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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14 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
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1answer
123 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) ...
1
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0answers
42 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 ...
4
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1answer
103 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
34 views

Definition of n independent event and example

Given a finite set of events $A_1,\dots,A_n$, the events are said to be independent if and only if for any subset of indices $I$ we have: $$\mathrm{P}\left(\bigcap_{i\in I} A_i\right)=\prod_{i\in I} ...
1
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1answer
27 views

SDE: Find $\mathbb{E}_{X_0 = x} X_\tau$ 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 ...
1
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1answer
55 views

Mixture of Discrete Binomial Distributions

Let $B\left(p,N\right)$ be a Binomial distribution with parameters $p$ and $N$. We define a Mixture of Discrete Binomial Distributions by $\left\{ \left(B\left(p_{i},N\right),\alpha_{i}\right)\right\} ...
0
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1answer
47 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
21 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 ...
1
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0answers
8 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 ...
-1
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1answer
51 views

A conditional probability question [on hold]

Let A and B two events and if $P(A)=0.5$ and $P(B)=0.4$ what is the $P(B\mid A^C)$?
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2answers
25 views

Understanding different definitions of bayes theorem

I am taking course on probability and reading about bayes theorem. In Sheldon Ross' book, it given as $$P(E) = P(E|F)P(F) + P(E|F^C)P(F^C)$$ with note: Equation above states that the probability of ...
3
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2answers
70 views

Uniform convergence in distribution

Consider a sequence of stochastic processes, $X_n(x)$ and a limiting process $X(x)$. For a fixed $x$, if $\mathbb{P}(X_n(x) \leq y)$ converges to $\mathbb{P}(X(x) \leq y)$ for continuity points of ...
2
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0answers
33 views
+100

Potential measure of the product of (independent $\alpha$-stable) subordinators

For a nondecreasing Levy process $\mathbf{X}$ with values in $[0,\infty)$ (i.e. a subordinator) Jean Bertoin defines the potential measure of $\mathbf{X}$ in his book "Levy processes" as follows (p. ...
2
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0answers
20 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 ...
0
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1answer
54 views

Why is $f(X_t)-\int_0^t Af(X_s) \, ds$ a martingale for a Markov process $(X_t)_{t \geq 0}$?

I think if $A$ is the usual generator for the Markov process $(X_t)_t$ $$A f (x) = \lim_{t \downarrow 0} \frac{\mathbb{E}^{x} [f(X_{t})] - f(x)}{t}$$ then we get that for any "nice" $f$ the process ...
1
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1answer
68 views

Reaching a level before another for a random walk

Suppose we are given a simple random walk starting in $0$, i.e. $(X_k)_{k\in\mathbb{N}}$ with $P[X_k=+1]=P[X_k=-1]=\frac{1}{2}$. What is the probability of hitting the level $a$ before hitting the ...
6
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1answer
51 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 ...
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0answers
20 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, \\ ...
3
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1answer
91 views
+50

Warren's proof for Benford's Law

Warren has a little proof of Benford's law in Hacker's Delight. To quote: Let $f(x)$ for $1 \leq x < 10$ be the probability density function for the leading digits of the set of numbers with ...
1
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0answers
10 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 ...
4
votes
0answers
82 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 ...
2
votes
1answer
48 views

$p$-stable Random Variables for $p>2$?

I will preface this by saying I am certainly no expert in Probability theory. My actual problem is an interpolation one, in which I am considering interpolation of bandlimited functions with shifts ...
4
votes
0answers
37 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 ...
2
votes
2answers
91 views

Convergence in distribution for $\frac{Y}{\sqrt{\lambda}}$

Given a sequence of independent r.v's $\{X_n\}_{n\geq 1}$ such that $P(X_n=x)=\frac{1}{2}$ if $x=-1$ and/or $x=1$ Let $N\in Po(\lambda)$ be independent of $\{X_n\}_{n\geq 1}$ and we set that ...
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0answers
15 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 ...
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0answers
23 views

Is the alias method “stable”?

The alias method is an algorithm for sampling from a discrete distribution. Let me describe it briefly. First there is a setup phase. You have $N$ values and associated probabilities. You introduce ...
1
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1answer
296 views

One-tailed two-sample T-test OK?

I'm trying to conduct a one-sided hypothesis test between two random variables which are both asymptotically normally distributed with different variances. The variances are not known but have been ...
6
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1answer
72 views
+100

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 ...
7
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1answer
143 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 = ...
4
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1answer
139 views

Central Limit Theorem is incorrect - where is my mistake?

Say I flip a coin 80 times and I ask for the probability to get over 48 heads. I then flip a coin 800 times and ask for the probability to get over 480 heads. Translating this into Central Limit ...
2
votes
1answer
40 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 ...
1
vote
1answer
291 views

Is maximizing the Shannon differential entropy equivalent to minimizing the predictability and/or minimizing the maximum density?

For a real-valued, 1-dimensional, continuous random variable X with density f(x), I am trying to determine if maximizing the Shannon differential entropy of f(x) is mathematically equivalent to ...
3
votes
1answer
45 views

Find probability of a Poisson process.

Given that $N=\{N(t)\mid t\geq 0\}$ is a Poisson process with parameter $\lambda>0$ I need to find $P(N(3)=2\mid N(1)=0, N(5)=4)$ So this is a conditional probability (can anyone clarify if this ...
1
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1answer
41 views

Is σ-finiteness necessary for the “in measure” version of the dominated convergence theorem to hold?

Let $(X,\Sigma,\mu)$ be a measure space, $g\in L_1$, $|f_n|\le g$ and $f_n\to f$ in measure. I want to prove that $\int f_n\to f$, and $f_n\to f$ in $L_1.$ Now, this may be already solved in the ...
2
votes
1answer
33 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: ...
6
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1answer
94 views

Show that $E(X\mid Y, Z) = E(X\mid Y)$ almost surely with condition Z is independent of $(X, Y)$

$(X, Y, Z)$ is a continuous random vector and $Z$ is independent of $(X,Y)$. Prove that $E(X\mid Y, Z) = E(X\mid Y)$ almost surely. I had been thinking this question tonight but couldn't figure out ...
3
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4answers
59 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 ...
2
votes
1answer
21 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 ...
1
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1answer
38 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 $f$ ...
1
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1answer
26 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 ...
3
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1answer
65 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 ...
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0answers
24 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
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0answers
17 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
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1answer
61 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
88 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
59 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 ...
1
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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 ...