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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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Finding a distribution function of random variable sum

Let $\xi_1, \xi_2, \xi_3$ independent random variables in $(\Omega, \mathcal{F},\mathbb{P}).$ Also, they are evenly distributed in $[0,1]$. I need to find a distribution function of sum $\xi_1+ \xi_2+...
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2answers
23 views

iid, and equal in distribution

Let X and Y be iid. If (X+Y)/2 is equal in distribution to XY, then what do we know about the distributions of X and Y? I feel like I can't say much about these distributions. I can only think of ...
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0answers
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Integration identity for nonnegative random variables

I'm trying to furnish a proof for the following identity, and would like to have my proof checked. Suppose $(\Omega, \mathcal{F}, \mathbf{P})$ is a probability space on which random variable $X: \...
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0answers
27 views

Showing that the Distribution of a Random Variable Must be Standard Normal using Characteristic Functions [duplicate]

Question Let $X$ and $Y$ be i.i.d with means $0$ and variances $1$. Let $\phi(t)$ be their common characteristic function and suppose that $X+Y$ and $X-Y$ are independent. Show that $\phi(2t)=\phi(...
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1answer
10 views

Why does $P(Q_t = q | X_{0:L} = i_{0:L}) = P(Q_t = q, X_{0:L} = i_{0:L})$?

This is a derivation of an equation used to maximize the posterior probability that $Q_m = i_m$ given a model and a sequence of observations. $Q_m$ is a RV which maps to some $q \in S$, the state ...
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0answers
9 views

Is there some probability measure associated to a markov chain?

Clearly there are some probabilities involved with the markov chains, but I cannot see how to extract a sigma algebra from it. Is it that the probability of getting from state $i$ to state $j$ is ...
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0answers
5 views

What is the distribution of the unit sphere multiplied by a uniformly distributed scalar?

We know that any rotationally-invariant distribution can be written as X = RU where R is |X|, U ~ Unif[S^(n-1)], and R is independent of U. In words, we can choose a direction vector using the unit ...
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1answer
14 views

Showing that the ratio of two standard independent normals is a Cauchy using Characteristic Functions

Question Let $X$ and $Y$ be independent standard normals. Use characteristic functions to find the distribution of $X/Y$. My attempt We will attempt to show that $Ee^{itX/Y}=e^{-|t|}$ (the c.f. ...
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1answer
79 views

Question involving Characteristic Functions and the Existence of a Distribution

Question Is it possible for $X$, $Y$ and $Z$ to have the same distribution and satisfy $X=U(Y+Z)$ where $U$ is uniform on $[0,1]$ and $Y$, $Z$ are independent of $U$ and of one another? The above ...
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Mean of normal CDF

Question- Let $X$ follows $N(\mu,\sigma^2)$. Show that $\mathbb{E}(\Phi(X)) \neq \frac{1}{2}$ for any $\mu \neq 0$, where $\Phi(X)$ is the cdf of $N(0,1)$ distribution. But the theorem of ...
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0answers
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If $X,Y,Z$ are PAIRWISE independent (i.e. only $X,Y$ and $Y,Z$ and $X,Z$ are) , does this imply that $X$ and $(Y,Z)$ are independent?

This question came to my mind when I was reading a theorem on Poisson Processes stating that the sum of two independent Poisson processes, $X_t$ and $Y_t$ is a Poisson process. In the proof, one has ...
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2answers
40 views

Using Adam's Law (Law of Total Expectation) to find expectation of residual

This may seem like a rather simple question, but I haven't been able to come up with an explanation myself or find one on the Internet. I've learned that Adam's Law states that $$E(E(Y|X)) = E(Y)$$ ...
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0answers
21 views

Computation of the expectation $E(2^{X-2})$ for $X$ negative binomial

If I have a coin with $\ 0.6$ probability of getting $\ H $and I throw it until I get $\ H $ for the second time. If $\ Y $ is the number of $\ T$ I get and $\ 2^Y $ is my revenue. how do I ...
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1answer
30 views

If $f$ is the pdf of a random variable, show that $g(x, y) = f(x+y)/(x + y)$ is a density function in the plane.

Let $f$ be the pdf of a positive random variable and write $$g(x, y) = \frac{f(x + y)}{x + y} , \text{ if } x, y > 0.$$ Show that $g$ is the density function in the plane. Clearly, $g(x, y) \...
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0answers
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Show completeness of sufficient statistic for continuous probability density function

The following probability density function: $$f(x)= \frac {(\log\alpha)\alpha^x} {\alpha-1} ;\qquad 0<x<1,\quad \alpha>1$$ Using factorization theorem the sufficient statistic would be $T=\...
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32 views

If $\chi_n \to \chi$ in distribution then $1/(n \log(1 - \chi_n/n))\to1/\chi$ in distribution

Suppose $\chi_n \to \chi, ~~ \chi \sim \mathrm{exp}(1)$ in distribution, and set $\rho_n = 1 - \chi_n/n$. Show \begin{align*} \frac{1}{n \log \rho_n} \end{align*} converges in distribution to $1/\...
2
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1answer
34 views

Can you compute $P(A)$ if you know $P(A|B), P(A|C), P(B)$ and $P(C)$?

The agent is described as either loyal or not-loyal. The probability that the agent is loyal is given by ‘p.’ The negotiations can reach two outcomes for the country: favorable and unfavorable. The ...
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1answer
20 views

Expectation and variance of travel time with several options for the transportation

A person is traveling between two places, and has 3 options for transportation. The jth option would take an average of µj hours, with a standard deviation of $\sigma_j$ hours. The person randomly ...
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0answers
24 views

Probability - scaled convergence in probability implies scaled almost sure convergence for an increasing sequence of RVs

I'm studying for a probability exam, and was having trouble with this problem: Let $X_n$ be an increasing sequence of random variables. Prove or disprove - If $X_n/n\rightarrow 1$ in probability, ...
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1answer
28 views

What is actually being asked here? (Measure theoretic probability)

I am reading a probability text that asks the following question. If $X_1, X_2, \dots$ are independent $\mathrm{Ber}(p)$ random variables, where $0 < p < 1$, then define $T : \min\{k : X_k = ...
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1answer
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If $\phi$ is a characteristic function, then $1-|\phi(2t)|\leq 8\{1-|\phi(t)|\}$

Question If $\phi$ is a characteristic function, show that $\text{Re}\{1-\phi(t)\}\geq \frac{1}{4}\text{Re}(1-\phi(2t))$ and deduce that $1-|\phi(2t)|\leq 8\{1-|\phi(t)|\}$. My attempt I have ...
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0answers
19 views

Uniform Law of Large Numbers - questions on supremum, infimum, and weak vs almost sure continuity

I am currently reading up on uniform convergence in probability and I have been trying to familiarize myself with the uniform law of large numbers. As I am not a mathematician or statistician by ...
2
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0answers
13 views

If a probability measure has two densities, then they agree a.e. (Proof verification)

Suppose that $f, g$ are two nonnegative measurable functions that integrate to one over $\mathbf{R}$, with respect to Lebesgue measure and are such that the probability measure $\mu(A) = \int_A f d\...
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1answer
28 views

Probability of winning with dices

I've been studying probability for a few couple weeks. Today, I've given a homework, and I am really struggling with it. The type of question is unfamiliar to me and I don't have any idea how to solve ...
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0answers
24 views

If $Y$ is a Markov chain and $h>0$, why is $(Y_{\lfloor t/h\rfloor})_{t\ge0}$ not a Markov process?

Let $\left(Y^{(n)}_k\right)_{k\in\mathbb N_0}$ be a Markov chain for $n\in\mathbb N$, $(h_n)_{n\in\mathbb N}\subseteq(0,\infty)$ with $h_n\xrightarrow{n\to\infty}\infty$ and $$X^{(n)}_t:=Y^{(n)}_{\...
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0answers
30 views

A construction of a Stratonovich type integral for fractional Brownian motion

I'm studying this article https://projecteuclid.org/download/pdf_1/euclid.twjm/1500574954 and I'm having problems understanding the proof of lemma 3. Let me recall some of the criminals involved. ...
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0answers
40 views

Random walk on $\{0,1,…,k\}$, find the average gain in 10 000 steps

I have the following problem which I can't seem to figure out. The problem is as follows. Consider simple random walk on {0, 1, ... , k} with reflecting boundaries at 0 and k, that is, random ...
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0answers
15 views

Finding the inverse of a CDF

I have a Beta(3, 1) distribution with CDF: Fx(x) = {0 if x <0 {x^3 if 0 < x < 1 {1 if x > 1 I need to find the inverse of that ...
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0answers
25 views

Does a sequence of random variables constructed in a certain manner converge in distribution to a Gaussian?

Let $\{X_n\}_{n \in \mathbb{N}}$ be a sequence of of IID random variables taken for simplicity with mean zero and variance one. The Central Limit Theorem give us that $$ \frac{X_1 + \dots + X_n}{\...
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0answers
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Calculate limit: Make a stopping time $T$ bounded $T\land n$ and take the limit $n \to \infty$

Say we have a martingale $X$ and a stopping time $T$. Instead of directly studying the stopped process $X_T$, many proofs employ a trick, namely, one considers the bounded stopping time $T\land n$ ...
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1answer
21 views

Prooving $\mathbb{I}_{\{A \Delta B \}}=(\mathbb{I}_{\{ A\}}-\mathbb{I}_{\{ B\}})^2$

Let $A$ and $B$ be two events from $\Omega, \mathcal{P}(\Omega),\mathbb{P})$. I need to show that next equal is true $$\mathbb{I}_{\{A \Delta B \}}=(\mathbb{I}_{\{ A\}}-\mathbb{I}_{\{ B\}})^2$$. I ...
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1answer
12 views

marginal probabilities, multivariate random variables

I want to solve the task below... However, I have a problem with the marginal probabilities not adding up to 1. what's wrong?
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1answer
22 views

Finding a distribution function of a random variable

Random variable $\xi : (\Omega, \mathcal{F}, \mathbb{P}) \rightarrow \mathbb{R}$ has Laplace distribution with density function $f_{\xi} (x)= \frac{1}{2}\exp \{-|x| \}$ where $x \in \mathbb{R}$. I ...
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0answers
19 views

Existence of a function $f$ with $\mathbb E[X^2f(X)]$ being finite for some $X$ with finite second moment

$\newcommand{\E}{\mathbb E}\newcommand{\PM}{\mathbb P}$This question is inspired by this question which is unfortunately closed. Anyways, I found it interesting, so I tried to solve it. The problem ...
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0answers
17 views

Repeatedly multiplying by numbers taken from a uniform distribution

Motivation: In a software project I know well, for testing purposes, the evolution of a particular price is simulated by repeatedly multiplying by a random number picked between $0.995$ and $1.005$. ...
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Can Convergence in probability in this problem be reinforced to Almost sure convergence

$X_1,X_2,…,X_n$ are independently and identically distributed and $E(X_i)$ exists, $\mu_n=E(X_n I(X_n \le n)),S_n=\sum_{i=1}^n X_i$. Proof:$$\frac{S_n}{n}-\mu_n\overset{p}{\to }0$$ My answer is: $$\...
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0answers
32 views

Computing Probability Using Central Limit Theorem

$X_1,X_2...X_{100}$ are IID random variables with variance equal to 1. The mean of the random variables is not known. We estimate the mean by taking the sample average $\hat{\mu}=\frac{\sum_{i=1}^{100}...
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Dependency between two experiment that are feedback to one Service provider

There is one service provider (SP) and two requesters. Each requester can get service from SP and at the end of communication return a number as feedback to SP. Any requester cannot see his feedback. ...
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1answer
30 views

Proving $E[\max(X^2,Y^2)]\le 1+\sqrt{1-\rho^2}$

The question is Let $X$ and $Y$ be random variables with mean $0$, and variances $1$ and correlation coefficient $\rho$. Show that $E[\max(X^2,Y^2)]\le 1+\sqrt{1-\rho^2}$. My attempt: $$E[\max(X^...
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1answer
21 views

Problem on Weak Law of Large Numbers

Question- $X_n$ can take only two values $n^a$ and $-n^a$ with equal probabilities. Show that we can apply weak law of large numbers to the sequence of independent random vatiables ${X_n}$ if $a<\...
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1answer
27 views

Suppose $X_n$ are iid with a symmetric distribution. Then $\Sigma_n \frac{X_n}{n}<\infty ~\mathrm{a.s. iff }~\mathbb{E}|X_1|<\infty$

It seems to be solved by using Kolmogorov strong law of large numbers. Why $X_n$ have symmetric distribution?
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1answer
29 views

Converges Uniformly Example

Folland's Real Analysis/Measure Theory textbook on page 61 states that, $n \chi_{[0,1/n]}$ converges uniformly to 0. I cannot see how this can be true as I thought that $\sup|f_n(x)-f(x)|$=$\sup |n \...
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0answers
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The problem is about the expection of the exitpoint distance for the symmetric random walk.

Let $\nu(x)$ be a symmetric probability measure with respect to the origin on $x\in[-1,1]$ such that $\nu(\{0\})\neq 1$. Consider a random walk started at $S_0=0$, denoted $S_n=X_1+\cdots+X_n$, where ...
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0answers
17 views

Clarification on a probability theory statement

In a text I'm reading, it says the following: Suppose that $(\Omega, \mathcal{F})$ is a measureable space and that $X_1, \dots, X_n: \Omega \to \mathbf{R}$ are $n$ (real-valued) random variables. ...
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0answers
19 views

Markov Chain whose state space $S=\{0,1,2,\dots,m\}$

I'm studying Markov chains and I came across the following exercise and I do not know how I should approach it. We note $(P_{x,y})_{0\le x,y\le m}$ the transition of the chain. Show that there is $0&...
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1answer
22 views

A closed (?) subset of the set of probability measures. Is my reasoning correct?

Let $f:\mathbb{R}^{d} \rightarrow [0,\infty) $ continuous function and denote by $P$ the space of probability measures on $\mathbb{R}^{d}$ and by $P_R \,=\, \{\nu \in P \, | \, \int f d \nu \leqslant ...
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0answers
22 views

Random time change for a Poisson process and convergence with respect to the Skorohod topology

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $\left(Y^{(n)}_k\right)_{k\in\mathbb N_0}$ be a time-homogeneous Markov chain on $(\Omega,\mathcal A,\operatorname P)$ and $$X^{(n)}_t=...
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0answers
21 views

Formalizing conditional expectation

I need help to translate into a conditional exepectation the following problem: We have an interval of $\mathbb{R}$ of size M (say $\mathcal{M} = [0, M]$). For each element $x \in \mathcal{M}$ I ...
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0answers
22 views

General techniques for coupling a set of random variables with mutual dependence

Disclaimer: First, the usage of "coupling" in the title is not of the usual definition in probability theory. Second, cross-posting from stats.stackexchange.com. Suppose I have a set of random ...
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+200

Let $P(X_j=j)=P(X_j=-j)=1/2j^{\beta}$ and $P(X_j=0)=1-j^{-\beta}$ where $\beta\in(0,1)$ then $S_n/n^{(3-\beta)/2)}\Rightarrow c_{\chi}$

Suppose $P(X_j=j)=P(X_j=-j)=1/2j^{\beta}$ and $P(X_j=0)=1-j^{-\beta}$, where $\beta>0$. Show that: (i) If $\beta>1$ then $S_n\to S_\infty$ a.s. (ii) If $\beta\in(0,1)$ then $S_n/n^{(3-...