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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32 views

kolmogorov equations for continuous markov chains

I'm trying to find the for for forward equations for a birth and death processes when all $\lambda$ coefficients are zero. The forward equation for a Birth and Death Process has the form: ...
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16 views

Convergence of Quantiles moments.

QUESTION: Let $F$ be an absolutely continuous distribution function whith density f, and $F_{n}$ be its nth empirical distribution. Suppose that $t\in (0,1)$ is constant. Is true the convergence ...
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1answer
31 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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1answer
22 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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10 views

Application of lindeberg/lyapunov condition

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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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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12 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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2answers
2k views

Maximum likelihood estimation of $a,b$ for a uniform distribution on $[a,b]$

I'm supposed to calculate the MLE's for $a$ and $b$ from a random sample of $(X_1,...,X_n)$ drawn from a uniform distribution on $[a,b]$. But the likelihood function, ...
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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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2answers
42 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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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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17 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
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
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
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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1answer
232 views

Aldous criterion for tightness in $D[0,1]$

Does anyone know where I can find some useful information about the Aldous criterion for tightness in the space of all cadlag functions $D[0,1]$?
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0answers
30 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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1answer
200 views

joint probability distribution of one discrete, one continuous random variable

This is a problem on the joint distribution of a discrete and a continuous random variable. Kitty Oil Co. has decided to drill for oil in 10 different locations; the cost of drilling at each ...
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1answer
197 views

Convergence in expectation problem

How do you prove that if X converges in probability and expectation that this implies convergence in mean? I think I have to use Chebyshev's Inequality, but am not sure how to incorporate the ...
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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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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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111 views

An exercise from Revuz, Yor; equality in distribution of 2 integrals.

Here is the exercise I have been struggling to solve. It is taken from this book by Revuz and Yor: link. Here is the full text of the problem ( Exercise 3.32, chapter 4). Exercise (3.32). Let $B$ and ...
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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
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
236 views

Unknown number of colours Bernoulli Urn

Okay, so, in the traditional Bernoulli Urn problem, we have an urn with a number N, possibly infinite, of coloured balls, and there are k possible colours. That one I grok. However, what if I don't ...
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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
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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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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451 views

Different versions of functional central limit theorem (aka Donsker theorem)?

I have seen several versions of functional central limit theorem (see the end of this post). I am confused, and hope someone could help to clarify their relations and differences. For example, I am ...
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1answer
19 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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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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2answers
169 views

Independent increments?

The questions are simple: Does the process $ X(t) = \int_0^t B(s)ds$ have independent increments? What about $X(t) = \int_{t-r}^{t}B(s)ds$? Here $B$ denotes the standard Brownian motion. ...
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284 views

Conditional distribution in Brownian motion

I need to prove the following: Let $X$ be a Brownian motion with drift $\mu$ and volatility $\sigma$. Pick three time points $s < u < t$. Then, the conditional distribution of $X_u$ given ...
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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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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
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 : ...
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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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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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206 views

Weak and Probability convergences

I have a question about this book, "Topics in Random Matrices Theory" of Terence Tao. He claims that if $$\int_{\mathbb{R}}\varphi \ ...
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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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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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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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1answer
226 views

Notationally, what is the difference between $\Pr(X = x)$ and $P(X = x)$? When should I use each?

I'm talking specifically about probability theory. I was reading some stuff about probabilistic graphical models, and they kept switching the notation in this book, but I couldn't discern the ...
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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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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 ...