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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5 views

Existing of a distribution of three random variables that have conditional mutual information with defined properties.

I have two similar questions: 1)Does exist a distribution of three random variables such that: $I(a:b) = 0$ and $I(a:b|c)>0$ (where $I(a:b)$ is a mutual information and $I(a:b|c)$ is a ...
3
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0answers
18 views

Guessing number in set 1-100 with weighted questions.

It is needed to guess number from 1 to 100. I can ask questions and get answers:"yes" or "no". For the "yes"-answer I must pay one dollar, for the "no"-answer - two dollars. How many dollars should I ...
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0answers
6 views

Multivariate Berry-Esseen/ Please help!

I've got a problem with understanding Berry-Esseen inequality for random vectors. You see, I keep coming across various forms of this theorem, all assuming a unit covariance matrix $I$, though it's ...
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0answers
15 views

what does it mean that a system is attractive?

What does it mean that a system is attractive in the context of Statistical Mechanics? Is this notion related to the presence of some monotonicity properties?
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14 views

5 point estimation

I want to perform Probabilistic load flow using 5 point estimation method based on method of moments. To estimate five points from a specific probability distribution along with their corresponding ...
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1answer
12 views

Lower bound on convergence in probability

Suppose that we have a sequence of continuous random variables $X_n$ that converges in probability to $X$. Given any $t$ and $\epsilon$ is it true that $$P(X \leq t- \epsilon) \leq P(X_n \leq t) ...
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1answer
26 views

Relation of $\mathbb P \{X\in B|\mathcal A\}$ and $\mathbb P \{X\in B|Y=y\}$

Consider a probabilty space $(\Omega ,\mathcal F, \mathbb P$) and two measurable random variables $X,Y:\Omega \rightarrow S$. Define $\mathcal A :=\sigma(Y)\subset \mathcal F$ and $\mathcal B ...
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9 views

How to find the number of transitions, after which the stationary distribution could be found in Markov chain?

Say I have the initial state space vector S = [1 0 0]. and I know both the transition matrix, P and final stationary distribution, S' = [0.3 0.5 0.2]. If I was asked to calculate after how many ...
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14 views

Question on Markov Chains

Let $S=\{1,2,\ldots,d\}$ for some $d\geq 2$. For $i\in(1,d)\cap \mathbb{Z}$ let $p(i,i+1)=p(i,i-1)=\frac{1}{2}$. Let $p(1,1)=p(d,d)=1$. How would I be able to find all invariant probability ...
3
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1answer
19 views

How to calculate $\mathbb{E}((B_3-B_2)(B_4-B_{\pi}) \mid B_1)$ for a Brownian motion $(B_t)_{t \geq 0}$

I want to find the best predictor of $(B_3-B_2)(B_4-B_{\pi})$ given an observation of $B_1$ Where $B_t$ is brownian motion for time $t \geq 0$. I am not sure how to approach this. I know it will be ...
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24 views

To determine probability distribution for large $N$ with mean at m

To show that the following expression turns to Gaussian for large value of $N$ $$\binom{N}{S}\binom{X-1}{a}\binom{Y-1}{a-1}$$ where X+Y+S=N. To show it shows normal distribution with mean at 'm' and ...
2
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0answers
21 views

Formal Expansion of another Expansion

Given a function $f(x)=\sum_{n=1}^{\infty}\frac{c_n}{x^n n!}$, where $c_n$ are constants, we want to find the formal series expansion of the function $g(x)=\exp(f(x))$ in terms of $x$. I want to ...
2
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1answer
11 views

Find asymptotic variance MLE heavy tailed distribution

$$\mathbf{X} = \{X_1,X_2,\dots,X_n\}$$ sequence of i.i.d. RV's. Let the distribution of the RV's be defined by $$f(x|\theta)=\frac{\theta}{x^{\theta+1}}, \quad x>1, \quad \theta>1$$ I am ...
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0answers
15 views

When Independence $\Rightarrow$ Independence of higher moments (Prob/ Stats)

suppose {$X_n$} is iid. Then, is $X_i$ independent of $X_j^3$ for j≠i? If so, why? Secondly, is $X_i^2$ independent of $X_j^2$ for j≠i? Intuition: yes no If there's a difference, why? (note: ...
2
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0answers
25 views

Joint Expectation of independent Random Variables given two sigma-algebras

We have a question regarding two random variables $X$,$Y$ on a probability space with sigma-algebra $\mathcal{F}$ and a sub-sigma algebra $\mathcal{M}$ such that $X$ is independent of $\mathcal{M}$ ...
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2answers
15 views

Conditional probability of a Joint distribution

Let $(X,Y)$ have joint density $f(x,y)=e^{-y}$ , for $0<x<y$, and $f(x,y)=0$ elsewhere. What is $f_{X\mid Y} (x,y)$ for $0<x<y$? I think that the answer is $1/y$, however, I am having ...
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1answer
25 views

Basic finite dimensional distribution question

I'm having trouble wrapping my head around the basic idea of a finite dimensional distribution. Suppose $(\Omega, \Bbb P, \mathcal{F})$ is a probability space. Let $(X_{t})_{t \geq 0}$ be a ...
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0answers
29 views

How do we improve the probability by repeating the experiment? [on hold]

In my class our sir told that probability of an experiment increases by repeating the experiment more times. for example i came through an example that Suppose a bin has white marbles and black ...
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0answers
31 views

Probability and stats question [on hold]

Im kinda confused on how to approach this question and how to solve for it. i know there are multiple ways but i was wondering if anyone could help me. thanks 1.) It appears that the mean commuting ...
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0answers
14 views

The mutual information rate spectrum

Definition: $\mathbf{X}$ denotes the random vector $({X_1},{X_2},...,{X_n})$. The mutual information between $X$ and $Y$, $I(X;Y)$, is determined by the joint law of $p(X,Y)$, Given two random ...
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0answers
45 views

How to randomly select a point from the surface of a unit sphere ?

Construct in $\Bbb R^k$ a random variable $X$ that is uniformly distributed over the surface of the unit sphere in the sense that $|X|=1$ and $UX$ has the same distribution as $X$ for orthogonal ...
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0answers
25 views

Why do we need to declare a probability measure for the definition of stochastic processes?

Let $(\Omega,\mathcal{A})$ be a measurable space $E$ be a Polish space and $\mathcal{E}$ be the Borel-$\sigma$-algebra on $E$ $I\subseteq\mathbb{R}$ $X_t$ be measurable with respect to ...
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1answer
45 views

Trying to understand the behaviour of i.i.d.

In a course called introduction to probability theorem we are covering now i.i.d. (independent and identically distributed random variables). I already know when two variables are independent: $X, Y$ ...
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0answers
16 views

When is a coupling ''natural''?

The definition of coupling is written below. In some articles, I found the term "natural coupling". When is a coupling said to be ''natural''? Definition of coupling between two random variables: Let ...
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0answers
27 views

Why the two expressions of total variation distance are equivalent?

In a stochastic processes textbook, I find the definition of total variation distance is $\|\pi - \nu\|_{TV} = \max\{|\pi(A) - \nu(A)|:A\subset S\}$ where $\pi$ and $\nu$ are two probability measures ...
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0answers
15 views

Definition of convergence rate of random variables

What is the definition of rate of convergence of a sequence of random variables? i.e. what does it mean that the convergence rate of the sequence of random variable $X_1,X_2,\ldots X_n,\ldots$ to $X$ ...
1
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1answer
40 views

Simple probability with dice

Suppose you roll a 6-sided dice 6 times. a. What is the probability that all of the rolls show either 1, 2, or 3? Would the answer be $(1/2)^6$? b. What is the probability that all of the rolls ...
2
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1answer
20 views

Continuity sets are neccesary for weak convergence.

Portmanteau theorem In particular if $\mu_n \to \mu$ weakly then $\mu_n(A)\to \mu(A)$ for each continuity set. I want an example to show that the hypothesis of $\mu(\partial(A))=0$ is neccesary. ...
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1answer
31 views

On the Preservation of Product Measurability under Partial Conditional Expectation.

Let $(X,\mathcal{X},\mu)$ and $(Y,\mathcal{Y},\nu)$ be probability spaces, $\mathcal{X}_{0}\subset\mathcal{X}$ a (sub)sigma field and assume that $f=f(x,y)\in L^{1}_{\mu\otimes \nu}$ where $(X\times ...
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2answers
28 views

Find probability of event

Task is: Find probability of 4 aces laying in row in a deck of 36 cards. All possible shufflings of 36 deck is $36!$ I can place 4 cards in a row with $33$ different ways. And each way can be $4!$ ...
2
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1answer
11 views

Show that $L^2(\Omega, \sigma(X),P)$ is a closed hilbert subspace of $L^2(\Omega, \mathbb{A},P)$ s.th $\sigma(X) \subset \mathbb{A}$

I was self-studying probability theory(conditional expectation). I know that a subspace is $U$ of $V$ is a set $U \subset V$ s.th $\forall x,y \in U$ and $\forall \alpha, \beta \in F$ we have that ...
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1answer
25 views

proving independence of stochastic integrals

Does anyone know how to show that the stochastic integrals \begin{equation} \bigg\{ \int_0^1 \cos \Big[ (n- \frac{1}{2}) \pi t \Big] \,dW_t \bigg\}_{n \in \mathbb{N}} \end{equation} are ...
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2answers
82 views

Clever way of finding $\int_0^\infty x\Phi(x)\phi(x)dx$

Suppose that $\Phi$ and $\phi$ are the Standard Normal c.d.f and p.d.f. respectively. Then, evaluate $$\int_0^\infty x\Phi(x)\phi(x)dx$$ There is no use of my trying to show my approach because ...
2
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1answer
17 views

Weak and vague convergence of normal distribution

Let $\mu_n = \mathcal{N}(0,n)$ be the normal distribution with mean $0$ and variance $n$ on $\mathbb{R}$, $\nu$ the zero-measure (which is defined by $\nu(A) = 0$ for any ...
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0answers
14 views

Probabilities for the repetition of the same experiment $N$ times

Sometimes one experiment we want to discuss in terms of probabilities is in truth the same as performing another experiment $N$ times. I have a doubt on how to relate the probabilities for the ...
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1answer
21 views

Decision-making with random term

Consider the following situation. There are multiple options to choose from based on an attribute related to those options. For example: ...
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1answer
35 views

Show that $\Omega\setminus A_1, Ω\setminus A_2,\ldots, \Omega\setminus A_n$ are independent

Let $(\Omega, \Sigma, P)$ be a probability space and let $A_1, A_2, \ldots , A_n$ be independent events in this probability space. Show that $\Omega\setminus A_1, \Omega \setminus A_2, \ldots , \Omega ...
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2answers
95 views

At time n, randomly choose a natural number ≤n. How long is it until a single number is chosen three times?

To clarify, the number ≤n is chosen uniformly at random at each step, and n chooses from the natural numbers beginning with 1. I wish to determine the expected value of $n$ at which a natural number ...
0
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1answer
18 views

product of two multivariate normal densities for the same vector, if one is only specified for a subset

A random vector x with n elements has a multivariate-normal density f(x). Another distribution is known for m linear combinations of elements of x. The linear combinations are given in the form ...
1
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1answer
23 views

Almost sure convergence, arithmetic mean, variance

I am stuck proving that this sequence $$\sigma^2_n:= \frac{1}{n-1} \sum_{i=1}^n (X_i - \frac{X_1+...+X_n}{n})^2$$ is converegent almost surely to $D^2X_i = \sigma^2$. We assume that $X_1, X_2, X_3, ...
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1answer
44 views

Help with two probability questions. Classic definition of probability.

The first can be done using condition probability, but was wondering how to do it just with the classic definition of probability? Both questions are in the same part of the book, and therefore i ...
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3answers
35 views

Problem on Baye's formula

I was reading A First Course in Probability by Sheldon Ross. I think I quite understood the below problem but I still feel fuzzy. Problem: In answering on a multiple choice test, a student either ...
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0answers
22 views

why does $X,Y \in L^2 $ and $E[X^2]=0 \implies X=0$ everywhere and not almost surely

If $L^2$ denote all (equivalent classes of almost sure equality) random variables $X$ such that $E[X^2] < \infty $. Note here we are identifying all random varibles $X,Y$ in $L^2$ that are equal ...
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1answer
45 views

Jee Main 2015 Question. Probabilty

If $12$ identical balls are to be placed in $3$ identical boxes, then the probability that one of the boxes contains exactly $3$ balls is: (1) $22 \times(\frac{1}{3})^{11}$ (2) $\frac{55}{3} \times ...
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0answers
17 views

Random Variables and Statistic

I'm studying Statistical Inference by Casella and I'm confused with the definitions of random variable & statistic. So let we have the probability space $(\Omega, F, P)$ where $\Omega$ is the ...
2
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0answers
31 views

Convergence in probability, expected value

I have problems with the following two sequences of random variables: We assume that $X_1, X_2, ... $ are iid. Let $m=EX_i$ The first one is: $$ \alpha_n := \frac{1}{n} \sum_{i=1}^n (X_i - m)^2$$ I ...
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0answers
20 views

expectation calculation problem small problem

a Continuous, positive random variable X, whose PDF is proportional to $(1+x)^{-4}$, where $0<x<\infty$, determine $E(X)$ i tried to solve it directly by integrating from 0 to infinity to get ...
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2answers
23 views

expectation calculation problem

I got the answers for this and i know its 1.05 but the way it explains is very difficult to understand so im seeking for some help here. A system made up of 7 components with independent, identically ...
1
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1answer
46 views

Tail field of random variables in $\mathbb{Z}$

Let $X_1, X_2, \ldots$ i. i. d. with values in $\mathbb{Z}$, define $S_0 := 0$, $S_n := X_1 + \cdots + X_n$ and $R_n := \{S_n = 0\}$ for $n \in \mathbb{N}$. Show that ...
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0answers
27 views

Is there a name for this stochastic process?

Let $(\Omega,\mathscr{F},P)$ be a probability space and $\{X_n\}_{n\geq 1}$ be a stochastic process. Assume each $X_n$ only takes two values $0$ or $1$, i.e., $X_n:\Omega\rightarrow \{0,1\}$. Of ...