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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17 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 ...
2
votes
1answer
11 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 ...
2
votes
1answer
52 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 ...
4
votes
1answer
55 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 ...
4
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0answers
59 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
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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 ...
5
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1answer
47 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
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0answers
47 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 ...
1
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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 $n$ ...
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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. ...
1
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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 ...
2
votes
2answers
54 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 ...
0
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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 ...
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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 ...
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 ...
1
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1answer
34 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 ...
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 ...
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 ...
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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 ...
3
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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
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 ...
0
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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 ...
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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 ...
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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 ...
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, ...
0
votes
1answer
41 views

Probability of Union of Events in a Probability Product Space By Counting Event Size

This question is about probability of union of events in a probability product space. Let’s say a fair die is thrown twice and we’re interested to find out probability of getting face value one in 1st ...
0
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0answers
29 views

Interpretation of integral as ratio of joint and conditional densities?

A common exercise in Bayesian statistics is specifying a prior $p(\theta)$ on some parameter $\theta$. We then observe a collection of data $D=(X_1,\dots,X_N)$, the distribution of which is ...
2
votes
2answers
35 views

Is there accepted notation for the pushforward measure that doesn't mention $\mathbf{P}$?

Let $(\Omega,\mathcal{F},\mathbf{P})$ denote a probability space, $(S,\mathcal{M})$ denote a measurable space, and $X : (\Omega,\mathcal{F},\mathbf{P}) \rightarrow (S,\mathcal{M})$ denote a measurable ...
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votes
2answers
92 views

What do you call this thing in probability theory? [closed]

I have studied it before but I forgot the name. It is like when the possiblity of something happens is so small, but you created the experience so so many times, then the probability of that thing to ...
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 = ...
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 ...
1
vote
1answer
24 views

What is the countable product sigma algebra of powersets of a countable set $E$? The powerset of the space of all sequences in $E$ or not?

Let $E$ be a countable set with power set $\mathcal{P}(E)$. $(E,\mathcal{P}(E))$ is a measurable space. Let $E^{\mathbb{N}}$ be the space of sequences in $E$ and $\mathcal{P}(E)^{\mathbb{N}}$ the ...
2
votes
1answer
22 views

Probability in knockout games.

Suppose in a knockout tournament 32 players p1 , p2 .....p32 participate. In each round players are divided into pairs at random and winner goes to the next round. If p5 reaches semifinal what is ...
0
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0answers
18 views

Calculating Variance of payment in patterns of balls.

We have five different bags labeled from 1 to 5 and several colored balls. There are 9 different possible colors. We know how many balls of each color there are in each bag. We have a grid of 5x3 ...
2
votes
0answers
67 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
37 views

Independent increments of a Poisson process

In the following $\{X_t\}$ is a Poisson process. Assume that I've proved that $P(X_s=i,X_t-X_s=k)=P(X_s=i)P(X_t-X_s=k)$ so that the two events are independent, does it follow that ...
0
votes
1answer
48 views

Show that for any random variable $X$, and any $a > 0$, $P(|X| > a) \leq {EX^4 \over a^4}$.

Show that for any random variable $X$, and any $a > 0$, $$P(|X| > a) \leq {EX^4 \over a^4}.$$ Maybe I need to use Markov's Inequality, but I don't know how.
2
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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
3answers
36 views

Understanding the definition of $P(Y = y \mid X = x)$

Let $X: \Omega \rightarrow E_X$ and $Y: \Omega \rightarrow E_Y$ be random variables. By definition, we have that $P(Y = y \mid X = x)$ is defined as follows: $$ P(Y = y \mid X = x) = \frac{P(X = x ...
2
votes
1answer
32 views

Intuition on Martin-Löf-Test for finite strings

The followng example is from An Introduction to Kolmogorov Complexity and Its Applications, Example 2.4.1. and is concerned with Martin-Löf-Tests for finite strings: A string $x_1 x_2 \ldots x_n$ ...
2
votes
2answers
45 views

What exactly is $\cap$-stable here?

From my lecture notes: Theorem: Let $(\Omega, \mathcal A, P)$ be a probability space, $A \in \mathcal A, \mathcal M := \{ M_1, \ldots, M_n \} \subset \mathcal A$. The following statements are ...
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votes
1answer
61 views

Prove the series converges a.s in Probability

I have an article as follows Why are they enough to prove that $ \sum_{n=1}^\infty \dfrac{X_n \textbf{1}_{\{|b_n|< |X_n|\}}}{b_n} $ converges almost surely? I want to know why must prove $ ...
3
votes
1answer
59 views

sign of the conditional expectation

I'm working on the following problem: Let $X$ be a random variable defined on $(\Omega,F,P)$ and $G$ a $\sigma$-algebra contained in $F$. Show that, if $E(|X|)<\infty$ and $E(X\mid G)$ has the ...
3
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0answers
32 views

Kolmogorov's sufficient and necessary conditon for SLLN - What about pairwise uncorrelated RV?

Kolmogorov proved, that, as one considers independent (not necessary equally distributed) Random Variables: $\{X_n\}_{n\ge0}\subseteq \mathcal L^2$ With $\mathrm{Var} (X_n)=\sigma^2_n$ and without ...
2
votes
0answers
26 views

What is the meaning of the cumulant generating function itself?

If we define the characteristic function for a random variable X as $\Phi(t)=<e^{itX}>$ then it seems like we can think of it as essentially a spectral decomposition that measures the ...
2
votes
0answers
37 views

Can Monotone Class Theorem be easier to check than $\pi$-$\lambda$ Theorem?

I've been working on problem 14.4 in Billingsley's "Probability and Measure", which says: "Let $C$ be the set of continuity points of $F$. Show that for every Borel set $A$, $P(F(X) \in A, X \in ...
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 ...