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-...

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Question about Supermartingales

I came across the following problem: In my setting I have two sequences of non-negative integrable random variables (measurable with respect to some filtration $F_n$) which are called $X_n$ and $Y_n$. ...
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0answers
21 views

Nash Equilibrium

Player A chooses a random integer between 1 and 100, with probability pj of choosing j (for j = 1, 2, . . . , 100). Player B guesses the number that player A picked, and receives that amount in ...
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3answers
22 views

Probability of increasing order permutation

Suppose I have n elements. What's the probability of a permutation such that the first half is increasing and second half can be ordered without any constraints? (A permutation can only have distinct ...
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1answer
65 views

Someone Ripped Me Off, Please Help Calculating Odds!! [closed]

I'm protesting a state contract, and one of the grounds for protest is that someone stole material from a past proposal my company submitted, and is representing it as their own. Besides leaving our ...
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0answers
20 views

Conditional expected value not mutually indipendent sets

Let be $E,G,H$ pairwise independent events but not mutual (e.g. $\mathbb{P}(E\cap H)=\mathbb{P}(E)\mathbb{P}(H),\,\mathbb{P}(G\cap H)=\mathbb{P}(G)\mathbb{P}(H), ...but \,\mathbb{P}(E\cap G\cap H)\ne\...
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0answers
13 views

Trouble with Bayesian Hypothesis Test Equation

A passage from Wasserman's All of Statistics: The Bayesian approach to testing involves putting a prior on $H_0$ and on the paramater $\theta$ and then computing $\mathbb{P}(H_0 \mid X^n)$. ...
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1answer
65 views

Monotone Class Theorem and another similar theorem.

I found different statements of the Monotone Class Theorem. On probability Essentials (Jean Jacod and Philip Protter) the Monotone Class Theorem (Theorem 6.2, page 36) is stated as follows: Let $\...
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0answers
28 views

The discrete Laplacian

I am working on the $d$-dimensional integer lattice. Let $S$ be a random walk with increment distribution $p$. Given the distribution $p$ we can define the discrete Laplacian just as in Wikipedia is ...
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0answers
19 views

Factory inspections on a budget

A factory inspector is testing the efficiency of $n$ machines. To pass the inspection, each machine is required to run at or above a certain standard efficiency. The inspector can measure the ...
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0answers
32 views

ergodic theorem for expectation of positive recurrent diffusion

Suppose $X_t$ is a positive recurrent diffusion on $\mathbb{R}$ with invariant probability measure $\mu$. There is an ergodic theorem (see V.53. in Rogers & Williams volume II) that states $$\lim_{...
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0answers
41 views

Transformations of two Laplace distributions resulting in a Laplace distribution

Suppose we have two independent identical random variables $X_1$ and $X_2$ with Laplace distribution \begin{align} f_X(x)=\frac{1}{2b}e^{-\frac{|x|}{b}} \end{align} I am looking for a non-...
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1answer
25 views

Voting with 3-way ties

From Peter Winkler's 'Mathematical puzzles' Ashford,Baxter and Campbell run for election and end up in a 3-way tie. To break it, they solicit voters' second preference and there is also a 3-way tie. ...
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0answers
21 views

Uniform Boundary for S.D.E with Lipschitz Coefficients

Edit of progress: Since the SDE is linear, I got a solution in the form of $e^\int...$$\cdot e^\int$. By Jensen's inequality I can change the order of the left factor to have the integral on the ...
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1answer
43 views

X random variable in $\mathbb{N}$ independence of events

If I have a random variable $X$ with values in $\mathbb{N}$, $$\mathbb{P}(X=n)=\frac{1}{n^s\zeta(s)}$$ where $s>1$ and $\zeta$ the Riemann zeta function, then how can I show that $$A_i=E_{p_i^2}=\...
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0answers
39 views

Lindeberg condition's counterexample (central limit theorem)

My aim is to find an example where the CLT is true but not the following (equivalent to Lindeberg's) condition: Find a sequence of independent $(X_k)\sim\mathcal{N}\left(0,\sigma^2_k\right)$, so ...
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0answers
24 views

Shifting the mean of a composite function of deterministic and random variables

For a project I am involved in relating to communication, I have the following model: $L = f(r).X$ where $X$ is a lognormal random variable with zero mean in the logarithmic scale and standard ...
2
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1answer
38 views

Transformation of random variables that preserves the distribution

Suppose we have a random variable $X$ with distribution $F_X$. Let $X_1$ and $X_2$ be two independent copies of $X$. My question: can we find a transformation $Z=g(X_1,X_2)$ such that the ...
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0answers
22 views

probability measures vs. probability distributions vs. measure of probability density

I am learning probability theory right now and am confused about some basic concepts. I have a few questions and am wondering if you can also check if the following is correct: Suppose we have a ...
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1answer
26 views

Weak convergence implies convergence on continuous functions

Let $X$ be a metric space, and let $\mu_n$ be a sequence of measures on $X$ converging weakly to a measure $\mu$, meaning for all bounded continuous functions $f$, we have $\int_{X}fd\mu_n \rightarrow ...
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0answers
15 views

Variance computed using Taylor series does not agree with numerical experiment [migrated]

I would like to estimate an angle $\theta\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$ given the noisy observations of its sine and cosine (this is related to my earlier question). My estimator is ...
0
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2answers
29 views

Independence of Random Variables From Expectation Counter Example

I know that if $X$ and $Y$ are independent random variables, then $\mathbb{E}(XY) = \mathbb{E}(X)\mathbb{E}(Y)$. I also know that the converse is not true, although I cannot seem to find an easy ...
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0answers
20 views

Convergence in law and sequence of reals

How to show that if I have $Y_n$ a sequence of random variables and $a_n$ a sequence of reals such that (i) $Y_n\to Y$ in law (ii) $a_n\to a$ i have $a_nY_n\to aY$ in law? I know that if I have (i)...
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1answer
30 views

Sum of Random Variables i.i.d. with $\mathbb{E}[|X_n|]=+\infty$

Let $(X_n)$ be a sequence of IID RVs (independent, identically distributed random variables) with $\mathbb{E}[|X_n|]=+\infty,\forall n$. Prove that $\sum_n \mathbb{P}[|X_n|>kn]=\infty$ with $k\...
2
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1answer
40 views

Does there exist a random variable $\xi$ and a constant $c \neq 0$ such that $\xi + c \stackrel{d}{=} \xi$?

Does there exist a random variable $\xi$ and a constant $c \neq 0$ such that $\xi + c \stackrel{d}{=} \xi$? For context, I'm re-reading Kallenberg and in Chapter 3, on page 49, in his proof of Lemma ...
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1answer
10 views

Independent events from any other

In $(\Omega,\mathcal{F},P$) probability space, how can I show that $\forall A\in \mathcal{N}=\left\{ A\in\mathcal{F}: \mathbb{P}(A)=0,or\,\, \mathbb{P}(A)=1 \right\}\Rightarrow$ $\forall E\in\Omega$...
3
votes
1answer
40 views

Minimal value of probability according to the difference of a Levy-process

Can we conclude for a Levy-Process, that for all $\epsilon>0$ it holds that $\min_{s\in [0,t]} \mathbb P\left(\left|X_t-X_s\right|\leq \epsilon\right)>0$? Stochastic continuity doesn't seem to ...
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votes
1answer
53 views

Using the Baire Category Theorem to prove $\mu$ is trivial

Suppose we have a probability measure space $(X,\mathcal A,\mu,T)$ where $T$ is measure-preserving. Then if for every $A,B\in\mathcal A$ we have $\mu\left(A\cap T^{-n}B\right)=\mu(A)\mu(B)$ for all $n\...
3
votes
1answer
118 views

Distributional equality

Let $(W_t)_{t\geq0}$ be a standard Brownian motion. I have to show that the following equality holds in distribution. Does someone has a good hint to show this? $\sup_{t \geq 0}( |W_t| -t) = \sup_{t \...
1
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0answers
43 views

One question regarding independence of $\pi$ systems.

Let G and H be sub-sigma-algebras of F and I and J are $\pi$ systems such that $\sigma(I)=G$ and $\sigma(J)=H$ Can anyone explain the following quote? Suppose I and J are independent, for fixed ...
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0answers
27 views

Probability of $n$ numbers picked from set to have greater mean than set

If we have (N) varying non-negative numbers , with a mean equal to X, and a median less than X, if we pick (n) unique numbers from the set, what is the formula for the probability that the mean of ...
0
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1answer
58 views

Probability for a leading candidate to eventually win

Two candidates contest a close election. Each of the $n$ voters votes independently with probability $\frac12$ each way. Fix $\alpha \in (0,1)$. Show that, for large $n$, the probability that the ...
0
votes
1answer
35 views

I am trying to find answer to this bivariate normal problem. Can anyone help. [closed]

The distribution of the heights of husband-wife pairs in a particular population is modelled by a bivariate normal distribution. The mean height of the women is 165cm and the mean height of the men is ...
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1answer
36 views

$X_1, X_2, \dots$ uncorrelated, $\frac{Var[X_i]}{i} \rightarrow 0$, then $\frac{S_n}{n} - \frac{\mathbb{E}[S_n]}{n} \rightarrow 0$ in $L^2$

Let $X_1, X_2, \dots$ be uncorrelated random variables with $\mathbb{E}[X_i]= \mu_i$ and $\displaystyle\frac{Var[X_i]}{i} \rightarrow 0$, when $i \rightarrow +\infty$. Show that $\displaystyle\frac{...
1
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1answer
26 views

20 identical balls to be distributed in 3 identical boxes with MAX & MIN balls in each box?

As the title suggests, In how many ways can 20 identical balls be distributed in 3 identical boxes with at most 8 balls in each box and minimum 1 ball in each box ?
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0answers
19 views

If $(W_t)_{t\ge 0}$ is a $L^2(D)$-valued Wiener process, then $W_t(x)$ is normally distributed

Let $d\in\mathbb N$ $\lambda$ denote the Lebesgue measure on $\mathbb R^d$ $D\subseteq\mathbb R^d$ be a domain $U:=L^2(D)$ and $\langle\;\cdot\;,\;\cdot\;\rangle:=\langle\;\cdot\;,\;\cdot\;\rangle_U$...
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0answers
12 views

Numerical scheme and boundary condition for 2D Fokker Planck equation

$\newcommand{\P}{\mathbb{P}}$ I have a 2D stationary Fokker-Planck equation $$\frac{\partial^2 \P(A,B)}{\partial A^2}+\frac{\partial^2 \P(A,B)}{\partial B^2}=\frac{\partial f_1(A,B) \P(A,B)}{\partial ...
1
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0answers
26 views

Return probability of a SRW in an even number of steps

I am looking for some references for the following problem. Consider a graph $G$ and a simple continuous time random walk $(X_t)_{t\geqslant 0}$ on this graph. Consider the family of events $(e_t)...
2
votes
1answer
120 views

Calculation of $\ln\left( \frac{S_{1}(t)}{S_{2}(t)}\right)$ where $S$ are stocks

Assume we have a probability space $(\Omega,\mathcal{F},\mathbb{P})$ where $\mathcal{F} =(\mathcal{F}_t)_{0 \leq t \leq \tau}$ is a Filtration of an incomplete finance market with stocks $S_j(t)$ for $...
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votes
1answer
46 views

In the answer to the question attached below, I don't quite see how step-3 is derived from step-2, Can anyone explain [duplicate]

Calculating the expected values of the min/max of 2 random variables Consider two fair $k$-sided dice with the numbers 1 through $k$ on their faces, obtaining values $X_1$ and $X_2$. What is $\mathbb{...
3
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0answers
23 views

Doob Meyer decomposition in an exercize

I have to find the Doob Meyer decomposition for the following process: $Y_t=e^{(1+B_t^2)}$ I think that the method is to derive with the Ito's formula the process and I've obtained: $dY_t=2B_te^{(...
0
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0answers
36 views

Function of random variable: Two ways to find the pdf

Suppose $X$ is a r.v with pdf $f_X(x)$. Let $Y = g(X)$. To find the pdf of $Y$ - $f_Y(y)$. I use one of two ways and I assume g to be a monotonically increasing function. Method I first using the ...
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1answer
43 views

Complicated probability question [closed]

There are 20 empty present boxes numbered from 1 to 20 are placed on a shelf, there are 4 men standing in front of the shelf. Each one asked to pick in his mind 3 numbers without telling any one then ...
0
votes
1answer
34 views

Condition expectation of functions: $E(fg\mid\mathcal{A})=gE(f\mid\mathcal{A})$ when $|g|<\infty$ a.e.

Let $(X,\mathcal{B},\mu)$ be a probability space, $\mathcal{A}\subset\mathcal{B}$ a sub-$\sigma$-algebra, then by an easy application of the Radon-Nikodym Theorem, letting $\nu(A)=\int_A\, f\,\mathrm{...
0
votes
1answer
40 views

Integration with respect to a Poisson random measure

Let $N$ be a Poisson random measure (PRM) on a Polish space, $\left(X,\mathcal{B}(X)\right)$, and let $\tilde{\nu}$ be its mean measure. Then, let $f$ be any non negative and bounded function on $X$. ...
1
vote
1answer
40 views

Why is the CLT stated like it is?

The CLT says that given finite variance of iid RVs, we have $$\sqrt{n}( \bar{X} - \mu) \rightarrow \mathcal{N}(0,\sigma^2),$$ but if this is true, then $\bar{X} - \mu$ should converge to $\mathcal{N}(...
0
votes
1answer
29 views

For an invertible measure preserving system, $\lim_NA_f^T(N)=\lim_N A_f^{T^{-1}}(N)$

For an invertible measure preserving system, show that $\lim_NA_f^T(N)=\lim_N A_f^{T^{-1}}(N)$. Here we consider the measure preserving system $(X,\mathcal A,\mu,T)$ where $T$ is invertible and $\mu$...
1
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1answer
49 views

prove that Doléans-Dade exponential is a local martingale

I want to prove that $Z_t$ the Doléans-Dade exponential is a local martingale i.e. that there exists a stopping time $\tau_n$ tending to infinity such that the stopping process $\mathbb{1}_{\tau_n>...
3
votes
1answer
53 views

Law of large numbers for moving mean

Consider the following process: For $n = 1,\ldots$ $U_n \sim U[0, 1]$, that is, uniformly distributed on $[0, 1]$, $X_n = U_n 1_{U_n > q_n}$, where $q_n = \frac{1}{n-1} \sum_{i=1}^{n-1} X_i$, ...
5
votes
3answers
74 views

Can $Y$ and $\frac{X}{Y}$ be uncorrelated if neither $X$ or $Y$ is constant?

Suppose I have two variables $X$ and $Y$ with $Y>0$. Can the random variables $Y$ and $\frac{X}{Y}$ ever be uncorrelated, i.e., $$\mathbb{E}(X)=\mathbb{E}(Y)\mathbb{E}\left(\frac{X}{Y}\right).$$ ...
0
votes
1answer
22 views

Question regarding probability density function?

I was going through the concept of probability density functions and had a small confusion about the notation that a pdf can take values greater than one.I found this How can a probability density be ...