Modern theory of probability is formulated on the footing of measure theory. Use this tag if your question is about this theoretical footing (for example probability spaces, random variables, law of large numbers, central limit theorems, and the like). Use (probability) for explicit computation of ...

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Application of lindeberg/lyapunov condition [on hold]

Suppose independent $X_n$ have density $|x|^{-3}$ outside $(-1, +1)$. I have to show that $\frac{S_n}{\sqrt{nlogn}} =>^d N$. How do I use Lyapunov here ?
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15 views

Prove that a metric is equivalent to levy metric. [on hold]

I have to prove that for distribution functions $F$ and $G$, define $d'(F,G)= \frac{\sup_t |\phi(t) - \theta(t)|}{(1+|t|)}$ where $\phi$ and $\theta$ are the corresponding characteristic functions. I ...
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17 views

Finitely Additive and Countably Additive Property of Probability Function, $\mathbb{P}$.

In Grimmett and Stirzaker's Probability and Random Processes (section 1.3), for two disjoint events $A$ and $B$, we have that $\mathbb{P} (A \cup B) = \mathbb{P}(A) + \mathbb{P}(B)$ From this ...
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1answer
17 views

Does weak convergence of $\nu_{n}$ imply convergence of $\int{f_{n}(x)d\nu_{n}(x)}$?

Suppose that we know that $ \int{ |f_{n}(x) - f(x)| d\mu(x)} \longrightarrow 0 \qquad (1) $ for every probability measure $\mu \in \mathcal{A}$ in a certain class. Also, suppose that $\{\nu_{n}\}$ ...
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1answer
6 views

Understanding PAC (probably approximately correct) bounds on the realizable case (and finite hypothesis class)

I was trying to understand PAC bounds on the realizable case (i.e. when there is some perfect $h^* \in \mathcal{H}$ and its generalization error is zero). Notation: Training data: $$S_n$$ Training ...
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1answer
20 views

What is the definition of sigma field generated by random variable $X$? [on hold]

What is the definition of $\sigma$-field generated by a random variable $X$? And what does it mean?
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1answer
40 views

Kolmogorov equations for death processes

I'm trying to find the forward equations for birth-and-death processes with no birth, that is, when all $\lambda$ coefficients are zero. The forward equation for a birth-and-death process has the ...
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1answer
33 views

Bounding second moment of entropy

Entropy is defined as $E(-\log(P(x))$. We know it is bounded by $\log(r)$ when $r$ is the size of alphabet. Defining the second moment as $E(\log^2(P(x))$, how to show it is bounded?
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18 views

A basic question on convergence in distribution [on hold]

If $Y_n =>^d Y$ and $D_n =>^d 0$, then I have to prove that $D_nY_n => ^d0$. Here all these are random variables. Hint needed.
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1answer
24 views

Average of IID Cauchy RVs

Suppose that $X_i$'s are iid Cauchy RV's with pdf $f_u (x) = \frac{1}{\pi} \frac{u}{u^2+x^2}$. I am aware that the RV $Y:=\frac{1}{N}\sum_{k=1}^N X_k$ has the same density as the $X_i$'s. I am trying ...
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1answer
25 views

Conditioning twice?

I know that $P(X, Y)=P(X|Y)P(Y)$. How can we apply this to $P(X,Y|Z)$? We have already conditioned on $Z$, so can we condition it again on $Y$? Thanks!
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16 views

Confidence Intervals Theory

The problem in the textbook is: Let $0 \le γ \le α$. Then a $100(1 – α)\%$ CI for $μ$ when $n$ is large is $$ \left(\bar{x} – zγ\frac{s}{\sqrt{n}}, \bar{x} + zα-γ \frac{s}{\sqrt{n}}\right) $$ ...
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1answer
28 views

A basic question on uniform distribution [on hold]

I want to know under what condition on random variable $X$, $\{\log_{10}X\}$ is uniformly distributed. Here $\{x\}$ is the fractional part of $x$.
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2answers
44 views

Forth Moment of Sum of Normal with Equal Correlation

I have $X_1,\dots,X_n$ identically normal distributed $N(0,\sigma^2)$ and $\operatorname{corr}(X_i,X_j)=\rho $ for all $i\neq j$. I'd like to compute \begin{equation} E\left(\sum_{i=1}^nX_i\right)^4. ...
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0answers
32 views

Trouble in proving the simplest case of central limit theorem from a convolution viewpoint?

I have once viewed an stanford video, which proves the CLT from a convolution viewpoint rather than using the moment generating function and characteristic function etc. I felt the convolution ...
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0answers
18 views

Prove if $A_1\supset A_2,A_1; A_2\in \Im$ then $\Pr(A_1)>\Pr(A_2)$

Let $(S,\Im,P(\cdot))$ be a probability space where $\Im$ is a field denoting a collection of subsets of $S$. How can I prove that If $A_1\supset A_2, A_1,A_2\in \Im$ then \begin{equation*} ...
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0answers
18 views

Sequences of refining partitions of a measurable space

Let $(\Omega,\mathcal F)$ be a measurable space. For $k\in\mathbb N$ let $(\mathcal F_k)_{k\in\mathbb N}$ be a filtration of $\mathcal F$ such that each $\mathcal F_k$ is generated by a finite ...
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1answer
24 views

Computing joint probability [on hold]

Let $X,Y\sim \text{Exp}(1)$ (exponential random variables with parameter $1$). Then prove that $$Pr(X> z_1, \frac{Y}{X} > z_2) = \dfrac{e^{-z_1 (1+z_2)}}{1+z_2}, \forall z_1,z_2>0$$
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1answer
41 views

How can I prove that $\: \operatorname{Pr}[Y=0] \leq (\operatorname{E}[Y^2] - (\operatorname{E}[Y])^2)/\operatorname{E}[Y^2] \:$?

How can I prove that $\: \operatorname{Pr}[Y=0] \leq (\operatorname{E}[Y^2] - (\operatorname{E}[Y])^2)/\operatorname{E}[Y^2] \:$? I know, $\: \operatorname{Pr}[Y=0] \leq (\operatorname{E}[Y^2] - ...
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1answer
20 views

Application of Ito's Lemma to integral expression

I have a problem applying Ito's lemma. I know that if: $dX_t= \mu_t \, dt + \sigma_t \, dB_t$ then for $f(t,x)$: $df(t,X_t) =\left(\frac{\partial f}{\partial t} + \mu_t \frac{\partial ...
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2answers
25 views

If $X$ is a continuous random variable uniformly distributed over $[a,b]$, then is $Y=2-4X$ uniformly distributed over $[c,d]$? Why?

I ran into this problem solving one of the problems on my course and if I knew that this applies and how to simply prove it, it would help me a great lot.
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0answers
15 views

Heavy tailed sum of iid light tailed random variables

I know that to get one, the number of summands has to be random with a heavy-tailed count variable. I am wondering how you prove the resulting sum is heavy-tailed and in particular wondering if there ...
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1answer
29 views

Clique factorization

I'm reading about Clique factorization in wikipedia: http://en.wikipedia.org/wiki/Gibbs_random_field#Clique_factorization but I'm unable to understand this: What is $X_C$ here? Ok I understood ...
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0answers
20 views

A bssic doubt regarding conditional probability

In Breiman, it is written that the following expression $$\lim_{h->0}\frac{P(A,X \in (x-h,x+h))}{P(X \in (x-h,x+h))}$$ intuitively describes the derivative of one measure w.r.t the other. How ? ...
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0answers
9 views

Factorizing about an undirected graph [on hold]

When do we say that a distribution factorizes about an undirected graph $G$ with maximal cliques $C$?
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1answer
35 views

Hammersley–Clifford theorem

I'm reading this paper http://image.diku.dk/igel/paper/AItRBM-proof.pdf and I got stuck in page 4 with equation (1) that's based on Hammersley–Clifford theorem. I'm not good in reading set theory ...
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1answer
24 views

How to Solve this Poisson Process? [on hold]

If TNT is receiving alot of mail request at a rate of 15 arrivals each second. Upon arrivals each request is found to be :that : ...
1
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2answers
16 views

Question on Doob's martingale convergence theorem

Let $(\Omega,\mathcal F,\mathbb P)$ be a probability space and $(\mathcal F_k)_{k\in\mathbb N}$ a filtration of $\mathcal F$ such that $\mathcal F=\sigma(\mathcal F_k\mid k\in\mathbb N).$ Let ...
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29 views

$E[X]< (\sum_{n=0}^\infty P[X>n]< E[X]+1$

If X takes only non-negative integer values then I figured out $$E[X]= (\sum_{n=0}^\infty P[X>n]$$ but I'm having hard time proving $$ E[X]⩽ (\sum_{n=0}^\infty P[X>n] ⩽ E[X]+1$$ for any ...
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0answers
12 views

probability of ranked paired events of games, like football matches and the league

I would like to improve my method of getting a prediction table for a league based on paired game events. The group of teams that are playing against each other build a set of matches each having a ...
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30 views

Show that $Y_i$ is independent of $Y_j$ for any $i$ not equal to $j$

Let $\{X_1,X_2,\ldots\}$ be independent, identically distributed, absolutely continuous random variables. Let $Y_n=I\{X_n>\max(1< i < n)\}$ for $n=2,3,\ldots$ a) Show that $Y_i$ is ...
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43 views

A basic question on density function [on hold]

Suppose $f$ is joint density function of random variable $X$ and $Y$. I have to prove that consider $G \in \sigma(X)$. Is the following true ? $$ \int_{\Bbb R} y (\int_{X(G)} f(x,y) dx )dy = ...
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0answers
68 views

Proof that a function is measurable

Suppose $f$ is a joint probability density function of random variables $X$ and $Y$. $Y$ is integrable. I need to prove that the function $g(x) = \int_{\Bbb R} f(x,y)ydy$ is measurable function. I ...
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0answers
21 views

Markov Chain Ergodic Theorem (Proof references)

Where can I find a proof of the erogidc theorem for Markov chains that doesn't use Birkhoff? The theorem states the following : Let $(X_n)_{n\in \mathbb{N}}$ be an irreducible and positively ...
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30 views

A basic question on dostribution of longitude and latitude

Let $\Theta$ and $\Phi$ be the longitude and latitude of a random point on the surface of the unit sphere in $\Bbb R^3$. I have to prove that $\Theta$ and $\Phi$ are independent, $\Theta$ is uniformly ...
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0answers
13 views

$X_n, n> 0$ is a Markov Chain, how to interpret $Z_n = (X_n,X_{n+1}), n > 0$?

Am a newbie to Markov Chain. So, this might be incredibly naive/stupid question. If $X_n, \, n > 0$ is MC, am having difficulty imagining/interpreting process $Z_n = (X_n,X_{n+1}), n > 0$. I ...
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1answer
17 views

finding conditional expectation under binomial distribution.

Suppose X and Y independent and are both binomial random variables with parameter N, p Compute E(X|X+Y).
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43 views

weak convergence and composition

Assume $X_n$ is a sequence of random variables defined on a common probability space and $X_n$ converges weakly (in distribution) to $X$ as $n \to \infty$. Assume $u_n$ is a sequence of integer valued ...
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0answers
40 views

Is the martingale propertey preserved by taking weak$^*$-limits?

Let $(\Omega,\mathcal F)$ be a measurable space, $X:\Omega\rightarrow\mathbb R^d$ a $\mathcal F$-measurable map. Let $(\mathcal F_k)_{k\in\mathbb N}$ be a filtration of $\mathcal F$ such that ...
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1answer
26 views

proof of property of exponential distribution, using taylor polynomial

I want to prove that if we have an exponential distribution with parameter $\lambda$, we have that $P(X \le x)=\lambda x + o(x)$. I want to do this by using Taylor-series and the lagrange remainder ...
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1answer
45 views

Signed finite Radon measures with vague topology

If $X$ is a locally compact and $\sigma$-compact metric space. Let $M(X)$ be the space of signed finite Radon measures on $X$. (1) Show that measures with finite support is dense is $M(X)$ in the ...
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7 views

Singular distributions: Applications and Instances

This is the duplication of the question I asked here. I repeat it here with hope of getting new answers. Singular distributions are special mathematical objects. They have an interesting property ...
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36 views

A basic question on sequence of random variables

Suppose for a sequence of mean zero finite variance random variables $X_n$ the following is true for all $\epsilon \gt 0$ : $$ \lim_{n->\infty} \frac{\int_{|x| \geq \epsilon \sqrt{r_n} \sigma_n} x ...
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0answers
18 views

Branching processes, extinction probability

Why do we assume that a branching process starts with one ancestor. What happens to extinction probability if we have more than one ancestor in generation Y(0)?
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14 views

Show $W_{\infty}$ is not tail measurable but {$W_{\infty}=0$} is

The random walk: ({$\omega_{n}$},$\mathbb{P}$) simple random walk on d-dimensional integer lattice $\mathbb{Z}^{d}$ and the random environment: $\eta$={$\eta(n,x):n\in\mathbb{N}, x\in \mathbb{Z}^{d}$} ...
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0answers
49 views

Bounds for sum of random variables

Let $A_1,...,A_M$ be random variables, not necessarily independent. For each one of them I know that $P( A_i \geq a )\leq B_i, \quad i=1,2,...,M$. How can I retrieve lower/upper bounds for ...
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0answers
23 views

probability of covariance proof [duplicate]

I started with covariance formula and then I got stuck. I don't know how to do further steps Let $X$ denote a random variable and $Y = g(X)$ where $g$ is increasing (on $\operatorname{Ran}(X)$). Show ...
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1answer
20 views

Joint distribution and Integration

I was trying to prove a problem in my notes and now I need to whether prove or disprove the following claim: Assume $X,Y,W,Z$ are random variables defined on $(\Omega,\mathcal{F},P)$. If $(X,Y)$ and ...
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1answer
16 views

Showing a process is a martingale from Ito's lemma

Suppose that we have the process $t-W^2(t)$ where $W(t)$ is a Brownian motion with filtration $\mathcal{F}_t$. It is easy to show that this is a martingale by computing ...
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30 views

Is there a standard proof for $\mathbb P(S^X_n\text{ hits }A\text{ before }B) >\mathbb P(S^Y_n\text{ hits }A\text{ before }B)$?

Let $X_i$ and $Y_i$ be two continuous random variables on $\mathbb{R}$ having distribution functions $F$ and $G$, respectively satisfying $G(y)>F(y)$ for all $y$. Let futhermore $S^X_n=\sum_{i=1}^n ...