The probability-distributions tag has no wiki summary.

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### Proving that a property holds for random sequences with given marginal distribution by rearrangement

I am currently investigating the property of random sequences with a special marginal distribution function $F(x)$. Given any random sequence $X_1, X_2, \cdots, X_n$, supposing their joint ...

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226 views

### Finding Decision Boundary from empirical distribution

Based on measuring a certain characteristic, we want to classify measurements as coming from either of two populations. The true population distributions are unknown (and we don't want to take any ...

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### Upper bound on expectation value of the product of two random variables [closed]

Hello,
I am trying to find an upper bound on the expectation value of the product of two random variables.
So suppose x, y are two non-independent random variables, given that I know the distribution ...

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114 views

### limit of functionals on weak convergent random variables

Suppose real value random variables satisfy
$X_{n} \Rightarrow X$ (convergence in distribution)
as $n\to \infty$ in the same probability space
$(\Omega, \mathcal F, \mathbb P)$.
It is well known that ...

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253 views

### Distribution of random vectors

Two positive numbers $\alpha$ and $\beta$ are given. We are going to describe a process of choosing a random vector on the unit sphere $S$ in $\mathbb R^3$ (given by $x^2+y^2+z^2=1$).
A vector $u\in ...

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### Ergodicity and convergence time in Probabilistic Cellular Automata

Has the following conjecture been prooved, or has any step in the direction of its proof been done?
"ANY Probabilistic Cellular Automata converge fast on the stationary probability distribution iff ...

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207 views

### Ergodicity for a Probabilistic Cellular Automaton on a finite space

Let's consider a Probabilistic Cellular Automaton on a one dimensional lattice $S$. Each site of the lattice can have two states, $0$ and $1$. The transition probability acting on each site is: ...

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### Cumulative distribution function and convolution.

Hello,
Given a probability distribution of a discrete variable p1(x) and a probability distribution of a discrete variable p2(y) defined by
p2(y) = Sum_{x,x'} p1(x) p1(x') * KroneckerDelta((x+x')/2 = ...

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### When can you describe a population and its component subpopulations with the same parametric family of distributions?

I believe that it is often the case that you are trying to select the best probability distribution to use to describe some phenomenon you are studying, and you have data not only for a population, ...

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133 views

### $Y|X \sim N(\mu X,X^2)$ and $X\sim N(\alpha, \beta)$. How is $Y$ distributed?

Dear all,
I have recently been breaking my head over this question. The idea is that a certain variable $Y$ is normally distributed with a parameter $X$ in both mean and variance.
$Y|X \sim N(\mu ...

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154 views

### Probability of first collision with replacement

Suppose I have a bag of N balls, each labeled with a distinct integer. The experiment starts when I draw a ball from the bag, record its label, and then return the ...

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352 views

### The Probability distribution of Random variable of Random variable

In my understanding, random variable is a measurable function from a probability space to a measurable space. Suppose $X$ is a random variable from $(A, \sigma_{A},P_A)$ to $(B,\sigma_{B})$. And $Y$ ...

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378 views

### What is the name for a non-normalized distribution?

For some analysis work with probability distributions, I remember a common trick being to drop the "integrate to 1" requirement, so the set becomes closed under addition and is more convenient to work ...

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205 views

### Tracking down locality assumption in CHSH inequality

CHSH inequality requires both locality and realism. I will equate here realism with counterfactual definiteness.
Now counterfactual definiteness tells us that given two different measurements on the ...

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### Stationarity of an Integral Process

Let $f$ be a continous deterministic function defined on $\left[0,c\right]$ and $(B_{t}^{H})_{t\geq 0}$ be a fBM with $H\in \left(0,1\right)$. We define a Process $\left(X_{t}\right)_{t\geq 0}$ with
...

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### Results regarding $E[\min X,Y]$. when $X$ and $Y$ are independent, of given distributions.

Working on fairly unrelated stuff, I needed to prove the following, fairly easy results, and I wonder if anyone can provide references to the literature. Not being a probabilist I wouldn't know where ...

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262 views

### Characteristic Function of a Non-negative Random Variable Evaluated at a Complex Value

Suppose we have a non-negative random variable $X$ with density $p(x)$,and its characteristic function, evaluated at a complex number $z$, being $\phi(z)=E[e^{z X}]=\int_{0}^{\infty}e^{zx}p(x)dx$.
It ...

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### Why this two model have same probability distribution?

(1)
Consider the following method of generating a random tree with $n$ nodes.
First expand the root node into two branches.
Then expand one of the two terminal nodes at random.
At time $k$, ...

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588 views

### Distribution of Maximum of a uniform multinomial distribution

Hello, I'm working with a data structure which uses a uniform distribution to bucket the inputs into $k$ buckets. The efficiency of the structure is bounded by the $\frac{k_{max}}n$, where $n$ is the ...

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289 views

### Azuma's Inequality when the conditions hold with high probability?

In Azuma's Inequality, is the statement true when $|X_k - X_{k-1}| < c_k$ almost surely rather than with probability 1? If not, is there another result which gives strong concentration when the ...

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### E[ | X - Y | ] where X and Y are independent Poisson random variable

What is the expected value of the absolute difference of two independent Poisson variables?
E[ |X - Y| ]
Seems like an easy question but I haven't found an easy solution.
I've split the double sum ...

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152 views

### Tail of solutions of a stochastic differential equation

As we know, solution to $dX_t=\mu dt+\sigma dW_t$ is normal distributed and is light tailed; solution to $dX_t=\mu X_tdt+\sigma X_t dW_t$ is log-normal distributed and is heavy tailed. Is there any ...

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### Product of densities of a wrapped normal distribution

The density of a wrapped normal distribution is given by
$$\frac{1}{\sigma \sqrt{2\pi} }\sum _{k=-\infty }^{\infty }\text{Exp}\left[\frac{-(\theta -\mu -2\pi k)^2}{2\sigma^2}\right]$$
Considering two ...

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### measuring distance between probability measures only at the tail

Is there any official (i.e., to be found in probability books) metric for the distance between two probability measures, defined only on a subset of their support?
Take, for example, the total ...

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143 views

### The degrees in a random subgraph

Fix some positive integers $N$ and $d_k$, $k=1,2,\dots$ with $N=\sum_{k=1}^\infty d_k$.
Suppose you have a graph $G$ taken randomly uniformly among the set of all (unoriented) graphs with $N$ ...

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### Distance metric between two sample distributions (histograms)

Context: I want to compare the sample probability distributions (PDFs) of two datasets (generated from a dynamical system). These datasets depend on a set of parameters, and I want a concise way to ...

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### Convergence of moments implies convergence to normal distribution

I have a sequence $\{X_n\}$ of random variables supported on the real line, as well as a normally distributed random variable $X$ (whose mean and variance are known but irrelevant). I know that the ...

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### Derivative of the CDF of a family of random variables

Suppose I have a r.v. $Z = X + \alpha Y$ and that $F_Z$ is the probability distribution function of $Z$. If we think of the probability $p = F_Z(q) = \mathbb{P}(X+\alpha Y < q)$ as a function $p = ...

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### Integration of the product of pdf & cdf of normal distribution [closed]

Denote the pdf of normal distribution as $\phi(x)$ and cdf as $\Phi(x)$. Does anyone know how to calculate $\int \phi(x) \Phi(\frac{x -b}{a}) dx$? Notice that when $a = 1$ and $b = 0$ the answer is ...

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### Continuity of hitting distributions

Hi everybody
Let $U$ be the domain (as shown in the picture) and $\bar{U}$ its closure, further more set $\partial_r U$ to be the reflecting boundary and $\partial_a U$ the absorbing one. The process ...

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### Is it known that every PDF continuous in all $R^n$ has a maximum? [closed]

I'm working with maximum a posteriori estimation and managed to show that every probability density function that is continuous in all $R^n$ always has at least one global maximum. I've search around ...

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526 views

### Marginalizing over discrete and continuous random variables

Suppose we have a joint distribution $P(D,X,L) = P(D|X)P(X|L)P(L)$. Here, D and L are discrete but X is a continuous random variable. I want to compute $P(D=d)$. How do I do this numerically? The fact ...

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### Multivariate Power Law distributions?

Is there a text books or publications that describes multivariate power law/pareto distributions?

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### Can you prove the monotonicity of the function (or find a counter example)?

Let $X$ be a non-negative random variable that is drawn from a cumulative distribution function $F(\cdot)$, pdf $f(\cdot)$ and mean $E[x]$. $k$, $c$, $v_l$ and $v_h$ $(v_h>v_l)$ are non-negative ...

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### on the difference of exponential random variables

Assume two random variables X,Y are exponentially distributed with rates p and q respectively, and we know that the r.v. X-Y is distributed like X'-Y' where X',Y'are exponential random variables, ...

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### If Mean Residual Lifetime is approximately constant, Residual Lifetime is Approximately Exponential in a Strong Sense

Suppose the "mean residual lifetime," $\mathbb{E}[X-x|X≥x]$ is approximately constant for large $x$. Then, I believe that the conditional tail distribution is approximately exponential, in the sense ...

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### Projection of a probability distribution according to another one,

Hi,
(Please forgive me if my question is vague or trivial)
Let a normal distribution $P(\mu, \sigma)$ and a poisson distribution $Q(\lambda)$.
I want to find a distribution $Q'$ that is :
a ...

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247 views

### Limit of a rescaled random sum of i.i.d. random variables

Consider a sequence of i.i.d. random variables $(X_i)_{i \in \mathbb N}$ and let $S_n=X_1+\dots+X_n$
For every $\alpha \in ]0,+\infty[$, let $N(\alpha)$ be a discrete random variable on $\mathbb N$, ...

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### a function of Bernoulli variables?

Let $X_1,X_2,...,X_n$ be a fixed number of Bernoulli random variables. My problem is to find a distribution for $Y$ such that for some function $f$, we have $Y=f(X_1,X_2,...,X_n)$. There are two ...

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331 views

### Composed function made Lebesgue integrable?

Let $p(x)$ be a probability density function on the unbounded set $X \subseteq \mathbb{R}^n$, so that $\int_X p(x) dx = 1$.
Let $F: X \rightarrow \mathbb{R}_{\geq 0}$ a measurable but non-integrable ...

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### what books to read to quickly understand adiabatic approximation

Hi group, I'm a theoretical ecologist with fairly adequate training in applied math (ODE, linear algebra, applied probability, some PDEs). In my current work, I've encountered the use of adiabatic ...

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### References for this game

Hello everybody,
I would like to know how the following game is known in the literature and, possibly, to have references for related papers.
Description of the game: Fix a space $X$ and two Borel ...

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### What does it mean to sample a value x* from f(x)?

This might be a really elementary question, but I'm not sure what it means. I have a density function f(x). How do I sample a value from f? For known distributions there are functions in R which do it ...

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### compound distribution in Bayesian sense vs. compound distribution as random sum?

I'm trying to sort out two different uses of the term "compound distribution" and figure out the relationship.
The Wikipedia article on compound distribution -- which I wrote -- defines a compound ...

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### Do these random variables follow Gaussian distribution?

Say that a random variable $X$ follows the Gaussian distribution $\mathcal{N}(\mu, \sigma)$. Then will the ceiling $\lceil X\rceil$, the floor $\lfloor X\rfloor$, and the rounding $\lfloor X\rceil$ ...

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### Montgomery's pair correlation function without RH?

In the theory of the Riemann zeta function, Montgomery's Pair correlation function is defined as
$$
F(\alpha) = \frac{1}{N(T)}
\sum_{T < \gamma, \gamma' < 2T} T^{i \alpha (\gamma - \gamma')} ...

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### Variance of exponential random variable

For a random variable $\xi$, what bounds can be achieved for Var $e^{\xi}$ in terms of E$\xi$ and Var $\xi$?

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### Copulas and marginals thereof

Hello everyone,
I recently became aware of the existence of the copula concept.
So, I have been reading a few things about copulas lately, but
I cannot seem to find information on the following ...

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### Entropy conjecture for distributions over $\mathbb{Z}_n$

Suppose we have two independent random variables $X$ (with distribution $p_X$) and $Y$ (with distribution $p_Y$) which take values in the cyclic group $\mathbb{Z}_n$. Let $Z = X +Y$, where the ...

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### Sum of Squares of Normal distributions

Given $X_i \sim \mathcal{N}(\mu_i,\sigma_i^2)$, for $i = 1,\dots,n$. How does one find the distribution of $D = \sum_{i=1}^n X_i^2$? In the case that all the standard deviations are the same (i.e. ...