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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33 views

$\sigma$-algebra. (A.1) (Sigma Axiom)

""Def. A family $\mathcal F$ of subsets of $\Omega$ is said to be a $\sigma$-algebra on $\Omega$ if: (A.1) $\Omega\in\mathcal F$ .... i am aware that this statement has to hold (a.1) ...
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1answer
37 views

What is a mathematically rigorous justification for multiplying edge probabilities of a tree diagram

I was trying to understand why it was mathematically justified to multiply edge probabilities in a tree diagra and I came across the following question: Why do we multiply in tree diagrams? The ...
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0answers
28 views

Expectation of a discrete random variable

This might seem to be a dumb question but here it goes : I am trying to derive the expectation of a random variable by two different means. The first one : \begin{align} E(X) &= ...
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1answer
21 views

How to prove the following proposition [on hold]

Let $\{X_n\}$ be a sequence of independent and identically distributed random variable's,$S_n=\displaystyle\sum\limits_{k=1}^{n} X_k$.Then we have $$\frac{S_n}{n} \rightarrow 0 \,\, {\rm in\,\, ...
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1answer
33 views

Partitioning the range of a function by numbers $a_i$ such that each set $\{x \mid f(x) = a_i\}$ has measure zero

Let $f \in C_b(S)$ (set of all bounded and continuous functions) and $\mu$ be a measure on $S$ where $S$ is a complete separable metric space. Then a book (Probability Theory by Borkar) claims that ...
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1answer
12 views

Geometric embedding of random variables

Given centered random variables $X_i \in \mathbb{R}$, $i=1,2,\ldots,n$ find $x^{(i)} \in \mathbb{R}^n$ such that $\langle x^{(i)}, x^{(j)} \rangle =E(X_i X_j) $ for all $i,j$.
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1answer
22 views

Problem on sequence of probability measures

Let $\mu_n$ and $\mu$ be probability measures such that $$\lim_{n\to \infty} \mu_n(A) = \mu(A)$$ for Borel $A \subset S$ satisfying $\mu(\partial A) =0$ (we call it a $\mu$-continuity set). I have ...
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4answers
38 views

Intuition for regression to the norm

So for regression to the norm it says if someone has a high score on a test (relative to the average) then they are likely to score lower and lower on each following test? This seems very counter ...
2
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1answer
20 views

Converging exponentially in probability implies convergence with probability one?

When I read the paper On the Strong Universal Consistency of Nearest Neighbor Regression Function Estimate, the theorem 1 in it states something like If for every $\epsilon > 0$ there exists ...
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1answer
14 views

Independent Probabilities for Multiple Events

I don't understand my professor's definition of multiple independent events. Does he mean that for $A_1,A_2,A_3,\ldots, A_n$ events to be independent that each $n$-tuplet (for $n = 2$ to $n$) must ...
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1answer
30 views

$E\left[\prod_{i=1}^nX_i\right]=\prod_{i=1}^nE\left[X_i\right]$ for all independent and real-valued random variables

Let $(\Omega,\mathcal{A},P)$ be a measurable space and $X_1,\ldots,X_n:\Omega\to\mathbb{R}$ be independent random variables with $\color{red}{\prod_{i=1}^nX_i\in\mathcal{L}^1(P)}$ $\;\Rightarrow$ ...
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0answers
26 views

Does a Gaussian Process have all the variables independent? [on hold]

Wikipedias definition of Gaussian process is that it is a set of Gaussian random variables indexed by a set T. I'm not sure if there are other conditions like independence of the varibles it feels ...
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0answers
19 views

better expression for simple random walk

Let $P_{k,j}$ be the probability that the probability that simple symmetric random walk starting from the origin reaches the point $k \in \mathbb{N}$ precisely in $j$ steps without ever returning to ...
3
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1answer
156 views

Please explain : “at least 95% of the time, the error does not exceed the reciprocal of the square root of the number of trials”

I am studying "Introduction to Probability" book by Charles M. Grinstead & J. Laurie Snell and the authors give a rule of thumb to arrive at the number of trials needed in a probability ...
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0answers
24 views

problem related to convergence of random variables [on hold]

Suppose two random sequences $X_{n}$ and $Y_{n}$ satisfy the following conditions: 1) $0<X_{n}<Y_{n}$, a.e. for all n 2) $X_{n}$ converges in distribution to $X$, and $Y_{n}$ converges in ...
2
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1answer
21 views

Variation on ergodic estimates

Let the sequence of random variables $\{X_{n}\}, n = 1,2, \ldots$ be a Markov chain, which is sufficiently "Ergodic" so that it has stationary distribution $\pi$ and for a function $f$ the sequence of ...
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0answers
14 views

Stopping Times and Directed Processes [on hold]

The book "Stopping Times and Directed Processes" uses the techniques of stopping times to convergence problems. Just wondering, what are the advantages of using this approach? Would it make the ...
3
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1answer
62 views

Find characteristic function of random variable

Let random variables $X, Y, Z$ independent. $X$ with uniform distribution on $[-a,a]$, $Y$ with Poisson distribution with parameter $ν$, $Z$ with Bernoulli distribution with parameter $p$. Find the ...
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1answer
18 views

Distribution function for sum of two random variables with geometric distribution

What would be a distribution function of a random variable $\zeta$, which is the sum of two independent random variables $\xi$ and $\eta$, which have following distribution: $P(\xi=k) = p_1 q_1^k$, ...
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0answers
18 views

Life annuity and the use of Gompertz-Makeham

First of all I posted this on the Quant stackexchange but haven't gotten any answers yet or hints/tip so I thought I might as well post it here and see if anybody here might be able to help! The ...
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2answers
28 views

How to see $P(X_{i_1} = x_{i_1}, X_{i_2}=x_{i_2}, \ldots, X_{i_j} = x_{i_j}) = P(X_{i_1} = x_{i_1})P(X_{i_1} = x_{i_1}) \dots P(X_{i_j} = x_{i_j})$?

Suppose $X_1, X_2, \ldots, X_n$ are independent random variables. Then by definition $P(X_i = x_i, X_j = x_j) = P(X_i = x_i)P(X_j = x_j)$ for $i \neq j$. I've the following questions: How do I see ...
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1answer
33 views

What is a good way to think about the continuous random variable paradox?

While transitioning from discrete random variables to continuous random variables, I came across the argument that a continuous random variable can take possibly infinite values. Therefore, the ...
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2answers
41 views

Borel $\sigma$-Algebra definition.

Definition: The Borel $\sigma$-algebra on $\mathbb R$ is the $\sigma$-algebra B($\mathbb R$) generated by the $\pi$-system $\mathcal J$ of intervals (a, b], where a < b in $\mathbb R$ (We also ...
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22 views

Transience question for Markov Chains

Let's suppose I have a countable state discrete time MC that is known to be transient, irreducible and reversible with respect to some measure that assigns positive finite mass to each singleton, but ...
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2answers
49 views

Definition of $\pi$ and d -systems.

Definition: Let $\Omega$ be a sample space. a) A d-system is a family of subsets containing $\Omega$ and closed under proper difference (if A,B $\in\mathcal D$ and A $\subseteq$ B, then B \ A ...
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1answer
34 views

exists $A \in \mathcal{F}$ such that $\mu(B\triangle A) < \epsilon$

Let $\mu$ be a probability measure on $(S, \mathcal{S})$, where $\mathcal{S} = \sigma(\mathcal{F})$ for a field $\mathcal{F}$. How do I go about showing that for each $B \in \mathcal{S}$ and $\epsilon ...
3
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1answer
55 views

Is there a closed form for this distribution (maximum difference between successes and failures in i.i.d. Bernoulli flips)?

Consider a series of i.i.d. coin flips: $$X_1,X_2,\ldots, X_n\sim \begin{cases} 1 &\text{w.p. } p \\ -1 &\text{else} \end{cases} $$ We define $$Y = \max_{i\leq ...
2
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1answer
20 views

Is the following process bounded (iterative normal sampling)

We define the following stochastic process: $X_0=1$ $\forall i\geq i:X_i\sim\mathcal N(0,X_{i-1}^2)$ That is, we first sample $X_1$ from the normal distribution with variance $1$, then in the ...
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1answer
21 views

If $(f_n)_n$ and $(g_n)_n$ converge stochastically to $f$ and $g$, then $(f_n+g_n)_n$ converges stochastically to $f+g$

Let $(\Omega,\mathcal{A},\mu)$ be a measurable space $(E,d)$ be a separable metric space $f,g,f_n,g_n:(\Omega,\mathcal{A})\to(E,\mathcal{B}(E))$ measurable $(a_n)_{n\in\mathbb{N}}\subseteq E$ We ...
2
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2answers
35 views

A probability function is determined on a dense set- Where is density used in the following proof?

A probability function is determined on a dense set- Where is density used in the following proof? Consider the following theorem and proof from Resnick's book A probability path. I cannot really see ...
5
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1answer
27 views

The expectation over an infinitesimal time interval

I have a deduction which I would like to formalize (I suppose with some additional measure theory): Let $N(t)\sim Pois(\lambda_t)$, where $\lambda_t$ is stochastic (we are thus looking at a Cox ...
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2answers
35 views

If $X, X_1, X_2, \ldots $ are positive and $X_n\stackrel{P}\to X $ and $E(X_n) \to E(X)$, then $X_n \stackrel{L_1}\to X$

Let $X, X_1, X_2, \ldots $ be positive random variables. Prove that if $X_n\stackrel{P}\to X $ and $E(X_n) \to E(X)$, then $X_n \stackrel{L_1}\to X$ My attempt: I tried to truncate $E(|X_n-X|)$ ...
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1answer
25 views

joint uniform distribution of two RV

i got two continuous R.V $X,Y$ their joint density is uniformly distributed in a bounded triangle, between the x,y axis and y = -x +1. lets mark the joint density as $f$. is it true that $f = 2 ...
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0answers
14 views

On why a limit of random variables implies $Z_k(w)\leq z+\frac{1}{m}$.

If $Z_1,Z_2,\ldots$ are random variables such that $\lim_{n \to \infty}Z_n(w)$ exist for all $w$ and $$Z(w)=\lim_{n \to \infty}Z_n(w)$$ and suppose $w\in\{Z^{-1 }\mid Z\leq z\}$, for $z \in \mathbb ...
5
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1answer
29 views

Unusual behavior in a conditional expectation

Show an example of random variables $X$ and $Y$ such that $X$ and $Y$ are not independent but still $$\textbf{E}(X\mid Y) = \textbf{E}X$$ I tried looking at the simplest discrete probability ...
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1answer
16 views

independence of stopping time and a sigma algebra

Let $(B_t)$ be a standard Brownian motion and $\{ \mathcal{F}_t \}$ be the filtration generated by the Brownian motion. For a stopping time $\tau$, we know that $\{B_{\tau + t} - B_{\tau}\}_{t \geq ...
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0answers
15 views

Alternatives to logit or probit models?

Let $W_{i}$, $X_i$, $Y_i$, $\epsilon_i$ be real-valued random variables . We assume that $W_i$ is explained according to the following linear statistical model $$ W_i=\beta X_i+\gamma Y_i+\epsilon_i ...
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0answers
23 views

Poisson Process RAID disk failure

I have the following question i am working on : Assume that the time it takes before a hard disk drive crashes is exponentially distributed with a mean of 5 years. Consider a RAID system consisting ...
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1answer
33 views

Understanding mean in Poisson Processes

Assuming I have a question like : Suppose that the amount of time one spends in a bank is exponentially distributed with mean ten minutes, that is, $λ = 1/10$. What is the probability that a ...
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0answers
25 views

How to prove $\int_{0}^{1/2}A(x,x)\,dx > 0$ where A is $D_4$- invariant copula? [closed]

I am interested in the article of Measure of dependence determined by $D_4$-invariant copulas. I have some questions. In page 3873, I want to know how to prove $\int_{0}^{1/2}A(x,x)\,dx > 0$. I ...
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1answer
33 views

Integral of a Brownian bridge with respect to time

Let $(W_s)_{s\geq 0}$ be a Brownian motion and $t$ a fixed point in time. What is the distribution of $$\Big.\int_0^tW_sds\Big|W_t$$ i.e. the integral of a Brownian bridge with respect to time? Is it ...
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0answers
12 views

Posterior of mean given an observation from a bivariate normal with unknown but common mean, and known variance

suppose the sample vector $(x,y)$ is generated from a bivariate normal: $$ \left[\begin{array}{c} x\\ y \end{array}\right]\sim N\left(\left[\begin{array}{c} \theta\\ \theta ...
2
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0answers
32 views

Laplace transform of stopping times

I am nearly done with a question: Let $(B_t)$ be a Brownian motion on $\mathbb{R}$. For a fixed $x >0$, let $\tau$ be a stopping time defined by $$ \tau = \inf \{t \geq 0 : B_t \not \in (-x,x) ...
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1answer
58 views

The correlation between two numbers drawn from a box without replacement

Question: if a box contains tickets numbered $1,4,4,7,$ and three tickets are selected at random from the box without replacement, and we let $X$ be the number on the first ticket and Let $Y$ be the ...
2
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0answers
29 views

Correlation of belief distributions from distinct signals

Anne and Bob are two Bayesians who initially share a non-degenerate prior about a binary state of the world. Anne observes some signal (i.e., an experiment in Blackwell's terminology) about the state ...
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9 views

Completeness of space of processes

Under what metirc, the space of continuous, non-decreasing, bounded processes becomes a complete mctric space? Thank you.
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12 views

Introductory study of Survival Analysis and Decision Theory

I'm pursuing a compact Masters degree in Mathematics, a 4 year program at BITS Pilani, India. Except for a couple of introductory courses on statistical and probabilistic analysis, and operations ...
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3answers
44 views

Why is $ -\sum_{i \in \text I} p_i \log_2(p_i)$ maximized for all $p_i$ equal? Is it true if $|\text I | = \infty$?

Reading a text it is stated without proof that $$ -\sum_{i \in \text I} p_i \log_2(p_i)$$ where $\sum_{i \in \text I} p_i = 1$ is maximized if $p_i$ is a constant. In the case of my theorem, the ...
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1answer
48 views

A question about independence of sigma algebras (generated by random variables)

Let $X_1, X_2, \ldots$ i.i.d random variables. Is it possible that $$\{X_{n+1} \in B\} \in \sigma({X_1, \ldots, X_n})$$ for some $B$? Why yes/not? I want to show that $\sigma(X_{n+1})$ and ...
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1answer
15 views

Distribution of a Gaussian Random variable vector [closed]

I read a slide on the internet which show that: If the random vector $w\sim N(0,I )$ then how can I prove: $x= A^{1/2}w+\bar{x}$ has the distribution $N(\bar{x},A)$ Here A is the covariance matrix ...