R}

    Thank you. I understand it much better now.