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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1answer
14 views

Variance of Transformed Random Vectors

Consider an $n$-dimensional normal random vector $\mathbf X:= (X_1, \dots, X_n)^T$ with mean $\mathbf 0$ and covariance matrix $\mathbf \Sigma$. Now define a new random vector $\mathbf Y:= (a_1X_1, ...
5
votes
1answer
79 views

Under what conditions does a specified conditional distribution exist

It is common to see conditional distributions specified in stats like: $$(X \mid \mu = t) \sim \mathcal{N} (t, 1)$$ (Of course, we can also use some other distribution here) How do you prove that ...
0
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0answers
23 views

Necessary and sufficient conditions for a probabilistic inequality

Let the following conditions be satisfied $0 \le p_1, p_2, \ldots, p_n \le 1$ $0 \le d_1, d_2, \ldots, d_n \le 1$ $0 \le p_1 + d_1, p_2 + d_2, \ldots, p_n + d_n \le 1$ $0 \le d \le 1$ $0 \le p_1 + ...
2
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1answer
31 views

Existence of probability measure, $\lim_{N\to\infty}\frac{1}{N}\sum_{j=1}^N f(x_j)$

Problem statement: Let $x_j$ be a sequence in $[0,1]$. Consider $$\lim_{N\to\infty}\frac{1}{N}\sum_{j=1}^N f(x_j)\hspace 2cm (\star).$$ If ($\star$) exists for all $f\in C[0,1]$ then there is a ...
1
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1answer
87 views

“Continuity” of minimum of a function

The following is a question I encountered while reading a topic on 'large deviations'. I abstract the problem here. I don't know whether the title is apt! Let $\mathbb{A}=\{a_1,\dots,a_d\}$ be a ...
2
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1answer
38 views

Show that $a_n>0$ for all sufficiently large $n$

Let $F_n, G$ be distribution functions on $\mathbb R$. Suppose that $F_n(a_nx+b_n)\to G(x)$ as $n\to\infty$ for each $x\in c(G)$ where $c(G):=\{x\in\mathbb R:G(x)-G(x-)=0\}$. Here $a_n,b_n$ are ...
2
votes
1answer
67 views

How to find the density of $Y=g(X)$ in this case?

I have a vector $X=(1,X_2,X_3)$, where $(X_2,X_3)$ is a random vector in $\mathbb{R}^2$. Now consider $Y=g(X)=X/\|X\|$. What is a density function of $Y$ with respect to the uniform spherical ...
0
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1answer
53 views

Probability as a function of time

I was really wondering when I have to select any one out of the n options available - the probability of selecting A (let's say) is 1/n. But then I'm confused. When I (or anyone/anything else) bring ...
1
vote
0answers
30 views

Measurability of a random function

Suppose $(U_t)_{t\in[0,1]}$ is a stochastic process such that for every $\{t_1,t_2,\dots ,t_n\}\subset[0,1]$, $$U_{t_1},U_{t_2},\dots ...
2
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1answer
43 views

Find $\mathbb{E}_{X_0 = x} X_\tau$ for an Ornstein-Uhlenbeck process $(X_t)_{t \geq 0}$ where $\tau = \inf\{t>0 \mid X_t \notin [a,b]\}$

Let $X_t$ satisfy the following SDE: $dX_t = X_t dt + \sigma dB_t$, $\sigma$ is a constant and $B_t$ is Brownian Motion. Find $\mathbb{E}_{X_0 = x} X_\tau$ where $\tau = \inf\{t>0 \mid X_t \notin ...
2
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0answers
22 views

Probabilistic interpretation for Fokker-Planck equation

It is well known that if $X_t$ is a stochastic process that solves the SDE $$dX_t = \mu(X_t,t)\,\mathrm{d}t + \sigma(X_t,t)\,\mathrm{d}W_t,$$ with $W_t$ a Wiener process, then the associated ...
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0answers
13 views

Conditional density of degenerate multivariate normal

Let $X_{12},X_{13},X_{14},X_{23},X_{24},X_{34}$ be identically normal $N(\mu,\sigma^2)$ such that every linear combination among $X_{ij}$'s is also normal, $corr(X_{ij},X_{rs})=\rho$ if ...
2
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0answers
22 views

Total variation distance and couplings

The total variation distance between two measures $\mu$ and $\nu$ can be shown to equal the infimum over all couplings $(X,Y)$ where $X\sim\mu, Y\sim\nu$ of $P(X\neq Y).$ What is the supremum of ...
1
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0answers
21 views

Simple clarification- big $O$ and small $O$ notations in Erdos-Kac theorem proof

From The Probabilistic Method by Alon and Spencer. Let $\nu(n)$ be the number of primes $p$ dividing $n$ and set \begin{equation} X_p= \begin{cases} 1, & \text{if}\ p|x, \\ ...
6
votes
1answer
60 views

Is this set of random variables a Hilbert space?

Consider a sequence of i.i.d. random variables $\left\{ {{\varepsilon _t}} \right\}_{t = 1}^\infty $ with $E\left( {{\varepsilon _t}} \right) = 0$ and $E\left( {\varepsilon _t^2} \right) = {\sigma ...
4
votes
0answers
52 views

Weak convergence of a sequence of stationary distributions to another stationary distribution

Let $\{X_n(t) \in \mathbb{Z}^+\}$ for each $t \in (0,1)$ denote a discrete time Markov chain (with time index $n$ and parameterized by $t$). For each $t$, the Markov chain $\{X_n(t)\}$ has a unique ...
0
votes
0answers
17 views

Writing event involving order statistics in terms of actual observations

For $X_1,X_2,X_3 \sim$i.i.d exponential ($\lambda$), how do I write the event: $$ \{X_{(2)}-X_{(3)} > x ~,~ X_{(3)} > y \} $$ In terms of $X_1,X_2,X_3$ Where $X_{(3)}$ is the third largest ...
2
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1answer
31 views

Symmetric function of two normal distribution implies bilinear

This question is related to my previous question which was partially answered my @MichaelHardy. Let $X$ and $Y$ be two independent standard normal random variables. Now, suppose that ...
2
votes
1answer
41 views

A more elegant approach to proving independence between $X_{(3)}$ and $X_{(2)}-X_{(3)}$

For $X_1,X_2,X_3 \sim$i.i.d exponential ($\lambda$), I am trying to show independence between $X_{(3)}$ and $X_{(2)}-X_{(3)}$ where $X_{(3)}$ is the third largest observation, i.e. the minimum in this ...
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0answers
12 views

Continuity of the Loewner flow (SLE theory).

In the SLE paper "Basic Properties of the SLE" from Rohde and Schramm, it is mentioned on page 898 that the map $$(y,t)\mapsto g_t^{-1}(iy+\xi(t))$$ is clearly continuous on $y>0,t>0$, where ...
2
votes
1answer
37 views

Sufficient condition for $E(wu\mid v)=0$ given that $E(u\mid v)=0$?

I'm trying to figure out what condition concerning $w$ and $v$ would be enough for me to infer that $E(wu\mid v)=0$ given that I already know $E(u\mid v)=0$. Clearly, $w$ is a constant works: ...
3
votes
4answers
66 views

Is conditional probability always meaningful

Problem: A bag contains $4$ red and $5$ white balls. Balls are drawn from the bag without replacement. Let $A$ be the event that first ball drawn is white and let $B$ denote the event that the ...
1
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1answer
28 views

Counter example: $X$ and $Y$ normal imply $(X,Y)$ bivariate normal

I vaguely remember this construction from one of my courses: Suppose that $X\sim N(0,1)$ and $Z$ is $\pm 1$ with probability $\frac{1}{2}$ each. If $X$ and $Z$ are independent, then $Y\equiv XZ$ is ...
1
vote
1answer
41 views

integrating product of PDF and CDF

I am trying to show that the following integral: $$ \int_{-\infty}^a F(x)~f(x)~dx = \frac{F(a)}{2!} $$ Where $F$ is the cumulative distribution function of some continuous random variable X, and $f$ ...
0
votes
0answers
18 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 ...
0
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0answers
28 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
22 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 $ i.i.d. random variables. Then the question is to ...
3
votes
1answer
67 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 ...
5
votes
1answer
93 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 ...
6
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0answers
103 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
1answer
43 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 ...
6
votes
1answer
82 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
0answers
58 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
39 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
vote
1answer
52 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
40 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
vote
2answers
37 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
68 views

Probability distribution of number of waiting customers in front of a counter [closed]

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
34 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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votes
1answer
66 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 $B(0)=1$ [closed]

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 ...
4
votes
1answer
112 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 $\left\{ S\left(j\right)\thinspace:\thinspace j=0,1,\ldots\right\}$ be a simple random walk on the ...
1
vote
1answer
46 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 ...
3
votes
2answers
422 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 ...
3
votes
5answers
144 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
39 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
votes
1answer
32 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
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0answers
30 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 ...
2
votes
0answers
54 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
28 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 ...