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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74 views

Simple Distribution Function

Given that you have a random variable $X$ and you know it's distribution is binomial, and its mass function: $\binom{k}{X}(\frac{1}{365})^X(\frac{364}{365})^{k-X}$, what is the distribution function? ...
5
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4answers
769 views

What is the probability you guess the number I am thinking of?

Probability is defined as the likely number of outcomes over all total outcomes. In this case, 1 over infinity; which would equate to zero. But, there is a chance you can guess the number I am ...
2
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2answers
66 views

Probability Proof.

Write a proof to show that $\mathbb{P}(X_1 \mid X_3) + \mathbb{P}(X_2 \mid X_3) - \mathbb{P}(C_1 \cap X_2 \mid X_3) = \mathbb{P}(X_1 \cup X_2\mid X_3)$ labeling theorems used for each step. My ...
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1answer
65 views

Is regular conditional distribution unique?

If $\kappa_1,\kappa_2:\mathfrak{B}\times\Omega\rightarrow\left[0,1\right]$ are two regular versions of the same conditional distribution $P\left(\left.X\in\cdot\right|\mathcal{A}\right)$, are they ...
5
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1answer
86 views

Universal measurability of a kernel

Let $X$ and $A$ be Borel topological spaces, that is it is they are homeomorphic to Borel subsets of a complete separable metric space. Let further $\pi$ be a universally measurable stochastic kernel ...
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1answer
31 views

probability space over the coin flips

I recently read the definition of differential privacy, which is as follows: a randomized function $K$ gives $\epsilon$-differential privacy if for all data sets $D$ and $D'$ differing in at most ...
2
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1answer
144 views

Almost sure convergence of sequences

Let $(a_n)$ be a sequence of positive real numbers such that $\sum_{n \geqslant 1} a_n < \infty $ and $$\sum_{n \geqslant 1} \Pr\left(\left|X_{n+1} - X_n\right| > a_n\right) < \infty $$ Why ...
2
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2answers
1k 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
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1answer
185 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, ...
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1answer
78 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 ...
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2answers
274 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 ...
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2answers
177 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 ...
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1answer
111 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 ...
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1answer
465 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}) = ...
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1answer
85 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 ...
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1answer
36 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, ...
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1answer
109 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
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1answer
120 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 ...
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1answer
39 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
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1answer
221 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 $(nM^{-1}_n)_{n\ge1}$ ...
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1answer
32 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
28 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
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1answer
68 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 ...
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1answer
187 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
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1answer
220 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 ...
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1answer
283 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
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1answer
217 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 ...
2
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0answers
129 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 $\mathbb{R}^n$ ...
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1answer
104 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| ...
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1answer
143 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
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1answer
115 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
224 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
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1answer
99 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
88 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
130 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
521 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
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1answer
149 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 ...
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1answer
290 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 ...
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1answer
973 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
103 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
194 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
115 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 ...
17
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2answers
688 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 ...
4
votes
1answer
157 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 ...
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0answers
52 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 ...
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0answers
263 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
235 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
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1answer
64 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
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1answer
110 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 $X|Y \sim N(a+bY,c+dY)$ ...