Modern theory of probability is formulated on the footing of measure theory. Use this tag if your question is about this theoretical footing (for example probability spaces, random variables, law of large numbers, central limit theorems, and the like). Use (probability) for explicit computation of ...

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2
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2answers
921 views

conditional distribution of random variable given its sum with another random variable

I am trying to figure out the following problem: I have two random variables: $X$ with pdf $f_X(x)$ on $[0,A]$ and $Y$ with pdf $g_Y(y)$ on $[0,B]$. Let's denote $Z=X+Y$. What should be the ...
4
votes
1answer
169 views

Application of central limit theorem for triangular arrays

A (1-dim) Brownian motion $(B_t)_{t \geq 0}$ satisfies the following properties: (B0): $B_0=0$ a.s. (B1): $(B_t)_t$ has independent increments (B2): $(B_t)_t$ has stationary increments, ...
1
vote
1answer
71 views

Can random variables have the same distribution but different conditional distributions?

Can two equally distributed random variables $X$, $Y$, defined over the same probability space have different conditional distributions relative to some $\sigma$-algebra $\mathcal{A}$, so that $X\sim ...
0
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2answers
227 views

Symmetric property for bivariate normal distribution

I'm trying to prove that the bivariate normal distribution has the symmetric property. I.E. N2(a,b;p)=N2(b,a;p) where a, b are constants (and the upper bound for their respective integrals.) and p is ...
2
votes
2answers
161 views

Coupon Collector's Problem

Let $\displaystyle (X_k)_{k\geq 1}$ be a sequence of random variables uniformly distributed on $\displaystyle \{1,...,n\}$. Let $$\displaystyle\tau_{n}=\inf\{m\geq 1:\{X_1,...,X_m\}=\{1,...,n\}\}$$ be ...
0
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1answer
104 views

Problems getting transformation function from source and destination random variables knowledge when handling the discrete case

In this question I asked about a way in order to find a specific transformation function $g(\cdot)$ in order to transform a random variable into another one. Thanks to the answer to that question I ...
1
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1answer
442 views

Expected value of a multivariate distribution

Given this random vector: $$ \mathbf{x} = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} $$ And this probability distribution function that takes it as argument: $$ f_\mathbf{X}(\mathbf{x}) = ...
0
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1answer
83 views

To what extent does conditional distribution determine conditional expectation?

Suppose $P\left(\left.X\right|\mathcal{A}\right)$ and $P\left(\left.Y\right|\mathcal{A}\right)$ are regular conditional distributions that satisfy: for any Borel set $B$, $P\left(\left.X\in ...
0
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1answer
35 views

Probabilistic statement about finitenes of a random variable

Suppose we have a r.v. $X\ge 0$ and a constant $c>0$. Then we look at the Laplace Transform: $$E[e^{-c X}]$$ We can suppose that this has a closed form $f(c)$. My question is now, why is it true, ...
1
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1answer
100 views

Interpolation result for Brownian Motion in Donskers Theorem

Suppose we have an increasing sequence of stopping times $\{\tau_n\}$ such that $\tau_n-\tau_{n-1}$ are iid. Furthermore let $B$ be a Brownian Motion and we define $S_n:=B(\tau_n)$ which gives a ...
2
votes
1answer
117 views

integer Random Walk with step size governed by a distribution.

This problem is for a final exam I am taking in a graduate probability class. Collaboration has been explicitly allowed, but with the remark that the professor felt he couldn't stop us even if he ...
1
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1answer
38 views

Probability with general solution

Let there exist box that a bee is trapped inside of. One of interior walls is black $\{B_1\}$, and the rest are yellow $\{Y_1,Y_2,Y_3,Y_4,Y_5\}$. Let $\{Y_1\}$ be opposite of $\{B_1\}$. If the bee ...
1
vote
1answer
207 views

Converge in Distribution

Let $(X_n)_{n\ge 1}$ be a sequence of i.i.d. random variables with standard Cauchy distribution, on the same probability space, and let $M_n = \max(X_1,...,X_n)$. Prove that $(nM^{-1}_n)_{n\ge1}$ ...
0
votes
1answer
31 views

Is my search for this distribution correct?

Let $D_j \sim \rm{Ber}(q_j)$ for $j=1,2,\dots,n$ and let $C_1,C_2,\dots,C_n$ be constants. Find the $f_{D_j,C_j}$.
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2answers
27 views

Central value of the partial exponential function [duplicate]

I need help calculating the central value of the partial exponential function : $$\lim_{n \to \infty} e^{-n} \sum^n_{k=0} \frac{n^k}{k!}$$ fd
0
votes
1answer
63 views

Why is $\lim_{n\rightarrow\infty}\frac{1}{n!}\sum_{\rho\in S(n)}\varphi\left(X^\rho\right)$ measurable w.r.t. the tail $\sigma$-algebra?

Let $X=\left(X_1, X_2, \dots\right)$ be an exchangeable family of random variables with values in a Polish space $E$. Fix $k\in\mathbb{N}$ and let $\varphi:E^k\rightarrow\mathbb{R}$ be measurable with ...
1
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1answer
172 views

Brownian Motion and the Functional CLT

Suppose we have a time series $(x_t\mid t\in \mathbb{Z})$ for which the partial sum process $X_T$ defined on the unit interval by $$ X_T(\xi)=\omega_T^{-1}\sum_{t=1}^{[T\xi]} ...
6
votes
1answer
207 views

Very basic doubt about Itô's lemma

While trying obtain the dynamics of $X_t = \exp( \int_t ^T \phi_s ds)$, where $\phi$ is an Ito process following $$ d\phi_t = \mu dt+ \sigma dW_t \ ,$$ I had some doubt concerning the application of ...
1
vote
1answer
238 views

Mean and Variance Convergence with r.v.

Let $(X_n)_{n\ge 1}$ be a sequence of random variables, with respective distributions being Gaussian, with respective mean $\mu_n \in \mathbb R$ and variance $\sigma_n^2 > 0$. Prove that if $X_n$ ...
1
vote
1answer
148 views

a.s. Convergence and Convergence in Probability

Let $(\Omega, \mathcal A,\mathbb P)$ be such that $\Omega$ is countable and $\mathcal A = 2^{\Omega}$. Prove that almost sure convergence and convergence in probability are the same on this ...
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0answers
123 views

Gamma Convergence of functionals on Probability measures

Would be grateful if someone could provide a hint or an appropriate reference for the following. Notation: $\mathcal{P}(\mathbb{R}^n)$- Space of probability measures on $\mathbb{R}^n$ ...
1
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1answer
92 views

Question about integration (related to uniform integrability)

Consider a probability space $( \Omega, \Sigma, \mu) $ (we could also consider a general measure space). Suppose $f: \Omega -> \mathbb{R}$ is integrable. Does this mean that $ \int |f| \chi(|f| ...
8
votes
1answer
129 views

Measure theory: existence of a monotone sequence of sets

Let $(X,\mathfrak B,\mu)$ be a probability space, and suppose that $\lim_n \mu(A_n) = m$ for some sequence of measurable sets $\{A_n\}_{n\geq 0}\subseteq \mathfrak B$. Is it true that there exists a ...
0
votes
1answer
111 views

Deriving the transformation function of a random variable from the original and the final distributions

Consider a random variable $X$ and consider that this variable can be either real or integral (so I would like to cover both cases: continuos and discrete random variables). Consider to transform this ...
3
votes
1answer
218 views

Probability computation involving order statistics

Let $X_1$, $X_2$.. $X_n$ be iid uniform random variables i.e. $X_i \sim U(0,1)$. We know that the order statistics, $X_{(i)}$ is beta distributed $X_{(k)} \sim B(k,n+1-k)$. Also let $Y_1$, $Y_2$.. ...
2
votes
1answer
95 views

Some preliminaries for the canonical construction of a Brownian Motion, help needed.

I have a lecture in stochastic analysis and I was given some facts, which are completely new to me and I do not really understand hot to understand/proof them. I would very happy if somebody could ...
2
votes
1answer
86 views

Density of $\frac{1}{X^2}$

I have to answer the following question: If $X$ is a standard Gaussian random variable, what is the density of $\displaystyle \frac{1}{X^2}$? Wikipedia states: ...
1
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1answer
123 views

Non-centered Gaussian moments

I would like to find a (closed nice) expression for the non-centered Gaussian moments with mean $\mu$ and variance $\sigma$. In found something in wikipedia: ...
2
votes
1answer
479 views

What is the analytic expression for PDF of joint distribution of two Gaussian random vectors?

I know that if $X$ and $Y$ are random variables with respective PDFs, $$ f_X(x) = \frac{1}{\sqrt{2\pi\sigma_x^2}}\exp\left\{-\frac{\left(x-\mu_x\right)^2}{2\sigma_x^2}\right\} \\ f_Y(y) = ...
5
votes
1answer
143 views

Exchangeability via Symmetric Functions

There are three ways to define the exchangeable $\sigma$-algebra of a stochastic process: one via symmetric functions, another via $n$-symmetric functions and a third via exchangeable events. Why are ...
0
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1answer
228 views

Interchanging the order of limit and expectation

Assume $\displaystyle\lim_{t\to0}X_t=\gamma\hspace{3pt}a.s.$ where $X_t\geq 0$. I would like to show that $\displaystyle\lim_{t\to0}E[X_t]=E[\lim_{t\to0}X_t]=\gamma$, i.e. that it's possible to ...
1
vote
1answer
757 views

Independent and mutually exclusive

Prove or disprove via proof that events $X$ and $Y$ can be independent and mutually exclusive if both of their probabilities are greater than $0$.
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3answers
99 views

proportion probability problem

I am just doing some olympiad exercises to practice my probability skills, but I have difficulties with this one: Lets consider a village where each resident gets off work at a random time. ...
11
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1answer
182 views

Calculate Asymptotics of Integral?

Let $f$ be a continuous function on $[0,1]$. How do I calculate the asymptotics, as $n\rightarrow\infty$, of $\displaystyle \int_{[0,1]^n}f\left(\frac{x_1+...+x_n}{n}\right)\text d x_1...\text d ...
2
votes
1answer
113 views

Maximum likelihood for $(\mu,\sigma)$ and other related questions

$$f(x)=\frac{1}{2\sigma}\exp\left(\frac{-|x-\mu|}{\sigma}\right)$$ $$\mu\in,\sigma>0$$ When trying to calculate the maximum likelihood for $(\mu,\sigma)$, I got as far as: $\log L(\mu,\sigma)=-n ...
13
votes
2answers
644 views

On the set of the sub-sums of a given series

Choose a sequence $(x_n)_{n\in\mathbb N}$ of nonnegative real numbers with finite sum $x=\sum\limits_{n\in\mathbb N}x_n$ and consider the set $X=\{x_I\mid I\subseteq \mathbb N\}$ where, for every ...
3
votes
1answer
147 views

A sequence of random variables which converges in distributon converges “to” some random variable

Let $(X_n)$ be a sequence of random variables on a probability space $\Omega$, with distribution functions $F_n$. Suppose $F_n \rightarrow F$ in distribution for some distribution function $F$. Must ...
1
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0answers
51 views

Expectation related to renewal measure

Let $X_1,X_2,\ldots$ be i.i.d. random variables, and $S_n=X_1+\cdots+X_n$. Assume that $0 < \mathbf{E}(X_1) < \infty$ (but don't assume that the $X_i$ are $>0$). Let $N$ be the almost surely ...
0
votes
0answers
247 views

Random Variable on a Sphere

Not sure where to start with this problem: For any $d\geq 1$, we admit that there is only one probability measure $\mu$ on $\mathcal S_d$, (the $(d-1)-th$ dimensional sphere embedded in $\mathbb ...
2
votes
2answers
200 views

$(X_n)$ an irreducible transient Markov chain. Is $f(x) = \mathbb{P}(X_n = x_0 \text{ for some } n > 0 | X_0=x)$ constant?

Let $(X_n)_{n=0}^{\infty}$ be an irreducible transient Markov chain with countably infinite state space $E$. Let $T_x = \inf\{n > 0 : X_n = x\}$. Let $\mathbb{P}_x$ be probability conditioned on ...
4
votes
1answer
63 views

$X_n - X_{n-1}$ is i.i.d. mean 1. Is $\frac{1}{n}X_n$ “nearly” decreasing a.s.?

Let $0=X_0 \leq X_1 \leq X_2 \leq \cdots $ be an increasing sequence of random variables with $X_n - X_{n-1}$ i.i.d. and $\mathbb{E}(X_n - X_{n-1}) = 1$ for all integers $n > 0$. I want to show ...
0
votes
1answer
99 views

What is the distribution of an unconditioned random variable knowing the conditional distribution?

I have two random variables $X$ and $Y$. I know that $Y$ can be approximated by a $N(\mu_1,\sigma_1^2)$ distribution (in particular $Y$ is not negative) and I also know that $X|Y \sim N(a+bY,c+dY)$ ...
4
votes
4answers
206 views

probability density function

In a book the following sentence is told about probability density function at point $a$: "it is a measure of how likely it is that the random variable will be near $a$." What is the meaning of this ? ...
3
votes
1answer
280 views

Central Limit Theorem for uncorrelated RV (counter example)

In Varadhan's probability book it is stated that there exists sequences of identically distributed uncorrelated (finite variance?) RV's such that the central limit theorem does not hold. The book ...
2
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0answers
74 views

Discontinuities of stochastically continuous Gaussian process

Can a stochastically continuous Gaussian process have essential discontinuities (non-jump)? Thanks!
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0answers
51 views

Separable processes

If $X_t$ is a separable (Gaussian) process, is the process defined by $Y_0=0, Y_h= \frac{\lvert X_{t+h}-X_t \rvert}{|h|^\gamma} $ for $h>0$, and $-t<h<0$ and some fixed $\gamma>0$ also ...
2
votes
1answer
121 views

Is Wishart Matrix?

Analyzing a system, I have faced a problem which is related to Random Matrices and in particular Wishart matrix. The problem is as follows: Lets assume $\boldsymbol{H}$ is an $m\times n$ random ...
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0answers
72 views

What's the probability of creating a Hello World Program?

Consider an application that has knowledge of all characters on the keyboard i.e. if asked to do so it can randomly choose any character and output it. Now, in the programming language Java consider ...
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2answers
30 views

r.v Law of the ratio

Earlier the following question was asked, I think by a classmate: r.v. Law of the min I posted my solution to it, but I am stumped on the second half of the problem: If the law of $(X,Y)$ is ...