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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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
12 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
20 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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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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27 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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11 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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41 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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22 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
42 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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14 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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26 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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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 $X$ ...
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1answer
38 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 ...
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1answer
19 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
30 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
24 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 $4!$ ...
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1answer
8 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
24 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
79 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
20 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
33 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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89 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 ...
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1answer
17 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 ...
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1answer
22 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
34 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
41 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 ...
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0answers
28 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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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 ...
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1answer
44 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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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 ...
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0answers
14 views

Integral of Constant Parameter Martingale

What is the $\int_{1}^{t}W_1W_sdW_s$. This is the question solved by Kuo in his paper an extension of the Ito's Integral (2008) but there limit runs from $0$ instead of $1$.
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Exercise on stationary measures.

This is a question from Durrett, exercise 6.5.4. Recall that $$ \mu_x(y) = E_x\left( \sum_{n=0}^{T_x-1} 1(X_n = y)\right) = \sum_{n=0}^\infty P_x(X_n = y, T_x > n)$$ is a stationary measure and ...
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12 views

Asking for helps about deriving arcsine distribution

I solved the above exercise. And the exercise below is based on the exercise above. Here, I managed to show the first equality of (i). But I can't find a way how to prove the second equality of ...
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1answer
30 views

Distribution of a transformed Brownian motion

Let $W$ be a standard Brownian motion. From an earlier proven result I know that $N_t = \exp\left\{a W_t - \frac12 a^2 t \right\}$ defines a martingale on the natural filtration of $W$ for all $a \in ...
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1answer
39 views

Finding Variance

I am a little confused on how to go about finding different parts of the Variance of a random variable. Here is the question. A total of $n$ balls, numbered $1,.. n$, are put into $n$ urns, also ...
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29 views

understanding darts probability

Note: this problem for who understands the game of darts Hello iam trying to compute the probability of a dart to hit a ring if you know that the opportunity to miss the ring is 10% what will the ...
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1answer
36 views

Divergent series of independent RV

I'm trying to prove that if $\{X_n\}_{n=1}^{\infty}$ is a sequence of independent random variables with the same distribution and $P(X_1 \neq 0)>0$, then the series $\sum_{n=1}^{\infty} X_n$ is ...
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11 views

Book recommendation needed: asymptotic behavior of non-stationary Markov chain

Is there any stochastic process textbook which covers some standard results for non-stationary Markov chain? For my purpose, countable state space is enough. Thanks!
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1answer
48 views

How to find $E[X^2\mid X+Y]$?

Suppose $X$ and $Y$ are independent Poisson random variables with rates $\lambda_1, \lambda_2$ respectively, then how would we go about calculating: $ E[X^2\mid X+Y] \text{ ?} $$
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1answer
16 views

Characteristic Function and Density Function

Consider a random variable $X$ with density function $f(x)$, moment generating function $M(t):= \int e^{tx}f(x) dx$ (existing in an interval containing $0$), cumulant generating function $K(t):=\log ...
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1answer
28 views

Calculate $\mathbb{E}(T^2)$ and $\mathbb{E}(\int_0^T X_s \,d s)$ for exit time $T$ of Brownian motion $(X_t)_{t \geq 0}$

Let $T$ be the exit time of from the interval $[-b,a]$ of a standard Brownian Motion $X_t$, then how would we go about calculating the following two expectations: $E[T^2]$ (and) $E[\int_0^T X_tds]$? ...
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29 views

Measurability of the event that Brownian motion hits a given set

Let $W$ be a Brownian motion in $\mathbb{R}^{2}$ on a probability space $\left(\Omega,\mathcal{F},\mathbb{P}\right)$ . Let us assume $\mathcal{F}$ is the sigma-algebra on the path space ...
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1answer
26 views

Deciphering proof of SLLN

I was looking at a proof of the string law of large numbers, and am having trouble finding where the proof uses the assumption that the random variables are identically distributed. I'll reproduce the ...
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1answer
23 views

How transformation of co-ordinates system relates to its vectors?

Consider a positive definite matrix. Can we consider that it has a underlying co-ordiante system? If we transform that co-ordinate system how the the vectors are transformed? Is this question even ...