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.
44,342
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Reference for a good multidimensional portmanteau theorem
I need references for a good n-dim portmanteau theorem that includes the following equivalent assertions:
The sequence $(X_n)_n$ converges in distribution to a random vector $X$ of $\mathbb R^\nu$;
$...
1
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1
answer
30
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Distribution of indefinite integral of Brownian motion multiplied by an exponential decay term w.r.t time
Consider the indefinite integral
\begin{equation}
I=\int_{0}^{\infty}f(t)e^{-t} W_t\mathop{dt},
\end{equation}
such that $W_t$ is a standard Brownian motion. To ensure convergence, we'll suppose that ...
0
votes
1
answer
97
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On the coupon collector problem
Originally from Proposition 8 of Tao's note: https://terrytao.wordpress.com/2015/10/23/275a-notes-3-the-weak-and-strong-law-of-large-numbers/comment-page-1/#comment-682885.
Let ${N}$ be a natural ...
2
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1
answer
78
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Martingale example: a martingale on the partitioned unit interval
For an $L^1([0,1])$ function $f$, if we partition $[0,1]$ into intervals by setting
$I_j = [x_j,x_{j+1}]$ with $0 =: x_0 < x_1 < \cdots < x_N := 1$ and let
$\mathcal{F}$ be the sigma algebra ...
0
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1
answer
37
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How to expand this recurrence?
Here is a recurrence relation I have derived:
$$E(Y_{r+1}^2) = E(Y_{r}^2) + 2E(Y_1)E(Y_r) + E(Y_1^2)$$
How can I get it in terms of $E(Y_1)$ and $E(Y_1^2)$?
I've expanded the first few terms and I'm ...
3
votes
1
answer
57
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Poisson's distribution for probability
This question is more for learning purposes than anything however I came across this while trying to solving the following problem:
The odds of winning the lottery are 1 to 50000 million. This week, ...
0
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0
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45
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Positive Correlation and the FKG Inequality
I am reading Chapter 2.2 of Percolation by Grimmett on the FKG Inequality and I am having a hard time visualizing the meaning of this inequality when I interpret it the statement in terms of integrals....
2
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1
answer
75
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Do the stochastic integrals of two processes with same distribution also have same distribution?
I am trying to prove: Given two continuous functions $f,b$, and $X$ a stochastic process on $(\Omega,\mathcal{F},P)$, satisfying
$X_t=\int_0^tb(X_s)ds+\int_0^tf(X_s)dW_s$, if $\hat X$ is a stochastic ...
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14
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Control of Characteristic Function in Bounded Region?
I have a certain $n$-dimensional multivariate random variable $X$ I wish to study.
I can compute its characteristic function, and can show that
$$
\lVert \vec t\rVert_\infty \leq 1\implies \exp(-\frac{...
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1
answer
46
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Construction of a coupling of a sequence of Bernoulli random variables
Let $(X_n)_{n\in\mathbb{N}}$ be a sequence of Bernoulli distributed random variables defined on a probability space $(\Omega,\mathcal{F},P)$ and $(\mathcal{F}_n)_{n\in\mathbb{N}}$ be a filtration such ...
3
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Random Walk Problem in Feller Probability Theory vol 2 Chapter XII page 425
Let $X_{i}$ be iid discrete random variable with distribution $P(X_{1}=-1)=q$ and $P(X_{1}=i)=f_{i}$ for all $i\geq 0$.
Then consider the random walk $S_{n}=\sum_{i=1}^{n}X_{i}$ and $S_{0}=0$. Let $\...
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0
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38
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'Compactness' result for independent sigma algebras
Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space and $A, B_1, B_2, \dots \in \mathcal{F}$. Let $\mathcal{F}_n := \sigma(B_1,\dots,B_n)$ and $\mathcal{F}_\infty := \sigma(B_i : i \in \...
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Additivity + $\sigma$-subadditivity implies countably additivity for set functions on semirings?
Let $X$ be a set, $\mathcal{S}\subseteq \mathcal{P}(X)$ be a semiring on $X$, and $\mu: \mathcal{S}\rightarrow [0,\infty]$ with $\mu(\emptyset)=0$. Assume $\mu$ is finitely additive and countably ...
7
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3
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222
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Supremum of average of i.i.d. stochastic processes
Suppose that $(X_t)_{t\in [0, T]}$ is a stationary stochastic process with continuous sample paths such that $\mathbb E[X_t] = 0$ and $\mathbb E[|X_t|^2] < \infty$, and assume that
$$
S := \mathbb ...
0
votes
1
answer
191
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Prove that the number that the root of a given tree connect is decreasing in exponential.
Consider a regular rooted tree, each vertex has $d$ offsprings. Now for every edge there's a probability $p$ choosing it and probability $1-p$ not choosing it, and the root of the tree can connect ...
1
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1
answer
119
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Conditional bakery problem
This question is from QuantGuide(Bakery Boxes):
A bakery manager uniformly at random selects an integer k between 1 and 4, inclusive. He then chooses from k distinct desert types at his shop to create ...
1
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1
answer
119
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How to mathematically compute the transition probabilities in the secretary problem
My model for the secretary problem is as in (1) in this Mathstackexchange question. More precisely, I look at the unit interval $I=[0,1]$, and $\Omega = I^n \smallsetminus D$ where $D \subset I^n$ is ...
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1
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66
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How to find $P\left(\sqrt{(X_2 - X_1)^2 + (Y_2 - Y_1)^2} > X_2 + X_1\right)$ where $X_1, X_2, Y_1, Y_2$ are uniformly distributed from [0,5]
How to find $P((X_2 - X_1)^2 + (Y_2 - Y_1)^2 > X_2 + X_1)$ where $X_1, X_2, Y_1, Y_2 \sim U[0,5]$ and iid.
Rearranging the probability, we have:
$$P\left(\sqrt{(X_2 - X_1)^2 + (Y_2 - Y_1)^2} > ...
1
vote
2
answers
82
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How to generalise inner product to measures without densities
Let $(E, \mathcal{E}, \lambda)$ be a metric finite measure space, and let $\mu, \nu$ be finite measures with densities $f,g$ with respect to $\lambda$.
Then, I am interested in considering the ...
4
votes
1
answer
71
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Distribution theory for random projections
Suppose $v$ is a fixed vector in $\mathbb R^n$, and let $u\in S^{n-1}$ (unit sphere in $n$ dimensions; $S^{n-1}=\{x\in\mathbb R^n:\|x\|=1\}$) be uniformly generated. What is the distribution of $\...
0
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0
answers
30
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Rearranging sums (by multiplying out)
I know that to reverse the summation requires the double sum to either be absolute convergent or non-negative (but in the second case, it may diverge to $\infty$)
I was wondering for something like ...
1
vote
1
answer
54
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Expectation for non-negative r.v
suppose I have
$$\sum_{n=0}^{0} P(X=k)$$
Would this evaluate to $P(X=0)$ or $0$?
In general, from my understanding:
$$\sum_{n=0}^{k} P(X=k) = (k + 1)P(X=k)$$. Is this correct?
I ask in this scenario, ...
0
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0
answers
36
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Kullback-Leibler Divergence between a random variable and the product of its entries
Problem Statement
I'm currently working with a result about Kullback-Leibler divergence. Let $X$ be an discrete random variable taking values in $\mathcal{X} := \{0,1\}^p$, with $X = (X_1, X_2,...,X_p)...
0
votes
1
answer
87
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Which is the domain of the probability density function of $X^2$?
Let $X$ be a random variable with probability density function
$$f_X(x) = \frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}} \text{ for each } x \in \mathbb{R}.$$
We want to find the probability density function ...
2
votes
2
answers
99
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Asymptotic convergence of sampling distribution of the sample variance
Let's consider a set $\{X_i\}_{i=1}^N$ of $N$ i.i.d. random variables drawn from the distribution $P_X(x) = \mathcal{N}(\mu, \sigma^2)$. Define the variable
$$\hat{\sigma}^2 = \frac{1}{N} \sum_i (X_i -...
0
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0
answers
28
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Decomposing stationary point process $Y$ as sum of two point processes $X,Z$. Can $X$ be chosen stationary as well?
Consider the following setup. We are given two $[0,1]$-marked point processes on $\mathbb R$ or $[0,\infty)$, denoted $X,Y$. Denote that the ground processes (i.e. the point processes ignoring marks) ...
3
votes
1
answer
52
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Estimating the expected supremum of the absolute value of a Gaussian process
I'm currently reading a famous paper of Talagrand and fail to "easily see" that for a Gaussian process $(X_t)_{t\in T}$ and any $t_0 \in T$ one has
$$ E \sup_T | X_t | \leq E |X_{t_0}| + 2E \...
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0
answers
34
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Finite-Dimensional Elementary Cylinder vs. Finite-Dimensional Cylinder
I'm currently reading Theory of Probability and Random Processes by Koralov and Sinai. In Chapter 2, the authors state the definition of a finite-dimensional elementary cylinder and a finite-...
2
votes
0
answers
75
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Dimension of a measure space?
Let's consider $\Omega = [0,1]^2$ and functions $f, g \colon \Omega \to [0,1]$ defined via $f(x, y) = x$ and $g(x,y) = y.$ Clearly, $f$ and $g$ are independent random variables that have uniform ...
1
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0
answers
38
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Adjoint operator and random variable
Let $(\Omega,\mathcal{A},\mathbb{P})$ be a probability space and $\mathbb{D}$ a dense subset of $L^2(\Omega,\mathcal{A},\mathbb{P})$.
I consider a linear map $D$ from $\mathbb{D}\subset L^2(\Omega)$ ...
4
votes
0
answers
106
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Why are Gaussian measures seen as the standard measure for infinite dimensional spaces?
I'm learning about infinite dimensional probability, and most resources I've consulted so far motivate things by saying there is no infinite dimensional Lebesgue/translation invariant measure that ...
3
votes
1
answer
57
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Best strategy for choosing valued items. (variant of secretary problem) [closed]
Suppose we have to fill our bag with items and each item has a value $V_i \sim \text{Uniform}[a,b]$ where each random variable is iid. There are $N$ items in total but only a fraction $x$ of all items ...
0
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5
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86
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If you roll a dice four times, what is the probability to get two consecutive threes?
I was reviewing brainteasers and this one is stumping me for some reason.
Let X be a non-3.
you have 3 cases.
33XX, XX33, and X33X where we have the successes. I suppose you also have 333X and X333, ...
2
votes
1
answer
55
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Integrability of random variable
I am interested in proving that
$$
\lim_{x \rightarrow \infty} E(|X| 1_{|X| > x}) = 0 \Longrightarrow E(|X|) < \infty
\tag{1}
$$
where $X$ is a real-valued random variable.
In this blogpost the ...
0
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0
answers
46
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Which spherical symmetric graphs are there?
I am interested which "relevant" infinite graphs G=(V,E) (or in my use cases called networks) are out there that are spherical symmetric,
i.e. for a fixed origin 0 and each pair $v,w \in V$ ...
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1
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37
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Continuous random variable, positive part and probability of $P(X=0)$
We all know that for a continuous random variable the probability of the variable taking a specific value is zero. However, I have come across the following example.
Consider a standard normal random ...
0
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0
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17
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Bivariate random variable and transformation
Let $X=(X_1,X_2)$ and $Y=(Y_1,Y_2)$ be non-negative absolutely continuous random vector and if $\phi(X_j)=Y_j$, $j=1,2$, are one-one transformation then $$H[Y;\phi(t_1),\phi(t_2)]=H(X;t_1,t_2)-E[\log ...
0
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1
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169
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Prove or disprove there exists a Borel subset $F$ of $[0,1]$, such that $\mu(F)=a$ and $\mathcal{L}(F)=\frac{1}{2}$ [closed]
Given $\mu$ a Borel probability measure on $[0,1]$, and $\mathcal{L}$ the Lebesgue measure restricted on $[0,1]$. $$a:=\sup \{ \mu(E) : \mathcal{L}(E) = \frac{1}{2}, ~E \text{ is a borel subset of [0,...
2
votes
2
answers
91
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Independence of a sequence converging in probability implies independence of the limit
Let ${X_1,X_2,\dots}$ be a sequence of scalar random variables converging in probability to another random variable ${X}$. Suppose that there is a random variable ${Y}$ which is independent of ${X_i}$ ...
2
votes
2
answers
62
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variance of $n$ bernoulli trials and central limit theorem
I'm trying to figure out how to calculate the variance of $n$ Bernoulli trials. I need this information to use the central limit theorem to calculate some probabilities. here are my thoughts so far. I ...
1
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0
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31
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What does it mean to deviate macroscopically from the mean? And what does it mean to deviate in terms of fluctuations?
Let $\{X_{n}\}_{n\in \mathbb{N}}$ be a sequence of random variables. Suppose that $\{X_{n}\}_{n\in \mathbb{N}}$ fulfills certain regularity conditions described below. The random variables $X_{n}$ are ...
3
votes
1
answer
71
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Am I computing conditional expectations correctly?
I am building a working model and would like to know whether my computations so far look correct.
I am working with a generally bivariate distribution of two variables $\displaystyle X\sim [ 0,1]$, $\...
0
votes
1
answer
32
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Find regular conditional distribution
Question 1.1)
Let $X_1 = 1_{A}$ for $A\in \mathbb{F}$ and $X_2$ be a R.V. Where $X_2$ takes value in an arbitrary Borel Space. Let $X_1,X_2$ take values in $(A_1, \mathbb{A}_1)$, $(A_2, \mathbb{A}_2)$
...
3
votes
1
answer
118
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Hilbert space under a mean value inner product
I am looking for a (Hilbert) space of (real-valued) functions on $\mathbb{R}^n$ where the following map defines an inner product:
$$
(f,g) \mapsto \lim_{T\to\infty} \frac{1}{\mathrm{Vol}[-T,T]^n}\int_{...
0
votes
0
answers
25
views
Independence of Hilbert-space-valued random variable
Let $X,Y$ be two independent random variables taking values in a separable Hilbert space $U$.
Prove that $X$ and $Y$ are independent if and only if for all $u,v\in U$, $(X,u)$ and $(Y,v)$ are ...
1
vote
1
answer
47
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Nonstandard Probability Axiom Construction
I am working on extending the concept of a probability space to the surreal numbers, the main reason which being that I think that having a set whose measure is nonzero while the measure of each ...
0
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0
answers
19
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Help understanding periodic Markov chain
I'm interested in a Markov chain with, say, 5 kernels $P_1, \dots, P_5$ that cycle deterministically over the time steps of the chain. This is nominally a time-inhomogeneous chain.
According to this ...
0
votes
1
answer
75
views
Let $(A_k)_{k \in \mathbb{N}}$ a sequence of events.If $P(A_k)$ does not converge to $0$ then $\exists$ an event belonging to an infinity of $A_k$
Question:
Let $(A_k)_{k \in \mathbb{N}}$ a sequence of events. Prove that if $P(A_k)$ does not converge to $0$ then $\exists$ an event belonging to an infinity of $A_k$
My answer:
1-Let writte $A'_1=...
1
vote
0
answers
21
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Rate of convergence of conditional expectation
Let $X_{0}\sim N(0,1)$, $X_{1}$ another RV independent of $X_{0}$.
Let $X_{t}=(1-t)X_{0}+tX_{1}.$ We know that as $t\to1,$ $\mathbb{E}[X_{0}|X_{t}]\to\mathbb{E}[X_{0}|X_{1}]=E[X_{0}]=0.$
Can we ...
0
votes
1
answer
70
views
Convergence in probability and $E(|X_n|) \to E(|X|)$ implies convergence in mean
It is well known that in probability theory that, if $X_n$ converges in probability to $X$, the following are equivalent.
$X_n$ converges in mean to $X$
$E(|X_n|) \rightarrow E(|X|) < \infty$
$\{...