Theorem: 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 ...

Two 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 ...

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

Where does the term "marginal" in "marginal probability" or "marginal distribution" come from?

Let $X$ 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...

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

I 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\,\,\...

The 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...

Let $X$ 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 $(\...

How 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 $E^{\mathbb Q}[Y]=E^{\mathbb P}[XY]$

Let $\alpha$ 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...

i'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...

Consider 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] t≥0 ,S_0 = s >0 $ where $\mu,\sigma$ are positive constants and a risk-free asset $dB_t = rB_tdt , B_0 = 1 $ $(W_t)_{t≥0}$ denotes a standard brownian motion with its ...

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

Where does the term "marginal" in "marginal probability" or "marginal distribution" come from?

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

I 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...

I 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...

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...

I 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: ...

I 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 >...