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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-distributions] for specific distribution functions, and consider [tag:stochastic-processes] when appropriate.

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11 views

Theorem 24.1 Billingsley “Convergence of Probability Measures” FIRST EDITION 1

I need Theorem 24.1 of Billingsley “Convergence of Probability Measures”. I can only find the second edition, first one checked out of the library . I really need this theorem, it should have ...
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1answer
28 views

A characterization of the probability density given information about a sum

I came across the following probability theorem. Let $X_1$ and $X_2$ be independent and identically distributed with density $f(x,\theta)$. If the random variable $Z=X_1+X_2$ is such that \begin{...
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2answers
20 views

Probability of a supremum of a sequence of independent random variables

I'm trying to prove the next: Suppose $\{X_n\}$ is an independent sequence of random variables. Show that $$P(\sup X_n<\infty)=1$$ if and only if $$\sum_n P(X_n>M)<\infty$$ for some $M.$ I'...
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20 views

Central limit theorem and integrability

If $(Y_n)_n$ is a sequence of independent random variables and identically distributed, and if $\frac{\sum_{k=1}^nY_k}{\sqrt{n}}$ converges in distribution to a random variable Y, does this mean that $...
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1answer
45 views

Deriving the Poisson from the Binomial

In my notes I have the following explanation: The probability function of the Poisson random variable is $P_X(k)={\alpha}^k \frac{e^{-\alpha}}{k!}$ A Poisson random variable with parameter $\alpha$. ...
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1answer
18 views

If $X,Y$ are random variables dependent on $Z, W$, but $Z$ depends on $W$, what is the proper way to represent the entire joint distribution?

Suppose that we have the following random variables $Y,X$ which are dependent on $Z, W$. However, $Z$, which is defined on a finite set $Z \in \{z_1, \ldots, z_n\}$ is further dependent on $W$. I ...
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16 views

How can we show that $\Phi\left(-\frac b{2\sqrt a}\right)+e^{\frac{a-b}2}\Phi\left(\frac b{2\sqrt a}-\sqrt a\right)$ are uniformly bounded in $a,b$?

Let $\Phi$ denote the cumulative distribution function of the standard normal distribution. How can we show that $$\Gamma(a,b):=\Phi\left(-\frac b{2\sqrt a}\right)+e^{\frac{a-b}2}\Phi\left(\frac b{2\...
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1answer
39 views

$X$, $Y$ independent if and only if $X$, $Y$ uncorrelated.

Suppose that the joint probability density function of $(X, Y)$ is given by $f_{X,Y}(x, y) =[1 - \alpha(l-2x)(l-2y)]I_{(o,1)}(x)I_{(o,1)}(x)$ where -1 < $\alpha$ < 1. Prove or disprove: $X$ and ...
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1answer
38 views

Probability of $\{X_1 < X_2 < \cdots < X_n\}$

Let $(X_i)_{i \in \mathbb N^*}$ be a sequence of independent and identically distributed random continuous variables having a density $f$. Since $X_i$'s are continous we have $$ \mathbb P(X_i=X_j) = ...
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1answer
39 views

Finding $L_n$ so that $\lim_{n \rightarrow \infty} P(L_n<\rho<1)=\alpha$

Let $(X_1,Y_1),(X_2,Y_2),..,(X_n,Y_n)$ be iid pairs of random variables with $E(X_1)=E(Y_1)$, $\text{Var}(X_1)=\text{Var}(Y_1)=1$,and $\text{cov}(X_1,Y_1)=\rho \in(-1,1)$. Given $\alpha>0$ , obtain ...
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27 views

Expected number of visits to $k$ before hitting 0

This problem is from Exercise 5.5.6 in Durrett's Probability: Theory and Examples, 5/E, Use Theorems 5.5.7 and 5.5.9 to show that for simple random walk on $\mathbb{Z}$, if we start from $k$ the ...
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0answers
23 views

Standard deviation of success

This problem is hard for me to understand mathematically and logically: To summarize this problem: In a takeover attempt you need $K$ shares in order to gain control. There are $N$ shareholders. $p$ ...
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0answers
18 views

If $K$ is compact, $x\in K$, $Y\sim\mathcal N_{x,\:σ^2}$ and $X∈[x∧Y),x∨Y)]$, then $\text E[|f(X)||Y-x|^k]\le\sup_K|f|\text E[|Y-x|^k]$ for small $σ$

Let $K\subseteq\mathbb R$ be compact $x\in K$ $Y\sim\mathcal N_{x,\:\sigma^2}$ for some $\sigma>0$ $X$ be a real-valued random variable with $X\in[\min(x,Y),\max(x,Y)]$ $f:\mathbb R\to\mathbb R$ ...
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1answer
18 views

KL-divergence of two distributions using probability measure

I am reading an article stated that: Given $\mu$ is a probability measure, a measurable set $A$, and $\hat{\mu}(\cdot) = \mu(\cdot \bigcap A)$, then $D_{KL}(\hat{\mu}||\mu) = - \log \mu(A)$. How do ...
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1answer
13 views

Find the constant $a$ such that $Y_t$ is a martingale.

Let $X_t$ be the solution of SDE $\text{d}X_t=3X_t\text dt+2X_t\text dB_t$ and $X_0=1$ which $B_t$ denotes the Brownian motion with $B_0=0$. Let $Y_t=e^{at}X_t$, Find the constant $a$ such that $Y_t$...
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1answer
25 views

Almost sure convergence of $\text{Poisson}(\frac 1n)$ to $0$

Let $X_n$ a sequence of random variables such that $X_n\sim \text{Poisson}(\frac 1n)$. Study the almost-sure convergence of $X_n$. Since $X_n$ is integer-valued and $P(X_n=0) = \exp(-\frac 1n)$ it is ...
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0answers
18 views

Product sigma algebra of countable cocountable sets

I want to prove the following proposition: Let $\Omega $ be an uncountable set and let $$\mathcal F = \{A \subset \Omega: A \textrm{ countable or } A^c \textrm{ countable} \}.$$ If $B \in \...
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0answers
12 views

Smooth function acting on continuous probability distribution

I got this task to do: "Prove continuous probability distribution transformed by smooth function theorem". I have no idea where to find such theorem, because TA only provided us with tasks. I don't ...
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3answers
19 views

A simple probability question but seemingly 2 answers, which one is the best?

A fair spinner with digits 1,2,2,3 marked on it is spun three times. The three numbers are added to give a score. Find the probability that the score is even. Answer 1 : There are 32 even outcomes ...
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28 views

Evaluate $P( \mid X+Y\mid \leq 2 \mid X \mid )$. [on hold]

Let $X,Y$ independent random variables, identically distributed and symmetric about $0$. Select some probability density function f(.), and evaluate $P( \mid X+Y\mid \leq 2 \mid X \mid )$.
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2answers
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Let $X,Y$ be independent, symmetrical, identically distributed, then $P( \mid X+Y\mid \leq 2 \mid X \mid )>1/2$. [on hold]

Let $X,Y$ independent random variables, identically distributed and symmetric about $0$. Prove that: $P( \mid X+Y\mid \leq 2 \mid X \mid )>1/2$
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0answers
10 views

Does marginal convergence in distribution + certain extra condition imply joint convergence in distribution?

Let $Y_n$ and $Y$ be random elements in some Polish space $F$ such that $Y_n \to Y$ in distribution and $Z_n$ and $Z$ be random elements in some Polish space $G$ such that $Z_n \to Z$ in distribution. ...
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1answer
20 views

CDF of derived distribution $|X-Y|$ when $X$ and $Y$ are exponential random variables

I recently had to solve this same problem, except $X$ and $Y$ were uniform on $[0,1]$. The joint probability distribution was uniform, so I just needed to find the proportion of the area inside the ...
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0answers
16 views

Irreducible Markov chain rotating cyclically

I am going through Knowing the Odds by John B. Walsh, and I am stuck at one of the exercises there (which is important to understand some next theorem). The exercise is as follows: Let $X$ be an ...
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0answers
33 views

Random mapping and entropy ordering

Let $X_1,X_2$ be discrete random variables such that $H(X_1)<H(X_2)$ where $H()$ is the entropy. We know that for any random mapping $T$ which is invertible ($T$ is a function of $\omega$ and $X$, ...
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1answer
29 views

Is all random variable finite (almost surely)?

I feel like what I've known are conflicting with each other, so I'd like to post it. When some prove that if $X_n \to X$ in probability and $Y_n \to Y$ in probability, then $X_nY_n \to XY$ in ...
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1answer
39 views

Expected value of a binary decimal generated by coin flips

A coin is flipped infinitely many times. Heads is $1$, tails is $0$. The string created by the heads and tails is turned into its respective string of ones and zeros. If I write $1.$ before the ...
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5answers
40 views

Probability of even number of events occuring

As part of trying to fresh up on my basic probability theory I came along Ex. 1.46 in Grimmet's probability book with the second part troubling me. If $A_1$, $A_2$ , . . . , $A_m$ are independent and ...
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1answer
45 views

Conditional expectation of the number of phone calls

There are twenty individuals numbered $1,2,...,20$.Each individual chooses 10 others from this group in a random fashion,independently of the choices of the others, and makes one phone call to each of ...
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1answer
32 views

a question about strong law for renewal process

Let $\{N(t); t > 0\}$ be a renewal counting process with inter-renewal rv s $\{X_n; n \geq 1\}$. Then, $lim_{t\rightarrow \infty} N(t) = \infty$ with probability 1. Proof: We only need to show $P\...
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15 views

Marginalise out success probability when inferring outcomes of binary trials

This is my first post and there's a bit of a preamble to this question, so happy to add further clarification! Edit: This felt quite maths-y, but this might be better suited to the statistics site. ...
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3answers
61 views

Does $X\le Y$ imply $\mathbb{E}[X|Z]\le \mathbb{E}[Y|Z]$

Let $X, Y, Z$ be random variables. If $X\le Y$, I wonder if $$\mathbb{E}[X|Z=z]\le \mathbb{E}[Y|Z=z]$$ holds. I have a contradiction between math and intuition. First, math. I think math shows ...
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1answer
21 views

Notation q(x) << p(x) in probability

I recently read an article on probability theory that use the notation: q(x) << p(x) where p(x) and q(x) are two density functions of two distributions. What does the operator << mean in ...
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1answer
28 views

Distribution of a function of Brownian motion

Q) Let $W = \int_{0}^{t}B_sds$, $B$ is a Brownian motion. Find $EW$ and $EW^2$. My attempt: $B_s \sim N(0,s) $ $$W = \int_{0}^{t}\frac{1}{\sqrt{2\pi s}}e^{-\frac{x^2}{2s}}ds , EW = \int_{-\infty}^...
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2answers
59 views

If X and Y are iid, is $E(X\mid X+Y)=E(Y\mid X+Y)$?

Just like question asked, my thought is they have the same mapping. My classmate gives me a counterexample: if $X$ and $Y$ are equal, then $X+Y=2X$. However, $\operatorname{Var}(2X) \neq \...
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1answer
42 views

Probability of events having to deal with a string permutation question

I am studying for a discrete math exam tomorrow and this is one of the review questions. I am having trouble answering the question as of now. If you could provide guidance on how to solve one or more ...
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3answers
44 views

Find distribution of $Z=\frac{X+Y}{2}$ given $f_{X,Y}(x,y)=e^{-(x+y)}$

Excercise Let $X, Y$ be random variables such that their joint density function is defined by: $f_{X,Y}(x,y)=e^{-(x+y)}, \enspace x,y>0$. Find the distribution of Z defined as: $Z=\frac{X+Y}{2}...
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3answers
37 views

measure and countable set [on hold]

how can we prove that if $\mu$ is a probability measure on $(\mathbb{R}^d,B(\mathbb{R}^d))$ then the set $E:=\left\{x \in \mathbb{R}^d,\mu(\left\{x \right\}) \neq 0\right\}$ is countable. Thank you.
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30 views

Example for Lévy's continuity theorem

I am searching a sequence of RV $(X_n)$ for which we prove a convergence in distribution to a RV X, using the fact that the characteristic functions $(\varphi_n)_n$ converges pointwise to some ...
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1answer
90 views

Let $X_1$ and $X_2$ be uniform on $n$-spheres. What is the distribution of $\| X_1+X_2\|$?

Suppose we have two independent random variables $X_1$ and $X_2$ distribution on $n-1$-sphere of radius $r_1 $ and radius $r_2$, respectivly. Assume $r_1>r_2$. Recall, that the $n-1$-sphere of ...
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1answer
23 views

Find the conditional distribution of $X$ given that $Y=y$

Let $X$ and $Y$ be two random variable with density $f_{X,Y}(x,y)=\begin{cases} \frac{1}{y}, & \text{for } 0<x<y<1, \\[8pt] 0, & \text{otherwise}.\end{cases}$. To find the ...
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0answers
26 views

Multivariate Kolmogorov distance bounded by Wasserstein distance

I'm trying to find a bound for the multivariate Kolmogorov distance in terms of the Wasserstein distance. Denoting by $F$ and $G$ two cumulative distribution functions (cdf) on $\mathbb{R}^n$ the ...
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2answers
39 views

Let $\mu(X)=1$, $0 \leq f \leq k$, and $m=\int_X f d\mu$. Show $\int_X |f-m|^2 d\mu \leq \frac{k^2}{4}$.

Let $\mu(X)=1$ for $\mu$ a positive measure. Let $0 \leq f \leq k$ for some $k\in\mathbb{R}$ and let $m=\int_X f d\mu$. Show $\int_X |f-m|^2 d\mu \leq \frac{k^2}{4}$. My attempt: I tried to expand ...
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1answer
45 views

Probability of a random subset of Z

I'm stuck in this question, could someone give me a hand? I'll post what I've done so far. Question 9: Let $A=(1,2,3,4)$ and $Z=(1,2,3,4,5,6,7,8,9,10)$, if a subset B of Z is selected by chance ...
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1answer
19 views

Find the density function of $(U,V)$

Let $X=(X_1,X_2)^T$ be a random vector with 2-dimensional normal distribution, $E(X_1)=E(X_2)=0 , \operatorname{Var}(X_1)=\operatorname{Var}(X_2)=1$ and $\operatorname{Cov}(X_1, X_2)= \nu$ with $|\nu| ...
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0answers
38 views

A lower bound on $|S_n - S_{n+1}|$

I am reading "Durrett: Theory and examples", fourth edition. In page 67, he proves theorem 2.3.7 (an application of Borel Cantelli lemma). During that demonstration, he arrives to a point where: $X_1,...
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1answer
13 views

Existence of random variable given infinite-dimensional probability measure

Let $\mathcal{X}$ be some possibly infinite-dimensional metric space and let $\mu$ be a Borel probability measure on $\mathcal{X}$. What theorem implies the existence of a probability space $(\Omega,...
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0answers
11 views

How to transform long term statistics into probability? [on hold]

My concrete problem is the following: If the long term goal average for a soccer team is one scored goal every 60 minutes, what is the probability of scoring a goal in 60 minutes? What is the ...
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1answer
39 views

Finding the conditional expectation of independent exponential random variables

Let $X$ and $Y$ be independent exponential random variables with respective rates $\lambda$ and $\mu$. Let $M = \text{min}(X,Y)$. Find (a) $E(MX|M=X)$ (b) $E(MX|M=Y)$ (c) Cov$(X,M)$ (a) I first ...
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0answers
18 views

Sufficient statistics and natural parameters of exponential family

I am studying some properties of exponential family distributions, i.e., distributions whose pdf/pmf can be written (in its "natural" form) as $$f_X(\mathbf{x}\mid\boldsymbol \theta) = h(\mathbf{x}) \...