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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-...

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Influence function of $\phi (F,G) = \int F \; \text{d} G$

I am reading Asymptotic Statistics and come across following result: Let $F, G$ be two probability measures. Define $\phi (F,G) = \int F \: \text{d} G$. The influence function is $$ \frac{\text{d} }{\...
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12 views

Laplace transform and tail probability

Let $X \ge 0$ be a non-negative random variable. I would like to know if the following statements are equivalent: $$ \lim_{\lambda \to 0^+} \frac{\mathbb{E} \left[X e^{-\lambda X}\right]}{\log (1/\...
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7 views

Why does this tight sequence of random variables also converge in probability?

Definition of convergence in probabilty We say that $X_n$ converges in probability to zero, written $X_n = o_p(1)$, if for every $\epsilon> 0$, $$ P(| X_n | > \epsilon) \rightarrow 0, \...
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0answers
33 views

Why the probability is concentrated on $\{0, \infty\}$

While I was reading Williams' "Diffusions, Markov Processes and Martingales" I found the following fact: Let $B_t$ be a Brownian Motion. Then $\mathbb{P}\left(\sup_tB_t=\infty\right)=1$ In the ...
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1answer
12 views

Show that $\frac{1}{n-1}\sum_{i=1}^n(X_iY_i-\overline{X}\overline{Y})$ is an unbiased estimator of $\text{Cov}[X,Y].$

Assume that $(X_1,Y_1),...,(X_n,Y_n)$ is a sample on a two-dimensional random variable $(X,Y)$ and that $E[X^2], \ E[Y^2]$ and $E[XY]$ are all finite so that the variances and covariance are well-...
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1answer
23 views

Inner-product of two Gaussian distributions.

Consider $2$ Gaussian PDFs $f_1$ and $f_2$ given as, $\log f_i({\vec x}) = - \frac {1}{2} \left [ ({\vec x} - {\vec \mu}_i)^T\Sigma_i^{-1}({\vec x} - {\vec \mu}_i) + \log \left ( (2\pi)^n \det(\...
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1answer
17 views

Sum of i.i.d. sequence converging to Normal Distribution

I provide most of the results without explicit calculations, but question is fairly short at the end. Let $X_i$ be i.i.d. with density $f(x) = |x|^{-3}$ for $|x|>1$ and $0$ else. Claim is; \...
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0answers
8 views

Coupling via push-forward from a source space

Consider a measurable function $g$ mapping a probability space $\left(\Omega,\mathcal{F},\mu\right)$ to a product measurable space $\left(T,\mathcal{T}\right)$ with cartesian product $T = X \times Y$ ...
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0answers
22 views

Lower bounding the expectation of an infimum

Consider the following sum: $$ \inf_{u \in \mathbb{S}^{n-1}} \frac{1}{m} \sum_{i=1}^m \left| \langle a_i, u \rangle \right|, $$ where $\mathbb{S}$ denotes the unit sphere, $$ \mathbb{S}^{n-1} := \...
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1answer
42 views

Interview Question on Probability: A and B toss a dice with 1 to n faces in an alternative way

A and B toss a dice with 1 to n faces in an alternative way, the game is over when a face shows up with point less than the previous toss and that person loses. What is the probability of the first ...
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14 views

Measure theory proof of the change of variables theorem for probability density functions

I'm preparing for a measure theoretic probability course, and I have some questions about some things that confuse me. First of all, I'm kind of confused as to how you prove the multivariable case of ...
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3answers
58 views

Why doesn't the Law of Large Numbers hold here?

This is from an old homework assignment of mine, which I've since turned in. Say you have an independent sequence of R.V.s such that $\mathbb{P}(X_n= 2^n) = \frac{1}{2^n} = 1 - \mathbb{P}(X_n = 0)$....
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10 views

Variance of a conditional distribution of a multivariate gaussian distribution

I am reading about multivariate Gaussian distributions and ran into the following equation about condtional distributions which says that any conditional distribution is also Gaussian. $$ p(x,y) = \...
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2answers
28 views

The probability of two independent random variables?

Let $X, X'$ be independent with $X \sim p(x)$, $X' \sim r(x)$ for $x, x' \in X$. I don't understand this equation: $\sum p(x)r(x)=Pr(X=X')$ What is intuitive to me is if $X \sim p(x)$, $X' \sim ...
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1answer
22 views

Convergence in probability, mean and almost surely

$X$, $Z_n$, $Y_n$ are independent random variables, where X is integrable, $Y_n$ has Bernoulli distribution $b(1,n^{-2})$ , $Z_n$ has Poisson disribution with parameter $n^2$. I need to check ...
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1answer
29 views

Examples non experiments in probability

I'm reading a book on probability theory and they say that an experiment is Any procedure that has not got a pre-determined outcome In the Wikipedia page they, instead, that an experiment should: ...
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33 views

How to compute the conditional probability of P(X|X^2) [on hold]

If X ~ N(0, 1), How to compute the conditional probability of P(X|X^2), is it equal to 1?
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2answers
26 views

Markov Property and FDDs

Let $X,Y$ be two discrete time $\mathbb{R}^n$-valued stochastic processes with the same finite dimensional distributions. It may be that $X,Y$ are defined on two different probability spaces. Now, if $...
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0answers
18 views

The capture event

Let $\Omega$ be a probability space. The capture event $\omega \in \Omega$ is given when the defender and attacker are located at the same state at time, i.e, $$ \sum_{j=1}^{N}\chi (\omega: s^{\ell}(...
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1answer
22 views

Convergence of a sequence of r.v. in probability and almost surely

Let $(X_k)_{k=1}^{\infty}$ be a sequence of independent r.v. where $\mathbb{E}X_k=0$ and $\mathbb{E}X_k^2=k$. Consider $Y_n=\frac{1}{n^2}\sum_{k=1}^{n}X_k$. Show $Y_n$ converges to 0 in probability ...
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0answers
41 views

Does strong convergence in probability of conditional probabilities imply countable additivity on open sets?

Please note. This question is a continuation of another question of mine. I'd really prefer hints to complete answers for this one. Let $\Omega = \{0,1\}^\mathbb{N}$, and let $\mathcal{B}$ be the ...
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36 views

Gaussian process properties

I am reading Gaussian Processes (GP) for Machine Learning (http://www.gaussianprocess.org/gpml/). The GP definition is usually like this: "A Gaussian process is a collection of random variables, any ...
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3answers
57 views

Conditional Expectation of Bernoulli R.V.

Let $X_1, X_2,\ldots, X_n$ be iid bernoulli r.v. with parameter $p$. Let $S=X_1+\cdots+X_n$ and $Y=X_1X_2$. Compute $\mathbb{E}(Y\mid S)$. I know that $\mathbb{E}(X_1\mid S) = S/n$. So If I could ...
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1answer
31 views

Defining unordered arrival times in Poisson process.

I am looking to precisely define the 'unordered' arrival times in a Poisson process. Say I have a one-dimensional, unit rate, Poisson process $(K_t)_{t\geq 0}$ and let the ordered arrival times be $(...
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2answers
34 views

Conditional expectation of $\mathbf1[X_1=0]\mid X_1+\cdots+X_n$ where $X_i$'s are i.i.d Poisson RVs

Let $X_1,\cdots,X_n$ be i.i.d Poisson random variables with mean $\mu$. Let $S=X_1+X_2+ \dots + X_n$. Set $Y=\mathbf{1}[X_1=0]$. Show that $$\mathbb{E}(Y\mid S)=\left(1-\frac{1}{n}\right)^S$$ $$\...
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3answers
29 views

How do I find the probability of occurrence of 2 events in this question?

What is the probability of occurrence of at least two of the three events if $P(A) = 0.38, P(A \cap B \cap C) = 0.1, P(A \cap B' \cap C') = 0.17$ and $ P(A'\cap B \cap C) = 0.12 $ I'm guessing ...
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1answer
23 views

Polya's distribution in Durret's Probability Theory book: A typo?

In Durret's book there's this theorem. But then in an example later in the book, we find the following. I think this is a typo, where the density has been switched with the Ch.f. However, we later ...
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2answers
27 views

Is $\lim_{n\to\infty} \mathbb{P}(A_n) = \mathbb{P}(\cup_n A_n)$?

On the bottom of page 22 of the lecture notes of R van Handel, part 4 of Lemma 1.2.1 states that, for a probability space $(\Omega,\mathcal{F}, \mathbb{P})$, \begin{equation} A_1 \subset A_2 \cdots \...
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3answers
44 views

Sampling from product of exponential distributions

I have a distribution who's moment generating function is the product of two exponentially distributed variables moment generating functions. If I wanted to generate samples from the distribution, ...
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2answers
55 views
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1answer
24 views

Finding the conditional entropy on the sum of independent random variables

I have two independent random variables $X_1$ and $X_2$. I want to find the differential entropy defined as $$H(X_1+X_2\mid X_1)=\int_{X_1} \int_{X_2} p_{X_1,X_2}(x_1,x_2)\log\left(\frac{1}{p_{X_1+X_2\...
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1answer
26 views

“The $\sigma$-additivity for measures is the equivalent of axiom of continuity for $\mathbb{R}$”. I read this in my notes.

I'm studying the very basics for probability theory and starting from axiomatic definition in the notes I have available. After some pages I find the quote in the title, which says: "The $\sigma$-...
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2answers
20 views

Convergence as surely Bernoulli r.v

Let $X_1,X_2, . . .$ be an i.i.d. sequence of Bernoulli random variables with parameter $p \in (0,1)$. Set $T=\sum_{i=1}^{\infty}\mathbb{1}[X_i=1]$. Prove that $P(T=\infty)=1$. Set $S_n=\...
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2answers
24 views

Question on convergence of a random sequence after conditioning on a specific event

Consider a random process $\{X_n\}_{n=1}^\infty$ where each random variable is continuous. Assume that the sequence of random variables converges almost surely to $\alpha >0$ i.e., \begin{equation}...
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1answer
45 views

Prove the inequality $m(x \in \mathbb{R}^d: \Vert x \Vert >r)\leq \frac{1+r^2}{r^2}\int_{\mathbb{R}^d}\frac{\Vert x\Vert^2}{1+\Vert x \Vert^2}m(dx)$ [on hold]

When I read the lecture notes about the Levy process, I get stuck in proving the inequality: $m(x \in \mathbb{R}^d: \Vert x \Vert >r)\leq \frac{1+r^2}{r^2}\int_{\mathbb{R}^d}\frac{\Vert x\Vert^2}{...
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0answers
11 views

Renewal equation with monotone density and hazard rate

Let $f$ be a density function with support $[0,\infty)$ and let $h$ be the associated hazard rate $$h(x) = \frac{f(x)}{\int_x^\infty f(s)\text ds}.$$ I conjecture that if f is decreasing then, if $h$...
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0answers
28 views

Help in understanding one detail of the proof of “An Elementary Proof of a Theorem of Johnson and Lindenstrauss” by Dasgupta and Gupta

I currently go through An Elementary Proof of a Theorem of Johnson and Lindenstrauss by Sanjoy Dasgupta and Anupam Gupta https://cseweb.ucsd.edu/~dasgupta/papers/jl.pdf and I got pretty confused. The ...
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1answer
41 views

Isoperimetric inequality for non-spherical multivariate Gaussian

Disclaimer: Sorry in advance, if the question is not very reasonable. Recently (like a few days ago...), I've started studying isoperimetric inequalities, and my thoughts on the subject are rather ...
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1answer
35 views

Is $F(x)=\frac{1}{\pi}\tan^{-1}(x),-\infty<x<\infty$ a distribution function?

Check whether $F(x)=\frac{1}{\pi}\tan^{-1}(x),-\infty<x<\infty$ is a distribution function? What I attempted:- There are three conditions for a function $F(x)$ to be a distribution function. \...
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1answer
25 views

Using the normal approximation, find the probability that the sum of the face values of the 100 trials is less than 300

A six sided die is rolled 100 times. Using the normal approximation, find the probability that the sum of the face values of the 100 trials is less than 300. This question is the second part to a ...
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31 views

Prove a system is not logically complete

Suppose $W = I \cdot X + ( 1 - I ) \cdot Y$, where $X$ and $Y$ are well defined real valued random variables on some probability space. Let $G$ be a fixed real number. Let $I$ equal $1$ if $W <G$ ...
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2answers
38 views

What is the cdf for a partially non-continuous pdf?

Suppose there is a pdf/pmf (?!) which places an atom of size 0.5 on x = 0 and randomizes uniformly with probability 0.5 over the interval [0.5,1]. Such that... \begin{equation} f(x)= \...
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1answer
27 views

Chebyshev's inequality application and convergence - practical example

Let $W_n$ be a random variable with mean $\mu$ and variance $\frac{b^2}{n^{2p}}$, with $p>0$ and $b$ and $\mu$ constants. Show that $$ \lim_{n\to\infty} P(|W_n-\mu| \leq \epsilon) = 1 $$ ...
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1answer
21 views

Definitions and properties of limits of stochastic processes in continuous time

Many books on stochastics take ample time to explain what it means for a sequence of random variables to convergence a.s., in $L_p$, in probability, in distribution, what $\limsup$ and $\liminf$ mean, ...
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1answer
25 views

Conditonal probability that person C won the game?

I had this problem on an exam a few weeks ago: The persons $A$, $B$ and $C$ are playing a game. What the game is about is of no concern, the only thing we need to know is that the person who obtains ...
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1answer
27 views

Probability Whole Sample Below Expectation

Let $X_1,X_2,\ldots,X_n$ be i.i.d real-valued random variables with finite variance $\sigma^2>0$. Can we non-trivially upper bound the probability $$ \mathbb{P}\bigl(\max_{1\leq i\leq n} X_i < \...
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0answers
42 views

About Conditional Probability, Circular Argument?

The following problem is about conditional probability. We want to know the value $P(A \cap B)$ by the formula $P(A \cap B) = P(A) \times P(B|A)$. The definition of $P(B | A)$ is $\frac{P(A \cap B)...
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0answers
30 views

Reconciling definitions of large deviations principle

I am reading some notes on the Large deviations principle and I want to reconcile the definitions I've seen in the abstract measure theoretic framework and one that is used on an introduction to the ...
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0answers
24 views

Maximum difference of Poisson process

I am trying to understand this remark in a paper by Bollobás and Riordan: Let $X_1, X_2,\dots$ be the points of a Poisson process on $[0, \infty]$ with rate $m$, so, setting $X_0 = 0,$ the ...
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1answer
38 views

Show as surely convergence (Borel Cantelli Lemma)

Let ${X_n}$ be independent r.v. taking values $n^2-1$ and $-1$ with $P(X_n=n^2-1)=1/n^2$ and $P(X_n=-1)=1-\frac{1}{n^2}$. Show that if $S_k=X_1+...+X_k$ then $S_k/k$ converges to -1 a.s but $E(S_k)=0$....