Use this tag only if your question is about the modern theoretical footing for probability, for example probability spaces, random variables, law of large numbers, and central limit theorems. Use [tag:probability] instead for specific problems and explicit computations. Use ...

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Expectation in measure theory

I'm reading a book on measure-theoretic probability, and the author defines the expectation of a random variable $X$ on a probability space $(\Omega,\scr H,\mathbb{P})$ as $\int_\Omega Xd\mathbb{P}$, ...
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Probability Law of Stochastic Process Definition

I am reading Probability and Stochastics by Çınlar, and am confused by the following definition in it: I must be missing something because this definition does not seem correct to me. For ...
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$\mathbb{P}(d(X,Y)>\alpha)<\beta$ if $\mathbb{P}(X\in E)\leq \mathbb{P}(Y\in E^{\alpha})+\beta$ for all measurable E

Given two random variables X,Y with measures P,Q (i.e. $P(A)=\mathbb{P}(X\in A)$). Show that if $P(E) \le Q(E^\alpha) + \beta$ for all measurable $E\subset\mathbb{R}$ then ...
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Almost surely diverging sum implies almost surely diverging sum of conditional expectations?

Suppose $\sum_{n=1}^\infty X_n = \infty$ almost surely for nonnegative $X_n$. Let $\mathcal F_n = \sigma(\{X_0, X_1, \ldots, X_n \})$. Can we show that $\sum_{n=1}^\infty \mathbf{E} (X_n | \mathcal ...
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Conditional independence expansion

I have four random variables A,B,C and S. A,B and C are conditionally independent given S. So, I need to obtain P(A,B,C,S) By the chain rule: $$P(A,B,C,S)=P(S)P(A|S)P(B|A,S)P(C|A,B,S)$$ By the ...
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Independence and imaginary events

Consider this experiment: A 6-sided fair die is rolled. If it is a $1$, a fair coin is tossed. Otherwise, a 4-sided fair die is rolled. Assuming results of all die rolls and coin tosses are ...
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1answer
20 views

Expectation of quadratic variation

I got stuck in a step of a proof and need some help. The situation is the following: Let $M$ be a continuous local martingale (which satisfies $\mathbb{E}[\langle M\rangle(T)]<\infty$ - I don't ...
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1answer
18 views

Probability measure on the space of $n \times n$ symmetric matrices with non negative integer coefficients

I know that there exists a particular measure, called Haar measure, defined on random matrices, i.e. $n \times n$ orthogonal complex matrices. My question is the following: can we define a ...
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1answer
35 views

Independence of random variables

Let $\{X_n\}$ be a sequence of independent random variables on some probability space. Then, by definition(according to the book that I am reading), I know that $\{\sigma(X_1),\sigma(X_2),\dots, \}$ ...
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1answer
28 views

Let $X$ denote the number of tosses required to get the 5th head and $Y$ the number between the 6th and 7th heads. Are $X$ and $Y$ independent?

Let $X$ denote the number of tosses required to get the 5th head and $Y$ the number between the 6th and 7th heads. Are $X$ and $Y$ independent? Y will always depend on X . NO ? i know geometric ...
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1answer
18 views

X denotes government will increase payment. x~Bin(2,2/3) . if one increment =9%. expected increment =?

If Government increases payment then they increase it by 9% . now if whether government will increase payment follows binomial distribution with parameters n=2 and p=(2/3) , then what percentage of ...
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1answer
77 views

Almost sure convergence of nonrandom sample

This is a question about almost sure convergence. Consider the following set-up: There are $B$ banks. Each has size $S_{b}$, which follows a size distribution $f_{S}$ with mean E[S]. $f_{S}$ is ...
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1answer
97 views

Can we have a Brownian motion under two different probability space?

Is it possible to construct a stochastic process $B_t$ such that $B_t$ is a Brownian under $(\Omega, F, P)$ and $B_t$ is a Brownian under $(\Omega, F, \hat{P})$? If not, how to argue that $P=\hat{P}$? ...
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1answer
38 views

how that $P(G)=1$ iff $\sum_n \Bbb P(A \cap E_n )=\infty$ for all events $A$ having $\Bbb P(A)>0$.

Two probability problems: 1. Let $a>0$ and let $X_n$, $n \geq 1$, be iid r.v. that are uniform on $(0,a)$ and let $Y_n = \prod_{k=1}^{n} X_k$. Determine all values of $a$ for which $\lim_{n ...
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1answer
20 views

Adapted random variable

Let $\{Y_n\}_{n=1}^{\infty}$ be a sequence of random variables, and let $F_n = \sigma(Y_1,Y_2,\dots,Y_n)$ for each $n$. Let $\{X_n\}_{n=1}^{\infty}$ be a sequence of random variables adapted with ...
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1answer
50 views

Uniformly integrability and convergence

Question 1: $X_n$'s are non-negative, uniformly integrable. Then $E\left[\dfrac{\max_{1\leq k \leq n} X_k}{n}\right]\rightarrow 0$. Question 2: If u.i. is dropped then above the may fail. My ...
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1answer
28 views

Existence of moments and slowly varying function at infinity

I have a somewhat advanced question involving the role of slowly varying functions and their relation to moments. I want to use them to derive certain results for domains of attraction. My problem is ...
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0answers
25 views

How to determine law of a random variable from its cumulative distribution

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space, and let $X$ be a random variable. Suppose that we are given the distribution of $X$, denoted by $F_X$. i.e. $F_X(x) = P(X \leq x)$. ...
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1answer
49 views

Probability via Geometry, applications and examples

People, I am going to teach a lesson including something about Probability via Geometry and, if you have, I would like to know some references or materials (or even some good ideas) that can help me. ...
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1answer
29 views

Independence of a random variable and its conditional expectation

Let $(\Omega, \mathcal{F},\mathcal{P})$ be a probability space. Let $\mathcal{H} \subset \mathcal{F}$ be a sub $\sigma$-algebra, and let $X\in L^1(\Omega, \mathcal{F},\mathcal{P})$ be a random ...
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3answers
37 views

Is convergence in probability sometimes equivalent to almost sure convergence?

I was reading on sufficient and necessary conditions for the strong law of large numbers on this encyclopedia of math page, and I came across the following curious passage: The existence of such ...
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1answer
62 views

Integration with respect to a concrete measure

I got the problem of integrating with respect to a measure in concrete detail. Im just finding formal stuff elsewhere. The measure $Q(A)=\int_0^\infty P(f(r,X)\in A)dr$ is given and i need to show ...
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1answer
53 views

Coupon collector problem doubts

The Coupon Collector problem off Wikipedia: Suppose that there is an urn of $n$ different coupons, from which coupons are being collected, equally likely, with replacement. How many coupons do you ...
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53 views

Bounding Expected Value of a piecewise function

Let X and Y to be two independent random variables with known pdfs. Get a bound for the expected value of the following expression in terms of $E[X]$, $E[Y]$, VAR[X] and VAR[Y]: \begin{equation} ...
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19 views

The domain of the sum rule(probability: The logic of science)

Anyone read Probability Theory: The logic of Science. Please help, I've been really stuck for ages at how the sum rule has it domain derived and I don't have any teacher to ask. Question How is ...
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1answer
38 views

Inequality in proof of SLLN

This comes from theorem 5.1.2 of KL Chung's A Course in Probability Theory. Suppose ${X_n}$ are uncorrelated and their second moments have a common bound. Then For each $n \ge 1 $, $D_n:= ...
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2answers
55 views

Average time until one disk drive fails

I have two hard drives. The given Mean to Failure Time is $100$ hours. So if I have one hard drive, the average amount of time until it fails will be $100$ hours. ...
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36 views

Finding a polynomial approximation of a PDF

I would like to find a polynomial $P(x)=\sum_{d=1}^D P_dx^d$ of degree $D$, where its derivative is larger than or equal to a given pdf $f(x)$ in $[0,1-\epsilon]$, for any $\epsilon>0$. Note that ...
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52 views

Question about Kolmogorov extension theorem

I need some help understanding the relationship between the following two theorems Theorem 1: Let $\{\mu_n\}$ be a sequence of probability measures on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$, where ...
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1answer
33 views

$\left \| \left \| f \right \|_{L^{p}} \right \|_{L^{q}} \leq \left \| \left \| f \right \|_{L^{q}} \right \|_{L^{p}} $ for $0<p\leq q$

Let f be bounded on $X\times Y$ measure space with $\mathbb{P}\times\mathbb{Q}$ probability measure, show that for $0<p\leq q$: $\left \| \left \| f \right \|_{L^{p}(\mathbb{P})} \right ...
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1answer
50 views

$\limsup \frac{|S_n|}{n}=\infty$

$X_n$'s are i.i.d symmetric with $E|X_1|=\infty$. Then $\limsup \frac{|S_n|}{n}=\infty$. How do I show $\limsup \frac{S_n}{n}=\infty$ and $\liminf \frac{S_n}{n}=-\infty$? My attempt: Let $c=\limsup ...
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36 views

Proof - Limits of CDF

For a cdf, defined as $F(x)=P(X\le x)$, in order to prove $\lim\limits_{x\,\uparrow\,\infty}F(x)=1$, I've two concerns: (1) Some concern about a proof from a book, and (2)Validity of a proof that I've ...
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2answers
40 views

What's the probability that a particle jumps out of an interval?

Suppose I have a small particle and put it on the center of an interval on a 1-D axis. If the particle undergoes a motion that satisfies: It chooses its direction freely and randomly It ...
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1answer
36 views

Existence of a random variable

Let $\mu$ be a probability measure on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$, where $\mathcal{B}(\mathbb{R})$ denotes the Borel sets. Then, is it true that there exists a probability space ...
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2answers
37 views

Concentration property of entropy

Let $X$ be a random variable taking its values in $A = \{a_1,\ldots,a_n\}$ such that $Pr[X = a_i] = p_i$ for all $1 \leq i \leq n.$ The entropy of $X$ is defined as $$H(X) = -\sum_{i=1}^n p_i ...
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23 views

Correspondence between AB-divergence and Kullback-Leibler divergence

I'm reading up on AB-divergence (alpha-beta-divergence) based mainly on the exposition given in Chichoki et al. (2011), "Generalized Alpha-Beta Divergences and Their Application to Robust Nonnegative ...
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1answer
28 views

For every probability $\mu$ on $(\Bbb R,\mathcal{B}(\Bbb R))$ exists at least a real r.v. $X$ s.t. $P^X=\mu$

Given a probability $\mu$ on $(\Bbb R,\mathcal{B}(\Bbb R))$, does exist always some random real-valued variable $X$ (defined on some probability space $(\Omega,\mathcal{A},P)$) such that its ...
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39 views

Help in understanding a probability calculation from a paper

Paper: A Framework for Investigating the Performance of Chaotic-Map Truly Random Number Generators download link = http://arxiv.org/pdf/1211.1234.pdf explains a method to determine if a random ...
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1answer
35 views

$\lim_{n \rightarrow \infty} \Bbb P(Y_n >c) =1$ for every $c>0$. Show that $\lim_{n \rightarrow \infty} \Bbb P(X_n+Y_n >c) =1$ or every $c>0$.

I have trouble in a probability problem. Let {$X_n,n \geq1$} and {$Y_n,n \geq1$} be two sequences of random variables such that $\lim_{n \rightarrow \infty}X_n =X$ in distribution for some random ...
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1answer
54 views

Deduce $\partial_tp=-\partial_x(b(x)p)+(1/2)\partial_{xx}(\sigma^2(x)p)$, for $p(x,t|y)$ of $X(t)$ and $dX=b(X)dt+\sigma(X)dW$, $X(0)=y$

I am stuck in this proof... I almost got it, but I must have made a mistake. It is part B that I am getting wrong. Thanks in advance for your help! QUESTION: Let $X$ satisfy the autonomous SDE ...
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Find the limit distribution of $Y_n =\frac{\Sigma_{i=1}^{n}X_i}{\sqrt{\Sigma_{i=1}^{n}|X_i|^2} }$.

Let $X_1,X_2,X_3,...$ be i.i.d. with uniform distribution on $[-1,1]$. Find the limit distribution of $Y_n =\frac{\Sigma_{i=1}^{n}X_i}{\sqrt{\Sigma_{i=1}^{n}|X_i|^2} }$. I think this is just a direct ...
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26 views

Book Recommendation for Studying Stochastic Optimization Problem with Almost Sure Constraint

I was begin to study the following type of stochastic optimization problem: Let $u(k)$ and $X(k)$ are discrete random variables for all $k =1,2,...,N$ and $f$ be any concave function and $g$ be ...
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29 views

How is $\mathcal F_\infty$ different from $\bigcup_{n=0}^\infty \mathcal F_n$? [duplicate]

Let $(X_n)_n$ be a sequence of random variables. Define $\mathcal F_\infty := \sigma(X_0, X_1, \ldots)$ and $\mathcal F_n := \sigma(X_0, X_1, \ldots, X_n)$. In the proof of the Kolmogorov's zero–one ...
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Rolling a die with n sides to get a cumulative score of n

I was told this problem a while ago, and recently someone explained the answer to me, which I didn't understand; could someone please explain in layman's terms (ish)? You have a die with $n$ sides. ...
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1answer
28 views

expectation of product of sums of normally distr. r.v.

Let $Z_1$ and $Z_2$ be i.i.d. standard normally distributed. $X_1=Z_1+Z_2$ and $X_2=Z_1-Z_2$. Apparantly E[|$X_1|*|X_2|$] = E$[|Z_1|*|Z_2|]$. Why?
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19 views

Question on Absolute Continuity of measures [closed]

Can I ask why does the question of absolute continuity of measure require the assumption of sigma-finiteness ? Thanks!
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31 views

Distribution or samples of a function of a random variable

OK I edited the question: I have the following setup: Stereo camera setup with two images I, I'. 4 1-dimensional random variables (each corresponding to the inverse depth value of a pixel on an ...
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1answer
108 views

Is there any $F \in \mathscr{F}$ such that $\mu(F)=x$?

Let $ (\Omega,\mathscr{F},\mu)$ be a probability space such that $\mu$ is non-atomic, and fix $x \in [0,1]$. Is it true that one can find $F \in \mathscr{F}$ for which $\mu(F)=x$? And what if $\mu$ is ...
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2answers
52 views

Differential Entropy

I'm a little temporarily confused about the concept of differential entropy. It says on wikipedia that the differential entropy of a Gaussian is $\log(\sigma\sqrt{2\pi e})$. However I was thinking as ...
3
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2answers
99 views

Stochastic variables independent given Tau

Say we have a filtration $(\mathbb{F}_s)$, and a stopping time $\tau$ w.r.t. to that filtration.Let $X_t$ be a continuous stochastic process (not required to be adapted to the mentioned filtration), ...