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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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24 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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13 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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0answers
12 views

Convergence of stopping times

Let $t > 0$ and let $\{ T_k \}_{k \geq 1}$ be a sequence of non-decreasing stopping times with $T_{k} \uparrow t$. Let $(X_s)_{s \geq 0}$ be some stochastic process. I am looking for a result of ...
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1answer
21 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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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
16 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
29 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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13 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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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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10 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 ...
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1answer
24 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
19 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 $...
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1answer
34 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
24 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, ...
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1answer
18 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
32 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}(...
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1answer
12 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
25 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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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 ...
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1answer
18 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
16 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\...
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1answer
25 views

“Independent sequences of independent random variables”?

I am re-reading Takács (1959) “On a sojourn time problem in the theory of stochastic processes”. At two points he wrote “Independent sequences of independent random variables”. Is the second “...
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0answers
18 views

Given a Markov process, are we able to construct another Markov process with the same transition semigroup but different inital law?

Let $E$ be a locally compact separable metric space $(T(t))_{t\ge0}$ be a strongly continuous contraction semigroups on $C_0(E)$ with generator $(\mathcal D(A),A)$ $(\Omega,\mathcal A,\operatorname P)...
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1answer
28 views

How can I calculate the Expected time to roll two dice and get all 1 through 36 combinations?

If I have two dice and I want to check the expected time it takes for all combinations to appear once, how can I calculate that? I am aware of the calculations for one die using geometric distribution,...
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1answer
74 views

How do I poof this Poisson distribution problem?

I have the following task. But first: The task is not a homework assignment, its just for me, I'm not a student yet. Task : P.S. I hope I did not use the wrong tags. I could not find somthink better ...
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1answer
38 views

Why is $Y_n$ not a bernoulli process?

for $n \in \mathbb{N}$, let $X_n$ be a Bernoulli process with parameter $p = \frac12$, let $N = \min \{n \geq 2: X_1 \neq X_n \}$ for $n \in \mathbb{N}$, let $Y_n = X_{N +n -2}$. in a question it ...
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2answers
77 views

What is the formalism that allows Random Variables to be treated algebraically like real or complex numbers?

We all know that if we have a variable x, then there is a meaning to - for example - $$y=e^x$$. And we all know how to manipulate that algebraically and to do calculus. For example, if $$y_1=e^{...
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1answer
15 views

Expectation of infimum of records

I'm trying to prove the next: Let $$L_{1}=\inf\{j\geq 2: X_j\space\text{is a record}\}.$$ Prove that $E(L_{1})=\infty.$ Here, we say $X_n$ is a record if $X_n>\max\{X_2,\ldots,X_{n-1}\}$ and $\{...
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2answers
30 views

convergence in distribution implies a.s and probability convergence

Let $(X_n)_n$ be a sequence of independent random variable. Let $W_n=\sum_{k=1}^nX_k.$ We suppose that $(W_n)_n$ converges in distribution. Prove that $(W_n)_n$ also converges almost surely and in ...
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4answers
40 views

Find pmf given a probability function

I'm learning probability theory and I am quite new in the concept. I'm stuck with the following problem: Consider a situation where people often get bitten by dogs (just as an example). Let $p_A(n)$ ...
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0answers
13 views

Asymptotic normality and multiplication by limit $1$ sequences

Suppose $X_n \sim \text{AN} (a,q^2/v_n^2)$. This means $v_n(X_n-a)\rightarrow_D N(0,q^2)$, where $\rightarrow_D $ stands for convergence in distribution, $a,q \in \mathbb{R}$ and $v_n$ is a real ...
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1answer
29 views

Is the sum of two (non-real) random variables necessarily a random variable?

Please note that I'm working with the following definition of random variable, which allows for a codomain other than $\mathbb{R}$. Definition: Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a ...
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0answers
20 views

Pushforward measure

Good evening, Let $\mu$ and $\nu$ two measures on $X$ and $Y$. Do you know when it exists a measurable function $h : X \rightarrow Y$ such as $ \nu = h\text{#}\mu$ with $ h\text{#}\mu(B) = \mu(h^{-1}...
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0answers
20 views

Convergence of Feller processes implies convergence of the resolvent operators of their generators

Let $E$ be a locally compact separable metric space $(T_n(t))_{t\ge0}$ and $(T(t))_{t\ge0}$ be strongly continuous contraction semigroups on $C_0(E)$ with generator $(\mathcal D(A_n),A_n)$ and $(\...
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0answers
22 views

Proving expectation and variance of a function of a random variable tends to a fix point

Given $f:\mathcal{X} \rightarrow \mathbb{R}$ is a continuous function and $\mathbb{E}_{Q(X)}[X] \rightarrow x^\star$ ($x^\star$ is a fix number), $\mathbb{V}\text{ar}_{Q(X)}[X] \rightarrow 0$. How can ...
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0answers
14 views

Proving convergence of expectation and variance given Rényi's $\alpha$-divergence tends to 0

I denote $p, q$ as density function of $P, Q$. Given $Y, X$ are random variables and \begin{align} \int q(x)\mathbb{D}_{\alpha}[p(Y\mid X=x)\,||\,p(Y\mid x^\star)] \,dx \rightarrow 0 \end{align} ...
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0answers
28 views

Proving the Asymptotic Equipartition Property (AEP) for i.i.d. random variables

I'm currently working on some assignment and I'm not sure if my solution would be considered as correct in any way: Let $(X)_{n \gt 1}$ be a stochastic process of i.i.d. random variables with ...
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0answers
16 views

Help me elucidate what “the vectors $(B(d):d \in \mathcal {D_n })$ and $(Z_t:t \in \mathcal {D-D_n})$ are independent” mean.

The following is taken from a proof of Lévy's construction of Brownian motion in a book by Peter Mörters and Yuval Peres. $\mathcal {D_n } := \{\frac {k } {2^n } :1 \le k \le 2^n \} $, the set of ...
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0answers
33 views

Showing uniform distribution on (0,1) of a random variable

Let $X,Y$ be two random variables and $Z:=(X,Y)$ and $Z\sim \mathcal{U}([0,1]^2)$. Show: $X\sim \mathcal{U}([0,1])$. The density of $Z$ is $1_{[0,1]\times[0,1]}(x,y)$ (indicatorfunction) and I want ...
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1answer
31 views

Show the map is measurable w.r.t. the product space

Let ${ ( X_t ) }_{ t \geq 0 }$ be an $\mathbb{R}^d$-valued stochastic process on $( \Omega, \mathcal{F}, P)$. I am trying to show that for any $A \in \mathscr{B} ( \mathbb{R}^d )$ the map $r : [ 0, \...
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0answers
18 views

martingale difference sequence

I am solving an exercise where I am using a martingale difference sequence, so here is my doubt. $\varepsilon_t$ is a stationary and ergodic martingale difference sequence. I get to one point where I ...
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0answers
29 views

Concentration in Gauss space

This is a theorem called Concentration in Gauss space. Let $f$ be a real valued Lipschitz function on $\mathbb {R}^{n}$ with Lipschitz constant $K$, i.e. $\left|f(x)-f(y)\right|\leq K\|x-y\|_{2}$ for ...
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1answer
42 views

How much do tails contribute to a Gaussian's total variance?

H${}$ello, if $X\sim \mathcal{N}(0,I_{n\times n})$ what is a good upper bound for $\frac{1}{n}\int_{A} \|X\|^2 d\mathbb{P}$ when $\mathbb{P}(A)<\varepsilon$? Thanks!
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0answers
40 views

Is there a deeper reason why the simple symmetric random walk on $\Bbb Z^D$ turns transient when increasing $D$ from 2 to 3?

Polya proved the following very well-known Theorem: A simple random walk on $\Bbb Z^D$ is recurrent if and only if it is symmetric and $D\le2$. Dropping simplicity (i.e. allowing jumps to non-...
3
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1answer
66 views

Expectation of Gaussian r.v. conditioned on positive r.v.s with positive covariances is positive

Suppose that $(X_1,\dotsc,X_K)^T \sim \mathcal{N}(0, \Sigma)$, with $\mathrm{cov}(X_i, X_j) > 0$ for all $i,j$. Prove that $$ \mathbb{E}[X_K 1\{ X_1> 0, \dotsc, X_{K-1}> 0 \} ] > 0$$ ...
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1answer
84 views

Figuring out the variance of the birthday paradox

Update to reflect what I think is the calculation for $E[X_1\cdot X_2]$ Given n people, if I want to estimate how many of them are likely to have an overlapping birthday with any other person, how do ...
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0answers
26 views

Obtaining a probability distribution after a change of variables.

I have two random variables $x_1,x_2$ whose distributions are unknown. I define $y_1=g(x_1,x_2)$ and $y_2=f(x_1,x_2)$ where $f(\cdot)$ and $g(\cdot)$ are known and the probability distributions $p(y_1)...
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0answers
15 views

Does convergence in distribution imply convergence of regular versions of the conditional expectation?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(E,d)$ be a locally compact complete separable metric space $D([0,\infty),E):=\left\{x:[0,\infty)\to E\mid x\text{ is càdlàg}\right\}$...