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 ...

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3
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1answer
55 views

How to resolve the issue of two sequences converging to zero for $n, m \to \infty$?

My question is motivated by the following exercise in probability theory: Let $X_n \to X$ in probability and $X_n \geq Y$ a.s. Show that $X \geq Y$ a.s. I noticed that for all $n, m \in ...
0
votes
0answers
22 views

Weighting the data by the history

I have a input stream 3D data that comes every time frame. Each point is defined by 3D vector of x,y,z. There is a evaluation function [say f(x)] that computes if the point at time t is valid or ...
0
votes
0answers
14 views

Probability Density function of E = exp(2), with a random variable of zero mean and unit variance

I'm having difficulty wrapping my head around some of the basic concepts surrounding the question: "Suppose $d$ is a Gaussian random variable with zero mean and unit variance. What is the probability ...
6
votes
1answer
54 views

Martingale converges to the boundary

I asked an almost same question before and it is solved by considering adjacent $Z_n$ can not be far away and obtain a contradiction. However, if the setting is altered a bit, I wonder whether it is ...
2
votes
3answers
87 views

How to comprehend $E(X) = \int_0^\infty {P(X > x)dx} $ and $E(X) = \sum\limits_{n = 1}^\infty {P\{ X \ge n\} }$ for positive variable $X$?

Suppose $X$ is an integrable, positive random variable. Then, if $X$ is arithmetic, we have $E(X) = \sum\limits_{n = 1}^\infty {P\{ X \ge n\} }$ and if $X$ is continuous, we have $E(X) = ...
6
votes
1answer
54 views

Convergence of $n^{-\gamma}T$ where $T$ a hitting time for uniform rvs, can I use CLT?

Let $X_1,X_2,\dots$ be iid uniform on $\{1,\dots,n\}$ and define $T=\inf\{k:X_k=X_r \text{ for some }r<k\}$. The objective is to figure out when $n^{-\gamma} T$ converges weakly to some ...
2
votes
1answer
15 views

proving converse of equality involving distribution of minimum observation

Suppose constants $v_n$ are such that: $\lim_{n \to \infty} nF(v_n) =d \in [0,\infty]$ where F is the Cumulative distribution function of $X_i \sim iid$ random variables. Then the question is to show ...
1
vote
1answer
43 views

Can a biased physical random source be post-processed to control the bias?

Let $X_i$ with $i\in\mathbb N$ be a sequence of independent 6-ary random variables with distribution $\operatorname{Pr}(X_i=e)=p^e_i$ where $e\in\{1,2,3,4,5,6\}$ and $\sum_{e=1}^6p^e_i=1$. Let's ...
4
votes
3answers
530 views

Expected Value of Local Maxima and Local Minima

Recently I came across this question: Given a random permutation of integers 1, 2, 3, …, n with a discrete, uniform distribution, find the expected number of local maxima. (A number is a local maxima ...
1
vote
1answer
517 views

Problem on continuous probability distribution

Problem:We pick two random numbers, x and y, between 0 and 2. What is the probability that x*y<1 AND y/x<1. I am familiar with continuous probability distributions for one variable, but it ...
5
votes
1answer
56 views

Basic question about $\sigma$-fields

Billingsley's text "Probability and Measure" has the following exercise problem: Problem 2.5(b): For a collection of sets $\mathcal{A},$ let $\mathcal{F}(\mathcal{A})$ be the intersection of all ...
0
votes
1answer
42 views

Convergence of expected values and integrability

I'm trying to prove a result for a homework assignment, and I got to a point that if the following result is true, then the result follows. Let $X_n$ be a sequence of positive random variables and ...
5
votes
1answer
48 views

Multivariate normal density function of function of random variable

Let $X_1,\dots,X_n$ be i.i.d random variables and $g$ be a symmetric function such that $$g(X_i,X_j)\sim N(\mu,\sigma^2)$$ for all $1\le i<j\le n$. I wish to know the density function of the joint ...
3
votes
1answer
36 views

If ${a_i} \to 0$ and $\{ {X_i}\} _{i = 1}^\infty $ is a sequence of iid random variables with zero mean, does ${a_i}{X_i} \to 0$ almost surely?

My problem is slightly more specific than the title of this question: Let $0 < \beta < 1$ and let $\{ {X_i}\} _{i = 1}^\infty $ be a sequence of i.i.d. random variables with $E({X_i}) = 0$. In ...
4
votes
0answers
60 views

Random variables that span copies of $\ell_p$

Consider the coin-toss measure $\mu$ on $\{0,1\}^\mathbb{N}$. Within this framework it is easy to construct a sequence of independent, symmetric Bernoulli random variables. Indeed the point-evaluation ...
3
votes
0answers
43 views

Generalizing the pull-out property in conditional expectations

Let $(\Omega, \mathscr{F}, P)$ be a probability space and $\mathscr{G}$ a sub-$\sigma$-algebra of $\mathscr{F}$. If $X$ and $Y$ are (integrable) random variables with $X$ being ...
2
votes
1answer
46 views

Intuition for probability density function as a Radon-Nikodym derivative

If someone asked me what it meant for $X$ to be standard normally distributed, I would tell them it means $X$ has probability density function $f(x) = \frac{1}{\sqrt{2\pi}}\mathrm e^{-x^2/2}$ for all ...
2
votes
2answers
55 views

Probability distribution of number of waiting customers in front of a counter

The number of customers arriving at a bank counter is in accordance with a Poisson distribution with mean rate of 5 customers in 3 minutes. Service time at the counter follows exponential distribution ...
1
vote
3answers
133 views

$\mathrm E [X \mid X=x] = x$?

I've gotten so caught up in measure-theoretic probability that I'm actually having trouble showing this simple result. Let $X$ be an integrable random variable. Then $$ \mathrm E[X \mid X=x] = ...
2
votes
3answers
70 views

probability of $k$ boxes contain exactly $1$ ball

Occupancy problem with balls and boxes. Suppose there are $N$ balls and $M$ boxes. The balls are thrown to the boxes at random. What is the probability of $k$ boxes contain exactly $1$ ball? where ...
1
vote
1answer
38 views

Proof of a classical Theorem of Martin-Löf on complexity dips for Kolmogorov complexity,

I have a question on the first Theorem from the article Complexity of Oscillations in Infinite Binary Sequences by P. Martin-Löf, which could be downloaded from the publisher or from here. Theorem ...
3
votes
0answers
49 views

A Markov Chain probability, conditioned on a random time.

My question: Upon reading theory about diffusion processes, i came across an argument which i believe simplifies to this: Say we have a Borel measurable set $A$ (if it matters you can set $A=\lbrace ...
2
votes
0answers
35 views

Expected value of multidinesional symmetric function is zero

Does anybody know a simple proof of this statement or reference to such proof? Statement Let $h: R^n \to R$ be a bounded function, symmetric in its arguments, i.e. for any permutation $\pi$ of set ...
2
votes
2answers
54 views

What conditional independence theorem is being used here

In stanford's machine learning lecture 1, linear regression is defined on page 11, section 3 as: For $i = 1, \ldots, m$, $y^{(i)} = \theta^T x^{(i)} + \epsilon^{(i)}$, where $\epsilon^{(i)}$ are IID ...
1
vote
1answer
45 views

How to use induction to show that $\delta(\mathcal G_1), \ldots, \delta(\mathcal G_n) $ are independent?

I have proven that if the systems $\mathcal G$ and $\mathcal H$ are independent then so are the Dynkin systems $\delta(\mathcal G)$ and $\delta(\mathcal H)$. Now I'd like to generalize it to $n$ ...
1
vote
2answers
37 views

Find the following probability

A bowl contains 16 chips, of which 6 are red, 7 are white and 3 are blue. If four chips are taken at random and without replacement, find the probability that there is at least 1 chip of each colour. ...
3
votes
5answers
121 views

Uncountable increasing family of $\sigma$-algebras

Could someone give an example of what an uncountable increasing family of $\sigma$-algebras $\{\mathcal{F}_t\}_{t\geq 0}$, $(\mathcal{F}_s \subset \mathcal{F}_t$ for $s<t)$ might look like? For ...
0
votes
1answer
329 views

Characteristic function and probability density function: Fourier or Inverse Fourier?

I have come across two contradicting definitions of characteristics function (CHF). In wikipedia CHF is defined as the inverse Fourier transform (FT) of probability density function (PDF) and at some ...
1
vote
2answers
34 views

Probability of being away from mean for independent random variables

Let $X_1,X_2,\ldots,X_n$ be independent random variables drawn uniformly from $[-1,1]$. The (weak) law of large numbers says that ...
-1
votes
1answer
55 views

Show that $\mathbb{P}(\tau_{0}>T)\approx\frac{1}{\sqrt{T}}$ where $\{ B(t) : t\geq 0\}$ is a linear brownian motion started at $B(0)=1$ [on hold]

I'd appreciate if someone could provide me with a solution for the following problem: Let $\left\{ B\left(t\right)\thinspace|\thinspace t\geq0\right\}$ be a linear brownian motion started at ...
0
votes
0answers
32 views

Show that the following set function is not a probability set function

If the sample space is $\mathfrak{C} = \{c : -\infty < c < \infty\}$ and if $C \subset \mathfrak{C}$ is a set for which the integral $\int\limits_C e^{-|x|}dx$ exists, show that this set ...
1
vote
1answer
36 views

Good introductory book for Probabilistic Number Theory

I have a decent high school knowledge of Elementary Number Theory and it is also a subject I love to study. I have a good background in Real Analysis (not Complex Analysis) and Abstract Algbera. I ...
1
vote
2answers
78 views

Condition implying tightness of sequence of probability measures

A sequence of probability measures $\mu_n$ is said to be tight if for each $\epsilon$ there exists a finite interval $(a,b]$ such that $\mu((a,b])>1-\epsilon$ For all $n$. With this information, ...
2
votes
2answers
400 views

Probability of inequality between random variables

In order to prove a theorem in my research, I would like to use a lemma on basic probability theory, but I don't know if it is correct. For three random variables $X,Y$, and $Z$ not necessarily ...
1
vote
1answer
53 views

Abstract enunciation of the Good Set Principle in measure theory

I am struggling with the Good Set Principle in Measure Theory, so is this rephrasing in the most abstract way ultimately correct? Good Set Principle Let $(X, \Sigma)$ be a measurable space. ...
1
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0answers
40 views

Normalized hit times of a simple RW converge in distribution to hit times of standard Brownian Motion

I would appreciate some hints or guidance towards solving the following exercise: Let $\left\{ S\left(j\right)\thinspace:\thinspace j=0,1,...\right\}$ be a simple random walk on the integers ...
-2
votes
1answer
24 views

The probability that two randomly selected $2$ year old male feral cats will live to be $ 3$ years old is? [on hold]

The probability that a randomly selected $2$ year old male feral will live to be $3$ years old is $0.82666$. (a) what is the probability that two randomly selected $2$ year old male feral cats will ...
0
votes
1answer
37 views

If independent r.v. converge in probability to a constant, do they converge almost surely?

I've seen several examples when a sequence of r.v. converge in probability but not almost surely, yet none of them had the sequence to be independent. Would additional conditions of independence and ...
2
votes
1answer
42 views

Does convergence in probability preserve the weak inequality?

Suppose I have two sequences of random variables $\{x_n\}$ and $\{y_n\}$ such that $x_n\leq y_n$ and $\text{plim}\;x_n=L_x$ and $\text{plim}\;y_n=L_y$, can I say $L_x\leq L_y$ (almost surely)? Does ...
2
votes
0answers
69 views
+50

Find a probability density

I am going through a paper trying to understand all the single steps, but I got stuck. I need to calculate $$p(x+\delta t) \mid x(t), t)= \int p(x(t+\delta t) \mid \mu , x(t), t)p(\mu\mid x(t), t) ...
2
votes
1answer
31 views

Interpretation of conditional expectation as a random variable

I have a couple problems understanding the conditional expectation as a random variable. Consider the fair dice roll as a random variable $X$. Let $C$ be the event that the dice shows a one and ...
0
votes
1answer
29 views

How to change the measure to make a non standard normal random variable standard normal

Given a probability space $(\Omega, \mathcal{F}, P)$ consider a standard normal random variable $X$. Let $\tilde{X} = X + c$, $c \in \Bbb R$. Now consider the following probability measure ...
1
vote
0answers
29 views

Expectation of a continuous function

Can someone help with the following? I have a continuous function $g: A_i \times A_{-i} \to \mathbb{R}^k$, and a probability measure $\mu \in \Delta(A_{-i})$. We can let $A_i=\mathbb{R}^n$ and ...
0
votes
1answer
415 views

Product of two exponentially distributed random variables

I am trying to find the close form expression of probability distribution of $Z$ such as $Z=X_1X_2$ where $X_1$ and $X_2$ are two independent exponentially distributed variables with PDF ...
1
vote
0answers
91 views

Proving $\sigma$-additivity and interchanging order of summation/integration just because positive

Let $\Omega = {\omega_1, \omega_2, ...}$ be some countable set. Let $\mathfrak{F} = 2^{\Omega}$. Consider a sequence {$p_n$} in [0,1] s.t. $\sum_{n=1}^{\infty} p_n = 1$. Define P: $\mathfrak{F} \to ...
-1
votes
0answers
27 views

Distribution Problem based on unknown function [on hold]

I got struck at this problems as Function is not given. Any help will be appreciated
2
votes
0answers
51 views

What is the General Central Limit Theorem?

General Central Limit Theorem says: Let $ \{(X_{n,j} , 1 ≤ j ≤ n), n ≥ 1\} $ be a triangular array of rowwise independent random variables, set $ S_n = \sum_{j=1}^n X_{n,j}, s_n^2 = \sum_{j=1}^n ...
7
votes
1answer
130 views

When does $\sum_{i=1}^{\infty} X_i$ exist for random sequences $\{X_i\}_{i=1}^{\infty}$?

Suppose $\{X_1, X_2, X_3, \ldots\}$ is an infinite sequence of random variables such that $E[X_i]=0$ for all $i$, and $E[X_iX_j]=0$ whenever $i \neq j$. Further suppose the variances $\sigma_i^2 = ...
0
votes
0answers
23 views

Conditional Expectation with respect to two Random Variables

Consider the quantity $$ \mathrm E[U \mid S,T]. $$ Is this shorthand for $$ \mathrm E[U \mid \sigma(S) \otimes \sigma(T)]? $$ If so, the defining characteristics are that $\mathrm E[U \mid S,T]$ is ...
3
votes
1answer
23 views

Almost surely on a subset

I often meet in the literature on probability theory statements like "$\phi$ almost surely on $S$", where $\phi$ is a property and $S$ a subset of the underlying complete probability space $(\Omega, ...