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10:18 AM
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A: Tag management 2018

Asaf KaragilaAs recently pointed on this thread, we are going to get rid of functions. Today I noticed that we also have partial-functions. I feel that this tag can go just as well as part of the process.

 
 
6 hours later…
4:28 PM
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A: Tag management 2018

Clarinetist2017 Outstanding Tag Management: From 2016 tag management Decision needs to be made on [tag:map-projections]. What to do about graded algebraic structure tags Pluralize [tag:division-ring] Semicontinuity Proposal to rename [tag:generalizedeigenvector] Discussion on [tag:probability], [tag:expec...

 
 
4 hours later…
8:42 PM
1
Q: Prove that there exists an another path $g$ with the same image as $f$ but length of $g = tL \forall t\in [0,1]$ where $L$ is the length of $f$

MathManTheorem: Suppose $(X,d)$ is a metric space and $f:[0,1] \rightarrow X$ is a path in $X$ with no-zero finite length $L$. Then, there exists a path $g:[0,1] \rightarrow X$ from $f(0)$ to $f(1)$ that has the same image as $f$ and satisfies $lth_t(g) = tL ~\forall ~t \in [0,1]$. In particular $g$ is ...

Is sufficiently different from the existing tag?
1
Q: Marginal p.m.f. of two random variables with joint p.m.f. $p(x,y) = 2^{-x-y}$

A. PortTwo discrete random variables $X$ and $Y$, whose values are positive integers, have the joint probability mass function: $$p(x,y) = 2^{-x-y}$$ I need to determine the marginal probability mass functions, which I believe to be defined as $p(x) = \sum p(x,y)$ for $y$ and $p(y) = \sum p(x,y)$ for ...

1
Q: Is there any situation where the joint probability is equal to the marginal probability?

user1936314Is there any situation in that joint probability $p(x,y)$ equals to marginal probability $p(x)$? What is the interpretation of this situation?

2
Q: Marginal PMF of $X$ given joint PMF of $(X,Y)$ is $k \cdot \frac {2^{x+y}}{x!y!}$

user111630Let $X$ and $Y$ be random variables with joint probability mass function $f(x,y) = k \cdot \dfrac {2^{x+y}}{x!y!} $, for $ x, y \in \{ 0, 1, 2, \cdots \} $ and for a positive constant $k$. How can I derive the marginal probability mass function of $X$? How do I evaluate $k$? Are $X$ and $Y$ ind...

1
Q: Joint distribution probability

Ayman Hi , I tried to resolve this question by taking the marginal pdf of $Y$ , then find it's expected value . But apparently this is not correct, they used the pdf of $f(x,y)$ in the solution and performed double integral to find $E(Y)$, can someone explain me why? Thanks

1
Q: Evaluating the lambda expression $(𝜆𝑥𝑦.𝑥𝑥𝑦)(𝜆𝑥.𝑥𝑦)(𝜆𝑥.𝑥𝑧)$

onurcanbektasIn the book of Haskell Programming by C. Allen, at page 39, it is given the following lambda expression $$(πœ†π‘₯𝑦.π‘₯π‘₯𝑦)(πœ†π‘₯.π‘₯𝑦)(πœ†π‘₯.π‘₯𝑧)$$ According to me, this equals to by applying the left two expression as an input for the rightmost expression $$(πœ†π‘₯.π‘₯𝑦)(πœ†π‘₯.π‘₯𝑦)(πœ†π‘₯.π‘₯𝑧)...

The tag also has tag-excerpt: "For questions involving the notion of the Radon-Nikodym derivative or the Radon-Nikodym theorem. Use this tag along with (probability-theory) or (measure-theory)."
1
Q: Radon-Nikodym Derivatives between Ito Processes

user225160I am curious about the following problem: Let $B_t$ be a standard Brownian motion on $(\Omega, \mathcal F, \mathcal F_t, \mathbb P_a)$, where the filtration is generated by $B_t$. On a finite interval $[0,T]$ we define $X_t$ as the one solving the SED $$\mathrm dX_t=\mu_a\,\mathrm dt+\sigma\,\,\...

7
Q: Given $\mathbb Q$ and $X_t$ is $\mathbb Q$-Brownian, find $\frac{d\mathbb Q}{d\mathbb P}$ / Uniqueness of Brownian or Radon-Nikodym derivative

BCLCThe problem: Let $T >0$, and let $(\Omega, \mathscr F, \{ \mathscr F_t \}_{t \in [0,T]}, \mathbb P)$ be a filtered probability space where $\mathscr F_t = \mathscr F_t^W$ where $W = \{W_t\}_{t \in [0,T]}$ is standard $\mathbb P$-Brownian motion. Let $X = \{X_t\}_{t \in [0,T]}$ be a stoch...

2
Q: Computing expectation under a change of measure

TA WongLet $X$ be a random variable on a probability space $(\Omega,\mathscr F, P)$. Define a new probability measure $$\tilde P(A) = E[1_A X]$$ for all $A\in\mathscr F$. Let $\tilde E$ be expectation taken with respect to the new measure $\tilde{P}$. Suppose now that $Y$ is also a random variable $(\...

0
Q: Prove $E^{\mathbb Q}[Y]=E^{\mathbb P}[XY]$ if $E^{\mathbb P}[X]=\mathbb P(X>0)=1$ and $ \mathbb Q(A)=E^{\mathbb Q}[X1_A] $

NatalioHow do I prove the following? I don't know where to start. If $X$ is a random variable with $E^{\mathbb P}[X] = \mathbb P(X>0)=1$ and $ \mathbb Q$ is the probability measure defined by $ \mathbb Q(A)=E^{\mathbb Q}[X1_A] $ then $E^{\mathbb Q}[Y]=E^{\mathbb P}[XY]$

2
Q: Show that the Radon-Nikodym density is $\mathcal{G}$ measurable iff $E_P[X|G] = E_Q[X|G]$

user428487Let $\alpha$ and $\beta$ be equivalent probability measures on $(\Omega, \mathcal{F})$, with Radon-Nikodym density of $\alpha$ wrt $\beta$ is $\eta$, i.e., for all $A \in \mathcal{F}, \beta(A) = \int_A\eta d\alpha$. Let $\mathcal{G}$ be a sub-$\sigma$-field of $\mathcal{F}$. Show that $\eta$ i...

1
Q: ergodic measure and absolutely continuous measure

Eduardo Alejandro Silva Múlleri've come across this problem in Petersen's "Ergodic Theory": Let $(X,\mathcal{B},T,\mu)$ be an ergodic dynamical system. Let $\nu\ll\mu$ be a measure un $(X,\mathcal{B})$ such that $\nu T^{-1}\ll\nu$. Show that $\nu=\nu T^{-1}$ and that $\nu$ is a constant multiple of $\mu$. I've tried solving...

1
Q: What is the Radon-Nikodym density $dP^βˆ— /d P$ of the unique $P^ βˆ— ∼ P$ such that the discounted price $S^*_t := S_t /B_t$ is a $P^βˆ—$-martingale

Moe21Consider the Black-Scholes Model where we have the following risky asset $dS_t = \mu S_t dt + \sigma S_t dW_t, t\in[0,T] tβ‰₯0 ,S_0 = s >0 $ where $\mu,\sigma$ are positive constants and a risk-free asset $dB_t = rB_tdt , B_0 = 1 $ $(W_t)_{tβ‰₯0}$ denotes a standard brownian motion with its ...

1
Q: Is there any situation where the joint probability is equal to the marginal probability?

user1936314Is there any situation in that joint probability $p(x,y)$ equals to marginal probability $p(x)$? What is the interpretation of this situation?

1
Q: Joint distribution probability

Ayman Hi , I tried to resolve this question by taking the marginal pdf of $Y$ , then find it's expected value . But apparently this is not correct, they used the pdf of $f(x,y)$ in the solution and performed double integral to find $E(Y)$, can someone explain me why? Thanks

3
Q: Forgetful functor from complete metric space to metric space

Brian CheungI am reading Mac Lane's Categories for the Working Mathematician. He mentioned that the usual completion of metric space is universal for the evident forgetful functor (from complete metric spaces to metric spaces). (p.57) I am not sure what is this forgetful functor 'forgetting'. I think this...

2
Q: A formal definition of forgetful functor

user107952I have read in many category theory textbooks the term "forgetful functor". But no has ever given me a precise definition of this term. I want a rigorous definition, not merely an answer that basically says, "I know it when I see it.". Has someone developed a perfectly rigorous definition of forg...

1
Q: Why $K$ is solid if its reproducing?

gloom Let $K$ be a closed convex cone. Then $K$ is solid if and only if $K$ is reproducing. Hint: If $K$ is a convex cone then $K-K$ has a nonempty interior. $K-K$ is the minimal subspace containing $K$. Definitions: A set $K$ in a Euclidean space $V$ is a convex cone if for every $x, y \i...

2
Q: Doubts about metrization theorems

FenrirI studying metrization and in different parts I encountered different formulations of theorems, for example in the Nagata–Smirnov metrization theorem I found: A topological space $X$ is metrizable if and only if it is $T_3$ and Hausdorff and has a $\sigma$-locally finite base, and other: ...

0
Q: Studying Muller's example of IVP with unique solution whose Picard iterates do not converge.

JavierI have read about the following example from Muller: $(M) \begin{cases} x' = f(t,x) \\[1mm] x(0) = 0 \end{cases}$ where $f: \mathbb{R}\times\mathbb{R} \rightarrow \mathbb{R}$ is the function: $f(t,x) = \begin{cases} 0 & t \leq 0, x \in \mathbb{R} \\ 2t & t>0,x < 0 \\ 2t - \frac{4x}{t} & t >...

 

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