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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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1answer
18 views

Validity of Proof of Wald's identity

$\newcommand{\E}{\mathbb{E}}$ Theorem (Wald's identity): Suppose $\{X_i\}_{n \in \mathbb{N}}$ is a sequence of i.i.d random variables with $\E X_1 < \infty$. Let $\tau$ be a stopping time with ...
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0answers
7 views

Poisson process on nonintersecting sets

I try to show that for a homogeneous Poisson process $N$ with intensity $\lambda$ and nonintersecting sets $A,B$ the amount of events happening in the two sets are independent. My idea is to prove the ...
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0answers
12 views

Prove that stochastic prcess trajectories are continous.

Consider gaussian process $\{X_{t}, t \in [0,1]\}$ with zero mean and covariance function $R(X_{s}, X_{t}) = \min (s,t) -st$. We want to know does this process has continous trajectories, i.e. $\...
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Confused about the expression relating the CDF and expected value of a random variable

Let X be a random variable that takes on nonnegative values and has distribution function $F(x)$. $E(X) = \int_{0}^{\infty}1-F(x)dx$ The proof my book gives is: This statement was included at the ...
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0answers
7 views

Distribution of a coupling time of 2d Brownian motions

I have a question about the distribution of a coupling time, which appears in this paper: GRV. Let $B=(B_t)_{t \ge 0}$ and $W=(W_{t})_{t \ge 0}$ are independent two-dimensional Brownian motions. $B$ ...
4
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1answer
24 views

Series Converging Almost Surely But Diverging in Mean

I am looking for an example of independent, non-negative random variables $X_1, X_2, \dots$ such that $$ \sum_{n=1}^{\infty} X_n \, \lt \, \infty $$ almost surely but $$ \sum_{n=1}^{\infty} \...
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0answers
9 views

Regression on trivariate data with one coefficient 0

Suppose {$(x_i,y_i,z_i):i=1,2,...,n$} is a set of trivariate observations on three variables:$X,Y,Z$, where $z_i=0$ for $i=1,2,...,n-1$ and $z_n=1$.Suppose the least squares linear regression equation ...
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1answer
39 views

$E(X_k|Y)$ for $X_k \sim Bern(\theta)$ and $Y := \sum_{k=1}^n X_k.$

Let $X_1, ..., X_n$ be i.i.d. random variables with $X_k \sim Bern(\theta)$ for $\theta \in (0,1)$. Furthermore, define $Y := \sum_{k=1}^n X_k.$ Determine $E(X_k|Y).$ Since $X_k$ and $Y$ are ...
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0answers
15 views

Error bound for Sparse Regression

My lecture notes state the following: We have, for all $t_0\geq 0$, $$ P\left(\|X(\beta-\beta^*)\|^2\geq 160k \cdot \log(2d/k)+8t_0\cdot k\right) \leq e^{-t_0\cdot k / 10}.$$ Then, using (for non-...
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1answer
9 views

A aperiodic state of a Markov chain has $N\geq 1$ such that $\forall n\geq N:p_{i,i}(n)>0$

The question I get asked is the following, I'm completely stuck on the problem: Let $i$ be an aperiodic state of a Markov Chain. Show that there exists $N\geq 1$ such that $p_{i,i}(n)>0$ for all ...
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1answer
32 views

Expectation and variance in martingale system

Suppose we are betting money on the outcome of a game of chance with two outcomes (e.g. roulette). If we guess correctly we get double our bet back and otherwise we lose the money we've bet. Consider ...
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0answers
22 views

Is the following always true: $\mbox{Var}[\mbox{Range}(X_1,\cdots,X_n)] = O(n^{-B})$ with $0\leq B \leq 2$?

Here $X_1,\cdots,X_n$ are i.i.d. The two extremes $B=0$ and $B=2$, and the standard case $B = 1$ are illustrated in the picture below. For the reference, see here.
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25 views

Constructing a Characteristic function

I'm working on an assignment and I am having difficulty with the following question: A particle is moving in one direction, covering each second 1 meter of distance starting at the the time $t = 0$ ...
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1answer
27 views

Understanding definition of random variable in textbook “Basic Stochastic Processes”

In Zdzislaw Brzezniak and Tomasz Zastawniak, "Basic Stochastic Processes" A random variable is defined as, How do I check the condition For every Borel set $B \in \mathcal{B}(\mathbb{R})$ ...
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0answers
21 views

Transformation of Bivariate Random variables

I am having trouble with this question and feel like my supports are not entirely correct: Given $f_u(u)= 1_{[1,0]}(u)$ and $f_v(v)= 2v*1_{[1,0]}$ with U and V being independent and $U,V \in \mathbb{...
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3answers
41 views

How to calculate covariance with an minus like V(X-Y)?

my task is this: Be $ X $ and $ Z $ independent with the same distribution and $ Y :=X-Z . $ Calculate $ \operatorname{cov}(X, Y) $ and $ \operatorname{corr}(X, Y) . $ My Problem is the minus in $...
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1answer
41 views

Yet another question on Expected number of flips to get 2 consecutive heads

So, there are quite a few solutions on the web to the problem of "what is the expected number of flips to get 2 consecutive heads for a fair coin". Many of theses solutions use the conditional ...
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0answers
42 views

How does Feller get this inequality?

I am currently loosing my mind trying to understand the proof Feller gives in his Book "An Introduction to Probability Theory and it's Applications" Vol2 about the general Berry-Esseen Theorem. What I ...
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1answer
52 views

Show that $X_n \to 0$ in probability under given condition.

Let $k > 0$. Suppose that $$\forall \epsilon > 0: \exists N: \forall n \geq N: P(|X_n| \geq \epsilon) \leq \epsilon k$$ Show that $X_n \xrightarrow{P}{ 0}$. Attempt: We have to show: $$\...
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0answers
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Combinatorial inequality in Erdös-Kac proof.

I am reading a proof of Erdös-Kac theorem, in Durrett, "Probability: Theory and Examples", fourth edition. In some point, it is stated that $(\sum_{m=1}^nEZ_{n,m}^2)^k - \sum_{i_j} EZ_{n,i_1}^2 . EZ_{...
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1answer
31 views

Using Markov's inequality to find upper bound of tail probability for Gaussian random variable(Q-function).

Suppose $X$ be the standard normal distriution. Its tail probability will be $$Q(t)=\frac{1}{\sqrt{2\pi}}\int_{t}^{\infty}e^{-\frac {x^2}2}\,{\rm d}x$$ I need to find the upper bound of $Q$ function ...
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0answers
16 views

Existence of a weak solution to this SDE?

I am looking at an SDE of the form $d{X_t} = \left( {{1_A}({X_t}) - {1_{{A^c}}}({X_t})} \right)d{W_t}$ such that ${X_0} = 0$, $A \in \mathcal{B}(\mathbb{R})$ and ${A^c}$ has a lebesgue measure of zero....
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0answers
12 views

Lp Wasserstein Distance for two distributions which have equivalent partial marginals

Consider two distributions $p(x_1,x_2)$ and $q(x_1,x_2)$ which have equivalent partial marginals, say $p(x_1)=q(x_1)$. I am wondering if there is any relationship between two Wasserstein ditances $W_{...
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0answers
25 views

Some questions related to expected value of a random variable

My book states that if $\phi$ is a continuous function that maps from the range of $X$ to $R$, then $E(\phi(X)) = \int_{-\infty}^{\infty}\phi(x)f(x)dx$ (for the discrete case $\sum_{x\in\Omega}\phi(x)...
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0answers
36 views

A question about estimating population mean

Consider conditional mean of $Y$ given $X$. Let $g(X)=E(Y|X)$, and $$\mu=EY=E(E(Y|X))=Eg(X)$$ Let $\hat\mu_1=\frac1n\sum_{i=1}^n g(X_i)$. It is easy to see that $$\sqrt{n}(\hat\mu_1-\mu)=\frac1{\...
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0answers
21 views

Probability: Poisson distribution; unsorted elements after random review

The problem I am trying to solve is formulated as following: Random number $\mathit{N}$ of unsorted elements in a set is Poisson distributed with $\lambda>0$. These elements are found ...
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0answers
21 views

Relationship between tightness and weak convergence.

I am getting confused with the following set of notes http://www.stat.umn.edu/geyer/8112/notes/metric.pdf. On page 10, the author states Prokhorovs theorem as follows: Let $X_1,X_2,\dots$ be a ...
2
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1answer
23 views

Proving this sequence converges in $L^2(\mathbb{P})$

We have some IID sequence, $\left\{ {{X_n}} \right\}_{n = 1}^\infty $, of standard normal random variable on the probability space $\left( {\Omega ,\mathcal{F},\mathbb{P}} \right)$. Also $\left\{ {{\...
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3answers
58 views

Two dimensional induction

I have the following problem: I need to prove that given the following integral $\int_{0}^{1}{c(k,l)x^k(1-x)^l}dx = 1$, we the constant $c(k,l) = (k+l+1) {{k+l}\choose{k}} = \frac{(k+l+1)!}{k!l!}$, ...
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0answers
29 views

Using mirrored sample data to improve estimates

I will ask my question through an example game: Each round we are given a blue coin and a red coin. We are either given a pair of fair coins or a pair of biased coins (these four coins are the set of ...
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1answer
23 views

Probability Theory - Combinations

Please, I would like some help with the following problem. I tried to use combinations but I am wondering if I have to use also the Bayes formula, in the process of solving it. The problem is at it ...
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0answers
7 views

Markov property for Simple Birth Process at random time

Let $(X_t)_{t \geq 0}$ be a simple birth process with rates $\lambda_n$, $n\geq 0$ starting from $k$. The Markov property states that the two processes $(X_t)_{0 \leq t \leq r}$ and $(X_{s+r})_{r \...
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1answer
20 views

understanding Krylov-Bogolubov Theorem

Could anyone tell me what is $P(x,dy)$ and $P^{n+1}(x,dy)$ means here in 1p- 30, in the proof of thm 4.17? And, why $\phi$ was taken bounded by $1$? Are all $P^k(x,A)$, $P$, $Q^n$ probability measure ...
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1answer
32 views

How to calculate E[Xi Xj]?

This question is from an example in the book of Bertsekas. (p240 of 1st edition). I would like to know why $$E[X_{i} X_{j}] = P(X_{i} = 1\text{ and }X_{j}=1)$$ and $$E[X_{i}] = P(X_{i}=1)$$. please ...
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0answers
14 views

Understand a probability distribution

I have a message transmision channel with input: a vector $x\in F_2^n$ and output: a vector $y\in F_2^n$ such that $y\neq x$. I call $E$ the set of positions where an error was transmitted, and $P$ ...
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0answers
6 views

EM algorithm for Probabilistic PCA: Complete-data log likelihood function

Consider the probalistic pca setting from "Pattern recognition and machine learning" by Bishop, where $x \in \mathbb{R^d}$ is an input vector drawn from $p(x)$, $z \in \mathbb{R^m}$ is an explicit ...
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1answer
6 views

Does the product of the CFs $\phi_{\mu}, \ \phi_{\lambda}$ of two finite measures correspond to the CF of the convolution of $\mu$ with $\lambda$?

It is known that if $X, \ Y $ are independent random variables then the distribution of $X+Y $, which we may get by convoluting the distributions of $X $ and $Y $, has characteristic function $\phi_{X+...
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0answers
15 views

Does a positive martingale necessarily converge to a finite limit?

The supermartingale convergence theorem says that if $X_n \geq 0$ Is a supermartingale, then $X_n \to X$ a.s for some $X$. Further, $EX \leq EX_0$. My question is whether the following is a valid ...
2
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1answer
33 views

Characterization of the geometric distribution

$X,Y$ are i.i.d. random variables with mean $\mu$ , and taking values in {$0,1,2,...$}.Suppose for all $m \ge 0$, $P(X=k|X+Y=m)=\frac{1}{m+1}$ , $k=0,1,...m$. Find the distribution of $X$ in terms of $...
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1answer
34 views

Recurrence formula for the moments / product moments of some order statistics

I am just interested in $E[L_n], E[U_n], E[L_n U_n], E[L_n^2]$ and $E[U_n^2]$ where $L_n =\min(X_1,\cdots,X_n)$ and $U_n=\max(X_1,\cdots,X_n)$. The $X_k$'s are i.i.d. In fact, I am only interested in $...
2
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1answer
40 views

Let $S_n = X_1 +\cdots +X_n$. Is $\sigma\left(X_j, 1 \leq j \leq n \right) = \sigma\left(S_j, 1 \leq j \leq n \right)$?

Let $X_1, \cdots, X_n$ be $\mathcal{L}_1$ random variables on a probability space $\left( \Omega, \mathscr{F}, \mathbb{P}\right)$. Define for $n \geq 1$, $S_n:= X_1 + \cdots + X_n$. Is $\sigma\left(...
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1answer
25 views

What is the probability that a random walk starting at 0 will reach +2 in 2 steps, 3 step, 4 steps, etc.? [duplicate]

The random walk I am referring to is a symmetric, unbiased, 1D random walk. In an answer given in the link below, the probabilities are given for S1, but I am trying to find out what it is for S2, ...
0
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1answer
24 views

Krylov Bogolubov Theorem in Unbounded space

Let $P$ be a Feller transition probability on an unbounded space $X$, if there exists $x\in X$ such that the sequence $\{P^n(x, \cdot)\}_{n\ge 0}$ is tight, then show that there is a probability ...
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2answers
35 views

Let $\bar X_n$ be the sample mean. What is the accurate rate of $\bar X_n-\mu$ convergence to $0$,

Suppose $X\sim N(\mu,\sigma^2)$ and $X_1,\cdots,X_n$ are samples from $X$. Let $\bar X_n=\frac1n\sum_{i=1}^nX_i$. Then it is well known that $$\bar X_n\overset{p}\to\mu\qquad\qquad(1)$$ and $$\sqrt{n}(...
0
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1answer
14 views

Conditional Expectation: How to Intergrate indicator function multiplied by the joint denisity?

I am currently reading "Measure, Integral and Probability" by Capinski, Marek (see p179). It includes some motivation for the definition of the conditional expectation. For example, given two random ...
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0answers
8 views

Taking the Derivative of independent RVs wrt to each other

Suppose that $(\Omega,\mathcal F,P)$ is a probability space and $X,E$ are (real-valued) rvs. Assume that $P(E) = P(E\mid X)$ (i.e. $E$ and $X$ are independent). Does that imply $E$ is not a function ...
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1answer
26 views

Will these two random variables be independent?

Assume you have two continuous random varaibles $X,Y$. Also assume that their joint probability density function can be written $f(x,y)=p_1(x)p_2(y)$, must they then be independent? The problem is ...
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0answers
26 views

understanding time-homogeneous markov chain

Could anyone make me understand the definition here 1 on page 7 definition 2.25, I quite do not understand the notation $P(a)(A)$, what does this mean? Also, is $P(a, A)$ a probability measure from ...
0
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1answer
19 views

Continuous mapping theorem, multivariate case, joint distribution.

I came across the following problem. Convergence in the following always means weak convergence, i.e. $X_n \rightarrow X$ if and only if $Ef(X_n) \rightarrow Ef(X)$ for all $f$ bounded, continuous ...
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0answers
17 views

Proving $P(y\in C\mid X=x)=\int_C \frac{f(x,y)}{f_1(x)}dy$

Consider $(\mathbb{R}^2,\mathscr{B}_{\mathbb{R}},P)$, $(X,Y)$ are random values such that $X:(x,y)\to x\\ Y:(x,y)\to y$ and $f$ is their density function. Let $$P(C)=\int\int_C f(x,y)dxdy;C\in\...