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 ...

learn more… | top users | synonyms (1)

2
votes
1answer
28 views

Poisson Approximation of Binomial

I have to prove the Poisson approximation of the Binomial distribution using generating functions and have outlined my proof here. Given, \begin{align} & \lim_{n\to \infty} np_n = \lambda \\ ...
1
vote
0answers
46 views

Sufficient condition for convergence in distribution in the plane

I'm trying to show convergence in distribution for a sequence $X_n$ of random variables in the plane. Here's what I know. I have a sort of squeeze theorem for the probability of the r.v.s being in a ...
1
vote
1answer
39 views

Does it mean “two successive tosses is the same” is same as “two successive tosses is either heads or tells”?

I got confusion ! Does it mean "two successive tosses is the same" is same as "two successive tosses is either heads or tells" ? I have two problems : Q. I have an unbiased coin , assuming ...
0
votes
0answers
14 views

Probability of Detection and pulse-pulse decorrelation time [migrated]

Not sure if this is the right forum, but thought I'd ask anyway: I'm trying to analyze the probability of detection ($P_d$) for a Swerling II target. I know that the Swerling II target assumes that ...
2
votes
1answer
31 views

Understanding the matrix normal distribution

A random $n \times p$ matrix $X$ is distributed according to a matrix valued normal distribution iff $\mathrm{vec}(X) \sim \mathcal{N}_{np}(\mu, V \otimes U)$, where $\mu \in \mathbb{R}^{np}$ is a ...
2
votes
1answer
29 views

Constructing dependent sequences of random variables

It is easy, given some random variable $X \colon \Omega \to \mathbb{R}$ on a probability space $(\Omega, \mathbb{P})$, to construct an i.i.d. sequence $X_1, X_2, \ldots$ distributed as the law of $X$. ...
2
votes
1answer
33 views

Show that is a probability space

Let $ \Omega:= \{(x,y) \in \mathbb{R^2}:0<x,y \leq 1 \}$, let $\mathcal{F}$ be the collection of sets of $\Omega$ such that $$ \mathcal{F}:= \{(x,y) \in \mathbb{R^2}:x \in A,0<y \leq 1 \}$$ ...
1
vote
1answer
37 views

'Markovian Property' vs 'Memoryless Property'

The two properties have the commonality in the sense that they predict the future based on the current state, not on the whole history of how the process wandered into the state. Then, what is the ...
1
vote
0answers
33 views

Show that $S_n/n$ converges almost surely if $S_{2^n}/2^n$ converges almost surely

Let $X_n$ be independent random variables such that $\dfrac{S_n}{n}\to0$ in probability and $\dfrac{S_{2^n}}{2^n}\to0$ almost surely. Show that, $\dfrac{S_n}{n}\to0$ almost surely. Here ...
0
votes
1answer
29 views

Can this probability be shown by using the properties of Lebesgue integration

(Grimmett and Stirzaker - Probability and Random Processes - Exercise 1.3.5) I am studying Lebesgue integration in parallel to probability theory, and my question is: Can the following be shown by ...
0
votes
0answers
23 views

Show that the likelihood ratio converges to $0$ a.s.

Let $S$ be a finite set, for simplicity assume $S=\{1,2,...,m\}$. Let $f_0$ and $f_1$ be two non-equal probabilities defined on $S$, with $f_0(j)=P_0(X=j)$ and $f_1(j)=P_1(X=j)$, such that ...
2
votes
1answer
27 views

measurability of a sum of random variables

Let $\{X_n\}_{n = 1}$ be a sequence of random variables. Let $Y_n = \sum_{i=1}^n X_i$ Then, I want to show that $\sigma(Y_1,Y_2,\dots,Y_n) = \sigma(X_1,X_2,\dots,X_n)$ for every $n$. It is clear to ...
1
vote
1answer
19 views

multivariate convergence in law

Suppose $X_n \overset{\mathscr{L}}{\longrightarrow} X$, and $Y$ is another random variable which may be depending on $X_n$. Then it seems not true that we have the following joint convergence in law ...
0
votes
1answer
22 views

Connection between autocovariances and Fourier series of a continous function.

Let $f(w)$ be a continuous function of period $2 \pi$ then it's Fourier series is $$f(w) = \sum_{k = 0}^j \left(a_k \cos(kw) + b_k \sin(kw)\right)$$ I wrote that the autocovariances $\gamma(k)$ (of ...
1
vote
0answers
54 views

If $X$ is such that $c P(X\geq c) \leq E(Y; X\geq c)$ for every $c$ then so is $X\wedge n$

I came across the following as I was reading the book Probability with Martingales by Williams (pg 143): Assumption: $X$ and $Y$ are non-negative random variables such that $$c P(X\geq c) \leq ...
1
vote
2answers
40 views

Resampling operation

I am reading from an arXiv.org paper the following math text: "Let $x\in \{−1, 1\}^I$ be random and uniform, and let $y$ be obtained from $x$ by resampling each coordinate with probability ...
1
vote
0answers
37 views

Show that $\frac1n\log X_n$ converges almost surely

Let $X_0$ follow $\mathrm{Uniform}(0,1)$. Define $X_{n+1}$ iteratively as $X_{n+1}$ follows $\mathrm{Uniform}(0,X_n)$, $n\geq0$. Show that $\dfrac{\log X_n}{n}$ converges almost surely and find the ...
2
votes
1answer
17 views

If $X_n$ are symmetric around $0$ then show $P(|S_n|\geq\max_{1\leq i\leq n}|X_i|)\geq\dfrac{1}{2}$

Let $\{X_k\}_{k=1}^n$ be iid random variables that are symmetric around $0$ i.e. $X=-X$ in distribution. Define $S_n=\sum_{i=1}^nX_i$. Then show $P(|S_n|\geq\max_{1\leq i\leq ...
0
votes
1answer
18 views

Show that $S_n$ converges almost surely to $\infty$

Suppose $X_n$ are independent random variables with $P(X_n=-n^2)=\dfrac{1}{n^2}$, $P(X_n=-n^3)=\dfrac{1}{n^3}$ and $P(X_n=2)=1-\dfrac{1}{n^2}-\dfrac{1}{n^3}$ for all $n\geq2$. Let ...
2
votes
2answers
51 views

piecewise weak convergence in $C[0,1]$

Let $P$ and the sequence $P_n$ be probability measures on $C[0,1]$ with the uniform metric. Fix $0<u<1$ and let $\Pi_1$ and $\Pi_2$ be the projections from $C[0,1]$ onto $C[0,u]$ and $C[u,1]$, ...
1
vote
1answer
21 views

Measurability of a version of a random variable

If $X$ is a ($\mathcal{F}$-measurable) random variable defined on a probability space $(\Omega, \mathcal{F}, \mathbf{P})$ and $Y$ is a version of $X$ in the sense that $\mathbf{P}(X \ne Y) = 0$ and ...
0
votes
0answers
34 views

Zero conditional entropy

This question is related to this math.se question but I need a bit more guidance. For two discrete random variables $X,Y$ we define their conditional entropy to be $$H(X|Y) = -\sum_{y \in Y} Pr[Y = ...
2
votes
2answers
44 views

expected values of identically distributed random variables

Let $X$ and $Y$ be identically distributed random variables on some probability space $(\Omega,\mathcal{F},\mathbb{P})$. Then, if I let $F_X$ and $F_Y$ denote the distribution functions of $X$ and ...
0
votes
0answers
43 views

Every random variable $X$ can be written as $X=\lambda Z_1+(1-\lambda)Z_2$, for $Z_1$ discrete and $Z_2$ continuous random variables.

Show that every random variable $X$ can be written as $$X=\lambda Z_1+(1-\lambda)Z_2$$ for a discrete random variable $Z_1$, a continuous random variable $Z_2$, and a real value $\lambda$. This ...
2
votes
1answer
54 views

Taylor expansion of characteristic function in probability theory

In probability theory, what is the Taylor expansion of characteristic function? I know this is a basic question but I couldn't find a full answer.
0
votes
1answer
26 views

Does the following result require the random variables to be independent?

I am sitting with the book Labelled Markov Processes by Prakash Panangaden, and on page 79 he defines what it means for a set of random variables on a probability space to be independent, and after ...
2
votes
2answers
51 views

Which one of the following versions of Bayes' theorem is correct?

I've seen two versions of Bayes' theorem: I've seen this very long version from a frequentist probability class: $$ P(B|A)=\frac{P(A|B)P(B)}{P(A|B)P(B) + P(A|B^c)P(B^c)} $$ where $B^c$ is the event ...
2
votes
0answers
30 views

A question on $\pi-\lambda$ system

I was reading Rick Durett's Probability Theory and Examples. In the proof of the $\pi-\lambda$ theorem i.e If $\mathcal{P}$ is a $\pi-$system and $\mathcal{L}$ is a $\lambda-$system that contains ...
1
vote
1answer
25 views

How is this an application of the independence property of events?

I'm currently working my way through Klenke's book on probability theory and do not understand a step in his proof of the Borel-Cantelli lemma (Theorem 2.7): The assertion is that for an independent ...
0
votes
1answer
13 views

Stochastic domination preserved by dilution?

Consider an at most countable set $S$ and the corresponding bit space $\{0, 1\}^S$ that is often considered in percolation, interacting particle systems, and other lattice models. Suppose that $\le$ ...
1
vote
0answers
31 views

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 $\int_\Omega Xd\mathbb{P}$, ...
0
votes
0answers
15 views

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 ...
0
votes
0answers
11 views

$\mathbb{P}(d(X,Y)>\alpha)<\beta$ 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 ...
1
vote
0answers
29 views

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 ...
-1
votes
1answer
21 views

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 ...
-1
votes
0answers
40 views

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 ...
1
vote
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 ...
0
votes
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 ...
1
vote
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 $\{\sigma(X_1),\sigma(X_2),\dots, \}$ ...
1
vote
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 ...
0
votes
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 ...
0
votes
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 ...
5
votes
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 $P=\hat{P}$? ...
2
votes
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 ...
2
votes
1answer
19 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 ...
4
votes
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 ...
3
votes
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 ...
1
vote
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. $F_X(x) = P(X \leq x)$. ...
3
votes
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. ...
1
vote
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 ...