Questions tagged [probability-theory]

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 [tag:probability-distributions] for specific distribution functions, and consider [tag:stochastic-processes] when appropriate.

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34 views

Is $P (C\mid A\;\text{and}\; B)$ the same as $\frac {P(A\cap B\cap C)}{P(A\cap B)}$?

Excuse me if this question is very basic: Is $P (C\mid A\;\text{and}\; B)$ the same as $\frac {P(A\cap B\cap C)}{P(A\cap B)}$?
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1answer
20 views

Discrepancy of answers between differing computations of $E[e^{W_s}e^{W_t}]$ ($W_t$ being the Weiner process)

I was looking at another thread, and the following two distinct solutions to $E[e^{W_s}e^{W_t}]$ (assume that $W_0 = 0$ and $t>s$) were given, with both gving identical answers (I have slightly re-...
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33 views

Rigorous justification for this inequality?

Let $(\Omega,\mathcal F,P)$ be a probability space. Consider a collection of bounded real random variables $X(\gamma)$, for $\gamma\in[0,1]$, defined on this probability space. Let $(\gamma_i)_{i=1}^\...
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1answer
38 views

The probability that $0$ occurs exactly twice in $x^n$

Consider a binary random variable $X$ taking value over $\mathcal{X} = \{0,1\}$ with probabilities $P_X(1)=aP_X(0)$ (with $a>0$) and an i.i.d. sequence of length $n$ denoted by $x^n = (x_1,...,x_n)$...
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20 views

Are those sets of measures identical?

Let $X$ and $Y$ be two random variables on the same probability space. We have $X \sim \mu$ and $Y \sim \nu$ and the measures $\mu$ and $\nu$ are convexly ordered with same mean. We denote by $L(X,Y)$ ...
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26 views

Can we estimate Max and Min of the $\frac 12 . P (B\mid A). P (C\mid A\;\text{and}\; B)$ for $3$ independent events $P(A)=P(B)=P(C)=\frac 12$?

I have three dependent events $A,B,C$, each one happens with probability $\frac 12$. I want to estimate the following probability $$ P_{tot}=P (A\;\text{and}\; B\;\text{and}\; C)=P (A) \cdot P (\;B\;...
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1answer
19 views

Introducing probability measures in $\left(\mathbb{R}^n, \mathbb{B}(\mathbb{R}^n)\right)$

I'm reading Shiryaev's Probability and I'm in the section where he introduces the probability measure in $\left(\mathbb{R}^n, \mathbb{B}(\mathbb{R}^n)\right)$. Now he uses a difference operator $$ \...
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Boundary characterization of harmonic functions on $\Bbb R^n$

I was reading some paper on Martin/Possion boundaries, but most of them concerned some kind of bounded cases. So I wondered, is there any boundary-like characterization of a kind there exists an $(n-1)...
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Understanding the “Découpage de Lévy” (or Levy decomposition)

I am looking for someone to help me understand what exactly the "Découpage de Lévy" (or Levy decomposition) of random variables is and why it works. The book I am reading is "The ...
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2answers
29 views

Can we estimate the minimum and maximum value of the probability of union of three dependent events (each with probability $\frac12$)?

I have three dependent events $A,B,C$, each one happens with probability $\frac 12$. I want to estimate the probability of the union of these events, ie. $$ P_{tot}=P (A\cup B\cup C)=P (A) + P (B) + ...
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2answers
64 views

A Law of Large Numbers for Conditional Expectations

Let $(\Omega,\mathcal F,P)$ be a probability space, and suppose that we are given, for each $\gamma \in[0,1]$, an iid sequence of real integrable random variables $\{X_n(\gamma)\}_{n=1}^\infty$. Let $...
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1answer
65 views

is this true $P ( A | B )= 1 \rightarrow B \subseteq A$

$$ P(A\cap B) = P (B) $$ Does this require $$B \subset A $$
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Trying to prove and equivalent definition of stochastic processes.

Explicitly I want to show, that a family of mappings $(X_t)_{t\in T}$ with $X_t :\rightarrow S$, $t\in T$ is a stochastic process, iff the mapping $X: \Omega \rightarrow S^T$, wich maps $\omega$ to ...
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25 views

Monte-Carlo Estimate of an Integral on a Riemannian Manifold?

Suppose I have a function $f:\mathcal{M} \rightarrow \mathbb{R}$, where $\mathcal{M}$ is a Riemannian manifold. I would like to compute the monte-carlo estimate of $\int_{y \in \mathcal{M}} f(y) \...
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12 views

Probability that the system works given two subsystems with series and parallel connections

Consider the system of components connected as in the accompanying picture. Components $1$ and $2$ are connected in parallel, so that subsystem works iff either $1$ or $2$ works: since $3$ and $4$ are ...
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28 views

Approximation by rectangles in metric spaces and Radon-Nikodym derivative

I'm reading a paper about metric measure spaces but I have some doubts on how those two things can hold: let $(X,d)$ be a Polish space and $\mathfrak{m}$ a non-negative Radon measure on $X$. Let $C\...
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Unions of Data Set, Conditional Probability

could I have help with this You are rolling 2 dice: For question a, state whether each scenario contains mutually exclusive or non mutually exclusive events and solve. For question b, state whether ...
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1answer
36 views

Conditional expectation of a random variable

Let $X$ and $Y$ be two real-valued random variables. The conditional expectation of X condion on $Y$ taking value $y$ is defined as $$ E[X | Y=y] = \int x \, p_{X | Y}(x | y) dx. $$ Let $\sigma(Y)$ be ...
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2answers
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If $X\sim U(0,1)$ then $Y=-\frac{1}{\lambda }\ln(1-X)\sim Exp(\lambda )$

If $X\sim U(0,1)$ then $Y=-\frac{1}{\lambda }\ln(1-X)\sim Exp(\lambda )$ Here is my solution : If $X\sim U(0,1)$ then $f(x)=1$ where $0\leq x\leq 1$. $0<x\Rightarrow -x\leq 0\Rightarrow 1-x\leq 1\...
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0answers
54 views

Property of KL divergence for Markov chains

We begin by showing the "start by showing". Is this right? I feel like something is missing when marginalizing out "Y". Was Y marginalized out correctly? $$\begin{split}KL_{X,Y}(\...
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0answers
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plz help.. Sally is traveling from city A to city C by way of city B… [closed]

Sally is traveling from city A to city C by way of city B. There are 4 flights from A to B and 6 flights from B to C. How many different routes are possible Answer Choices A)10 B)12 C)24 D)15 Please ...
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0answers
29 views

Density of noncentral $F$ distribution

Suppose $Z\sim N(\mu,1)$ and $V$ is independent of $Z$ with distribution $\chi^2_m$. Then $T=\frac{Z}{(V/m)^{1/2}}$ is said to have a noncentral $t$ distribution with noncentrality $\mu$ and $m$ ...
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8 views

Guessing a random variable from moments

Define $\mu_m$ as follows: \begin{align*} \mu_{2m + 1} &= 0 \\ \mu_{2m} &= \frac{c^{2m}}{2m + 1}. \end{align*} Is there a continuous random variable $X$ such that $E[X^m] = \mu_m$? The ...
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10 views

Why do homogeneous Markov chains with a transition probability matrix coinciding with a given stochastic matrix exist?

Why do homogeneous Markov chains with a transition probability matrix coinciding with a given stochastic matrix exist? And why would such a Markov chain, given a stationary distribution as the initial ...
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23 views

Find $\lim_{x\uparrow 1}\mathbb{P}(F_Y(Y)>x|F_X(X)>x)$ where $X,Y$ poisson

Let $X=Y_1+N,Y=Y_2+N$ where $N\sim \text{pois}(\lambda), Y_1\sim \text{pois}(\lambda_1), Y_2\sim \text{pois}(\lambda_2)$ and $N,Y_1,Y_2$ are independent. I'm asked to find $$\lim_{x\uparrow 1}\mathbb{...
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0answers
27 views

How to show that $N(t)^2 - \lambda t$ is a submartingale when $N(t)$ is a Poisson process with rate $\lambda$?

How to show that $N(t)^2 - \lambda t$ is a submartingale when $N(t)$ is a Poisson process with rate $\lambda$?
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30 views

CDF of noncentral $t$ distribution

Suppose $Z\sim N(\mu,1)$ and $V$ is independent of $Z$ with distribution $\chi^2_m$. Then $T=\frac{Z}{(V/m)^{1/2}}$ is said to have a noncentral $t$ distribution with noncentrality $\mu$ and $m$ ...
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0answers
37 views

(Probability : Theory and Examples, Durrett ) Theorem 5.2.2 Monotone Class Theorem

Theorem 5.2.2 is as below. (Monotone class theorem) Let $\mathcal{A}$ be a $\pi$-system that contiains $\Omega$ and let $\mathcal{H}$ be a collection of real-valued functions that satisfies: (1) If $...
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12 views

Example of a risk measure that is not law-invariant

In some theorems about risk measures, the property of law invariance is required. Let $\mathcal{Z} = \mathcal{L}(\Omega, \mathcal{F}, P)$. A risk measure $\rho\colon \mathcal{Z}\to \mathbb{R}$ is law ...
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0answers
25 views

Conditional distribution under Random Unitary Transformation

Let $\mathbf{A}$ and $\mathbf{B}$ be random unitary matrices. Is the following relation satisfied: \begin{align} p(\mathbf{y}|\mathbf{x})=p(\mathbf{A}\mathbf{y}|\mathbf{B}\mathbf{x}), \end{align} ...
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1answer
60 views

A question on Williams' “Probability with Martingales” Theorem 12.2 (b).

Theorem 12.2 (b) of Williams' "Probability with Martingales" states that the convergence of a series of zero-mean independent and bounded random variables $X_k$ (say bounded by $K$) implies ...
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0answers
30 views

Applying the Girsanov to use the reflection principle

I would like to compute the probability of a Geometric Brownian motion exceeding a certain value somewhere in a given period. We define the process by \begin{align*} d S_t = \mu S_t dt + \sigma ...
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1answer
20 views

Expectation of dirac measure of a function

Suppose $\pi(dx)$ is a measure, $\phi$ is a deterministic function and $\delta_{\phi(x)}(A)$ is the dirac measure. The dirac measure can be considered a measurable function (I think? Should be a ...
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1answer
24 views

Uniqueness of the representation of Ito processes for $G \in \mathcal{L}_2^\text{loc}$

Let $T > 0$. Let $(\Omega, \mathcal{F}, P, \{ \mathcal{F}_t \}_{t \in [0, T]})$ be a probability space with a filtration. We write \begin{align*} \mathcal{L}_2 &= \{ \Phi = \{ \Phi(t) \}_{t \in ...
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11 views

probability of liminf of a sequence

I wonder what's the difference between the probability below.(maybe (1) is greater than (2)?) P($\varliminf_{n \to \infty} {x<\epsilon}$), note (1) and $\lim\limits_{n \to \infty} P(x<\epsilon)...
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1answer
13 views

probability of exponential distribution question

Suppose $X_n$ follows Exponential distribution with parameter $\alpha$. Find P($X_n$ < log(n)*$\epsilon$). I got my answer equals 1 - $\frac{1}{n^(\alpha\epsilon)}$, which is quite different from ...
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1answer
19 views

Show $(a, b], (-\infty, a]$, and $(b, \infty)$ are fields, but not sigma fields

Show that all finite unions of intervals in R of the form $(a, b], (-\infty, a]$, and $(b, \infty)$ are not $\sigma$-field. I think it's a simple question but I have just started studying probability ...
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23 views

limit of inf of a sequence

I wonder whether the followings are equivalent. $P(\liminf_{n\to \infty} X_n \leqslant \epsilon) = 1$, for all $\epsilon >0$, and $\liminf_{n\to \infty} X_n = 0$
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1answer
33 views

Expected number steps in a Birth and Death Chain

I've been working on this problem for several days now, and I'm completely stuck on what to do next. Here's the problem: Consider a birth-and-death chain $X_t$ on $S = \{0, 1, 2, . . .\}$ with the ...
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0answers
47 views

Integrate with an other measure than Lebesgue

Let $\mu : \mathbb{B(\mathbb{R})} \to \mathbb{R}$ such as $$\forall A \in \mathbb{R},\, \mu(A) = \frac{1}{4}\int_A e^{-|x|} \, \mathrm{d}x \, + \frac{1}{2}\mathbb{1}_A(0)$$ Let $f : \mathbb{R} \to \...
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1answer
38 views

Proof of Borel-Cantelli Lemma explanation

I am trying to follow the proof of the Borel-Cantelli lemma as shown below: Could you please explain me how to go from: Thus $\sum \limits_{n = 1}^{\infty} 1_{A_n}$ is almost surely finite to: ...
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1answer
29 views

Convergence almost surely and in $\mathcal{L}^1$ for square of sequence of random variables

If $Y_n \rightarrow Y$ almost surely, then is $Y_n^2 \rightarrow Y^2$ almost surely too? If $Y_n \rightarrow Y$ in $\mathcal{L}^1$, then is $Y_n^2 \rightarrow Y^2$ in $\mathcal{L}^1$ too? I am ...
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1answer
24 views

Probability density function notation question

My question: What do the parts in red mean? I have never seen this notation before. Does this change what I have to show? I need to determine the coefficients $C_b$ and $C_c$ (if possible) so that $...
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0answers
17 views

Inferring bounds from joint typicality on three variables

Consider the following exercise from Cover and Thomas: And the given solution from the solutions manual: It is reasonably clear that these bounds are valid (one simply follows the counting argument ...
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0answers
19 views

Under what conditions does the median of an $n$-sample of a random variable remain below its expectation

Let $X$ be a real-valued random variable satisfying $\mathsf E X = 0$ and $\mathsf P(X\le 0)\ge\frac12$. In other words, $X$ is a (centralised) random variable whose median is below its mean. For ...
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1answer
24 views

If the median of a random variable is below its expectation, is this still true for a repeated sampling of this random variable?

Let $X$ be a real-valued random variable satisfying $\mathsf E X = 0$ and $\mathsf P(X\le 0)\ge\frac12$. In other words, $X$ is a (centralised) random variable whose median is below its mean. For ...
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0answers
13 views

Effect of linear transformation on Gaussian Complexity

I'm reading Vershynin's tutorial on High dimensional Probability (linked at the end). Vershynin also has a book, where this and related concepts like Gaussian width are also discussed. In it, he ...
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0answers
16 views

Avoidance of double loop of Monte Carlo integrations for nested expected values with indicator function [closed]

I have (potentially high dimensional) random variables $X$ and $Z$ and a constant $\rho \in [0,1]$. I can sample from $p(X,Z)$, $p(X|Z)$, $p(Z|X)$ , $p(X)$ and $p(Z)$, but don't know more about these ...
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1answer
58 views

Visual Representation of $X_n \sim U[0, \sin^{2}n]$

Trivially, I understand how the cdf/pdf of an uniform distribution with interval $[a,b]$ looks. But when there's a variable within my interval itself, say a sequence of RV's $X_n \sim U[0, \sin^{2}n]$...
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0answers
43 views

How did we get the probability of the site being reached?

"Since a site can be reached by one of its k links, its probability of being reached is kP(k)/(N< k >), where N is the number of nodes, P(k) is the fraction of nodes having degree (number ...

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