Skip to main content

Questions tagged [probability-theory]

For questions solely 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.

Filter by
Sorted by
Tagged with
0 votes
0 answers
10 views

probability of maximum of gaussian processes

Let $X$ be a $d$-dimensional Gaussian vector with mean $\mu$ and covariance matrix $\Sigma$. A standard Gaussian concentration result would allow me to write for any fixed $d$ dimensional vector $v$, $...
rostader's user avatar
  • 477
0 votes
0 answers
23 views

On two-sided Laplace transform in probability; how to show it is injective?

I am studying a theorem in probability that the Laplace transform of a (nonnegative) random variable determines the law of that random variable, which is equivalent to its injectivity. The book (...
psie's user avatar
  • 813
0 votes
0 answers
23 views

For $X_t$ a continuous-time Markov chain, show $M_t := \frac{f(X_t)}{f(X_0)}\exp\left(-\int_0^t \frac{Gf(X_s)}{f(X_s)} \; ds\right)$ is a martingale

Let $(X_t)_{t \geq 0}$ be a continuous-time Markov chain on finite state space $E$ with generator $G$ and let $f: E \to \mathbb{R}_{> 0}$ be a function. Then I would like to show that \begin{...
chessman's user avatar
0 votes
1 answer
15 views

simplified proof of Sanov's theorem ala Mohri et al.

I am trying to understand the following proof provided in "Mehryar Mohri, Afshin Rostamizadeh, and Ameet Talwalkar. Foundations of machine learning. MIT press, 2018. Appendix D.2 p.438 ". ...
v.tralala's user avatar
  • 299
1 vote
0 answers
13 views

Conditions for a Random Variable to Satisfy a Probability Bound on Boundary Points.

Let $X$ be a random variable supported on $\mathcal{X}\subset\mathbb{R}^{d}$, and let $\mathcal{X}$ be compact. Consider $ f $ as the probability density of $ X $. My question is: What conditions ...
Diego Fonseca's user avatar
-1 votes
0 answers
22 views

Elementary assumption question on Jaynes' Sum Rule derivation (2.48 - 2.50)

In Jaynes' book "Probability Theory: The Logic of Science", chapter 2.2, Jaynes shows how to derive the Sum Rule from "common sense". Even though I was able to follow most of the ...
lostintimespace's user avatar
2 votes
3 answers
89 views

Law of Total Probability and Conditional Probability

There is something that's a bit confusing for me in the topic of Conditional Probabilities. Assume we have the Universal Set, which is partitioned into $F$ and $F^c$. Now if we want to calculate the ...
Vacation Due 20000's user avatar
1 vote
0 answers
45 views

Show multivariate version of Slutsky's theorem: $Z_nX_n\overset{d}{\to}AX$

Let $Z_n$ be a $(d \times d)$-probability matrix and $A$ be a $(d \times d)$-constant matrix. If $X_n\overset{d}{\to}X$ and $Z_n\overset{p}{\to}A$, then $Z_nX_n\overset{d}{\to}AX.$ my attempt If $A$ ...
ytnb's user avatar
  • 600
2 votes
0 answers
24 views

Continuity of $L_p$ norms

I am solving Exercise 3.21 of section 4 in Erhan Cinlar's book - Probability and Stochastics. The question is: Fix a random variable $X$. Define $f(p)=\|X\|_p$ for $p\in[1,\infty]$. Show that the ...
Mshirur's user avatar
  • 33
2 votes
0 answers
26 views

Spaces of Probability Measures with Weak Convergence

I am not fluent in measure theory but have to address a question that I have come across. Let $X_i$, $i=1,2,\dots, K$ be compact metric spaces. So, for each $i$, by Prokhorov's theorem, the space of ...
GA-Student's user avatar
0 votes
0 answers
35 views

Different approach to calculating longest run of heads probability

In need of implementing a solution for this, I've read through this prior stack exchange discussion that describes an approach to the problem. I've also read this paper The Longest Run of Heads Author(...
koalacombatsystems's user avatar
0 votes
1 answer
25 views

Is it possible to decompose the joint conditions?

Given three events $E$, $A$ and $B$, is it possbile to decompose the joint-conditional probability $P(E \vert A, B)$, as a expression in terms of non-joint-conditional probabilities, and marginal ...
Phil's user avatar
  • 43
0 votes
0 answers
19 views

Does is hold that $E[f(X_t)1_{\{ s \geq T_1\}}| \mathcal F_{T_1}] = P_{t-T_1}(X_{T_1})1_{s \geq T_1}$ for $s\leq t$ and X is a strong Markov process

Let $X=(X_t)_{t\geq 0}$ be a homogeneous cadlag Markov process taking values in a finite state space $S$. Let $T_1$ be its first jump time and $f$ be a bounded measurable function. I would like to ...
mathnoob's user avatar
  • 169
1 vote
1 answer
36 views

On moment generating function of generalized gamma distribution

I'm reading An Intermediate Course in Probability by Gut. I am confused about a statement made concerning the generalized gamma distribution and its existence of a moment generating function. I quote: ...
psie's user avatar
  • 813
1 vote
0 answers
18 views

Data Augmentation Scan Order with Multiple Latent Variables

I am trying to extend the traditional definition of the data augmentation algorithm to more than one auxillary variable and with a random scan order. The Data Augmentation algorithm Consider a random ...
gtd3368's user avatar
  • 11
-1 votes
0 answers
25 views

Known correlations [closed]

A bit lost on where to get started on this problem — any thoughts would be greatly appreciated! I know X + Y is normally distributed with N(0, 2), not sure how to compare two normally distributed ...
navierstokes24's user avatar
0 votes
0 answers
42 views

Modelling of a Markov Chain via Subsequences

I'm running into an interesting problem, in that I've found out that for the Markov Chain I'm trying to model I can improve my optimization results if I model unique subsequences of the process. The ...
E.S.'s user avatar
  • 11
0 votes
1 answer
61 views

Convergence of the martingale $M_t = \frac{1}{\sqrt{1-t}}e^{-\frac{B_t^2}{2(1-t)}}$ to zero, as $t \to 1^-$

I am self-learning applied stochastic calculus from the text: A first course in Stochastic Calculus, by L.P. Arguin. Exercise problem 5.18 in my text, asks to prove the almost sure convergence of a ...
Quasar's user avatar
  • 5,450
1 vote
2 answers
107 views

Optimal strategy for uniform distribution probability game

There are 2 players, Adam and Eve, playing a game. The rules are as follows: $n$ and $d$ are chosen randomly. Adam samples a value $v$, distributed uniformly on $[0,n]$, and can either cash out $v$ or ...
jimsimons's user avatar
2 votes
1 answer
37 views

Definition of mixture of two distributions

What is the formal definition of a mixture of distributions? Usually one says that we flip a coin, and then choose a distribution out of the two to follow. Is it formally correct, then, to say that a ...
xyz's user avatar
  • 1,022
0 votes
0 answers
29 views

Median of a Poisson-Binomial distribution

Let $Z$ be a random variable following a Poisson-Binomial distribution with parameters $(p_1, \dots, p_n)$ such that $\sum_{i=1}^{n}p_i > \frac{n}{2}$. Consider the following two properties: -If $n$...
Ibra's user avatar
  • 175
1 vote
2 answers
52 views

Discretized Distributions on Rationals?

Consider the measure space $(\mathbb{Q}, 2^{\mathbb{Q}}, \nu)$, with $\nu$ being the counting measure. The space is then $\sigma$-finite. Is there any attempts made to define analogues of continuous ...
温泽海's user avatar
  • 2,497
5 votes
0 answers
96 views

Why are Martingales the Objects to Look at in Banach Spaces/Understand their Geometry?

Perhaps I need to read some more, but could anyone provide basic intuition as to why (Banach valued) martingales are the right things to look at when performing vector valued analysis/understanding ...
rubikscube09's user avatar
  • 3,925
-1 votes
0 answers
31 views

Distribution of two combined ML models

Due to the complexity of the problem, the problem was divided into two models: a stationary model and a model that corrects the stationary model for temporal effects, i.e. $X = X_{stat} + X_{time}$ ...
xbc68's user avatar
  • 1
6 votes
1 answer
363 views

"Peeling Technique" in Probability

So I am reading "Bandit Algorithms" by Lattimore wherein for one of the proofs he uses a technique called as "Peeling Device" which he says is a widely used tool in probability. I ...
tango's user avatar
  • 75
1 vote
1 answer
70 views

What's the probability that $n + k^2$ is a perfect square, for fixed $n$?

What's the probability that $n + k^2$ is a perfect square, for fixed $n$ where $k$ runs from $1$ to $N$, where we can take the limit as $N \rightarrow \infty$. We can write this as $$P\left({n}\right)...
Lorenz H Menke's user avatar
1 vote
0 answers
18 views

How restrictive is the single-crossing property for a mean-preserving spread?

I am trying to prove some results based on a mean-preserving spread of a distribution, and they basically depend on the distributions satisfying a single-crossing property, like the figure here: ...
radaic's user avatar
  • 11
0 votes
1 answer
32 views

Infinite Summation of Almost Sure Convergent RVs

Suppose we have random variables $X_{n,i}$ for $n\geq1,i=1,\dots,b_n$. Here we suppose $b_n$ is non-decreasing. We know that for each $i$, the sequence $X_{n,i}$ converges to $X_i$ almost sure. Now we ...
Percy Wong's user avatar
6 votes
1 answer
138 views

When is $\mathbb E[F(S)\mid S=s]= \mathbb E[F(s)]$ true?

Let $S$ be a discrete random variable on a set $\mathcal S$. Moreover, let $F(s)$ be another random variable (say in $L^1$) for each $s\in\mathcal S$. Now consider some $s\in\mathcal S$. I am looking ...
Joseph Expo's user avatar
0 votes
3 answers
147 views

Confused about a counting problem

This question is reproduced from a text by Sheldon Ross: Example 5k. A football team consists of $20$ offensive and $20$ defensive players. The players are to be paired in groups of $2$ for the ...
Vacation Due 20000's user avatar
2 votes
1 answer
51 views

Expected hitting time of 2-d brownian motion with $B_0 = (x_0, y_0)$

I wonder if it is possible to calculate the expected time of hitting a ball of radius $R$ while the initial position of the 2-d Brownian motion is not at the origin but $(x_0,y_0)$ inside the ball. I ...
LOREY CHU's user avatar
2 votes
2 answers
103 views

The distribution of $XY+(1-X)(1-Y)$ for $X,Y$ sampled uniformly from [0,1]

Let $X,Y$ be sampled uniformly from the interval $[0,1]$ and $Z=XY+(1-X)(1-Y)$. I would like to know the exact distribution of $Z$. I conjecture it should be uniform as well, but was not able to prove ...
user50394's user avatar
  • 429
0 votes
2 answers
31 views

What is the chance of two events happening after X attemps?

Let's say two favorable events have a 5% and a 10% chance of happening. Each time you try, you can only get one favorable event. How would you then calculate the probability that both events happened ...
Александр Ананьев's user avatar
0 votes
0 answers
42 views

Converge of iterated average posterior to high entropy distribution

Setup Assume $p_Y \in \Delta^n$ is a discrete probability distribution obtained by $p_Y=L_{Y|X}p_X$, where $L_{Y|X} \in \mathbb{R}^{n \times m}$ is an arbitrary likelihood (i.e, a column stochastic ...
backboltz37's user avatar
-1 votes
1 answer
29 views

Probability - Find the expected number of item throws into N containers until one of the container reaches k items [closed]

There are $N$ empty containers, which are unlabeled and are exactly the same. Each time, with equal probability, a ball is throwed into a random container. Q: What is the expected number of throws $f(...
Froest's user avatar
  • 11
0 votes
0 answers
25 views

Non-parametric and parametric statistical manifolds: Equivalence of score functions in tangent spaces [closed]

Below is the framework to give the manifold structure to the space $M_{\mu}=\{f \in L^{1}(\mu): f>0 , \mu a.e , \int f d\mu=1\}$ Statistical Model and its Topology: The Statistical Model and ...
Andyale's user avatar
  • 181
0 votes
0 answers
36 views

Convergence of element of random vector if the random vector converges weakly [closed]

Let $\left\{Z_n\right\}_{n=1}^{\infty}$ be stochastics proces, where $Z_n=\left(Y_n,X_n\right)$ for each $n\in\mathbb{N}$ and $\left\{X_n\right\}_{n=1}^{\infty}$ and $\left\{Y_n\right\}_{n=1}^{\infty}$...
Waney's user avatar
  • 624
1 vote
0 answers
19 views

Show convergence of $\sqrt N \int (\hat G_m - G)\,\mathrm d(\hat F_n - F)$ to zero in probability

Let $\hat F_n$ be the empirical CDF of a probability measure with CDF $F$. I want to prove that $$\sqrt N \int (\hat G_m - G)\,\mathrm d(\hat F_n - F)$$ converges to zero in probability as $\min(n,m)\...
Quertiopler's user avatar
-1 votes
0 answers
40 views

$Var(\bar{X}) \leq Var(X_i)/n$ for $X_i$ iid? [duplicate]

Is this inequality right for $\bar{X} = \frac{1}{n}\sum_{i=1}^nX_i$ and $\{X_i\}$ iid and if so why?
Sen90's user avatar
  • 453
1 vote
2 answers
152 views

Is such a property for conditional expectation true?

If $X$ and $Y$ are random variables defined on $(\Omega,\mathscr{F},\mathbb{P})$. Is it true that:For $\mathbb{P}$-a.s $\omega$, we have $\mathbb{E}[X|Y](\omega) = \mathbb{E}[X|Y=Y(w)]\quad$ which $\ ...
SiFei Yang's user avatar
-1 votes
0 answers
44 views

The moment of multivariate normal distribution

This is a computational problem I ran into while reading an article. I describe my question below: Let $\boldsymbol{Z}\sim N(0,I_{p\times p})$ and $\boldsymbol{y}_{i}\in \mathbb{R}^{p}$. We need to ...
Lop's user avatar
  • 1
1 vote
1 answer
40 views

Application of Wald's equation to bound a sum

I'm wondering if Wald's equality can be modified and applied as an inequality, given that we have $\forall i\; \mathbb{E}[X_i]\leq c$ with $c$ being a constant. More specifically, given that $N$ and $...
won_jun's user avatar
  • 21
0 votes
0 answers
23 views

Understanding of Probability density function and Radon–Nikodym derivative

In the measure-theoretic formalization of probability theory: A random variable $X$ is defined as a measurable function $X$ from a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ to a ...
MathAccount12's user avatar
-2 votes
0 answers
31 views

How to prove the number of zeros of a polynomial is the same as that of its linearisation with high probability in a small interval around zero? [closed]

Let $f(x)=a_0+a_1x+a_2x^2+\dots+a_nx^n$ be a real valued polynomial, i.e. $f: \mathbb R \rightarrow \mathbb R$. Let $a_0,a_1,\dots,a_n$ be independent standard normal random variables, with $a_0$ not ...
arechteron's user avatar
0 votes
0 answers
10 views

Bayesian network representation of uniform distribution over binary vectors of an even number of 1s

On slide $4$ of Jose Hernandez-Lobato's introduction to sum-product networks, he states in the first bullet point that some compact distributions, e.g. the uniform distribution over binary vectors ...
smoking_big_ole_doinks's user avatar
0 votes
0 answers
33 views

Averaging the expectations of stationary sequences

Consider the Hilbert space $$ \ell^2 :=\ell^2(\mathbb{Z}) = \left\{ a = (a_k)_{ k \in \mathbb{Z}} : \sum_{ k \in \mathbb{Z} } a_k^2 < \infty \right\}, \quad \langle a, b \rangle := \sum_{ k \in \...
Holden's user avatar
  • 1,557
1 vote
0 answers
32 views

Intuitive understanding of the definition of the $\sigma$-algebra of a stopping time $\tau$

I would like to better understand the basic intuition behind the definition of the $\sigma$-algebra of a stopping time $\tau$. Definition. Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space ...
Quasar's user avatar
  • 5,450
0 votes
1 answer
30 views

Question on the expected number of same color balls left in a urn

I'm working on a problem where I am given an urn with $a$ white balls and $b$ black balls. One ball at a time is selected randomly until there is only balls of the same color. I am asked to find the ...
Kham Bodrogi's user avatar
-3 votes
0 answers
23 views

Lyapunov Inequality [closed]

Lyapunov's Inequality states that: For a random variable $X$ and numbers $0<r<s<\infty$, $E(|X|^r)^{1/r} \leq E(|X|^s)^{1/s}$. I am working through a proof of the central limit theorem using ...
Alex's user avatar
  • 142
-1 votes
0 answers
31 views

independent random variables with rotationally invariant joint distribution are centered gaussians

This is from Foundations of Modern Probability by Olav Kallenberg, chapter 4 exercise 14 (chapter 6 in the latest edition): if $\xi$ and $\eta$ are independent such that $(\xi,\eta)$ has rotationally ...
Budge's user avatar
  • 1

1
2 3 4 5
900