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0
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0answers
140 views

softmax activation function with infinite support ?

Hi, How do we calculate the terms of a softmax activation function with an infinite support ? That is, finding the $\{p_i\}_i$ with $p_i = {{e^{q_i}} \over {\sum_{j=1}^\infty e^{q_j} }}$ (how to ...
1
vote
1answer
139 views

First moment of a function of a normally distributed random variable

I'm trying to find the first moment of the following function: $f(x) = \frac{(-ax+\sqrt{1-a^2})(-bx+\sqrt{1-b^2})}{\sqrt{x^2+1}}H(-ax+\sqrt{1-a^2})H(-bx+\sqrt{1-b^2})$ where $H(x)$ denotes the ...
1
vote
1answer
89 views

Equivalence between choosing a subspace and choosing its orthogonal

Hi, We consider subspaces of $\mathbb{R}^N$. Suppose that we have a property called $\mbox{Prop}$ that apply to subspaces of $\mathbb{R}^N$. That is to say a function from the set of subspaces of ...
0
votes
1answer
285 views

Simple markov chain problem

I know this is an easy problem, but I can't figure it out. A particle takes discrete steps $σ_1,σ_2,σ_3,…,σ_n$ which take on values +1 or −1. However, $P(σ_i=+1)=p$ and $P(σ_i=−1)$ will be $1-p$. ...
6
votes
1answer
205 views

Trasportation metric (AKA Earth-Mover's, Wasserstein, etc.) as “natural” / “induced”?

Context: Given a discrete finite metric space $X$ (in my case X={0,1}$^n$ with the Hamming/L$_1$ distance), I need to define the natural or canonical metric on the set of all probability distributions ...
0
votes
1answer
174 views

Limit of the stochastic process at time 0

This is not a homework question so please be kind not to remove it right away. I am working on some research but have to justify the following argument: Assume $S_t$ is a continuous stochastic ...
0
votes
0answers
520 views

Derivative of the most probable value (of a gaussian variable) VS most probable value of the derivative

Let $x$ be a random variable with gaussian probability distribution $P(x)$. We assume that $x$ depends parametrically on a parameter $t$ so that : ...
0
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0answers
98 views

power law distribution of time events

Suppose you have the logs of a web server. In these logs you have tuples of this kind: user1, timestamp1 user1, timestamp2 user1, timestamp3 user2, timestamp4 user1, timestamp5 ... These ...
0
votes
0answers
79 views

Marginal Distribution.

Consider a couple of real random variables $(X,Y)$ and $\mu_{X,Y}$ the induced probability distribution. Denote by $\mu_X$ and $\mu_Y$ the distributions of $X$ and $Y$. Is it true that ...
2
votes
2answers
368 views

Sampling without replacement until hitting a subset

I randomly sample uniformly from $ \{1,..,N \}$ without replacement until drawing a number $ \leq k$. Denote the expected number of draws by $R(N,k)$. I want a good approximation for $\sum_{k=1}^N ...
2
votes
1answer
170 views

If two probability distributions have the same weak limit and one of them satisfies Large Deviation Principle, what can we say about the other?

If the probability distribution function of two sequences of random variables have the same weak limit and one of the sequences satisfies a Large deviation principle, then does it imply that the other ...
4
votes
2answers
190 views

Sampling from a recursively defined distribution

I'd like to know if there are techniques for sampling from a recursively defined probability distribution, assuming that solving the recursion for a formula for the distribution is too difficult. As ...
1
vote
1answer
271 views

Product of probability densities of the form x^{-t} exp (-ax)

I have two probability distributions $p(x) = N_1 x^{-\tau} \exp(-\frac{x}{x_0})$ and $p(y) = N_2 y^{-\kappa} \exp(-\frac{y}{y_0})$. $N_1$ and $N_2$ are just normalization constants and $x>0$, ...
1
vote
1answer
154 views

Expectation under a t-distribution

Given parameters $\lambda, \nu>0$, a covariance matrix $R$, a mean vector $\mu \in R^p$, the Arellano-Valle and Bolfarine's generalized $t$ distribution is given by (see, for example, the book by ...
0
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0answers
83 views

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 ...
1
vote
2answers
234 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 ...
3
votes
2answers
1k views

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 ...
1
vote
2answers
116 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 ...
1
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0answers
260 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 ...
1
vote
1answer
127 views

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 ...
3
votes
2answers
210 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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2answers
1k views

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 = ...
3
votes
1answer
389 views

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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3answers
134 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 ...
1
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2answers
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 ...
0
votes
1answer
384 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 $Y$ ...
3
votes
3answers
383 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 ...
2
votes
1answer
211 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 ...
0
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0answers
99 views

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 ...
3
votes
1answer
266 views

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 ...
1
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2answers
268 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 ...
1
vote
0answers
176 views

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 $k$, ...
1
vote
2answers
623 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 ...
1
vote
1answer
304 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 ...
3
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0answers
758 views

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 ...
1
vote
1answer
153 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 ...
1
vote
1answer
323 views

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 ...
1
vote
2answers
379 views

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 ...
1
vote
1answer
146 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 $N$ ...
2
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2answers
5k views

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 ...
4
votes
2answers
1k views

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 ...
0
votes
1answer
172 views

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 = ...
2
votes
3answers
4k views

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 ...
3
votes
2answers
285 views

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 ...
0
votes
1answer
242 views

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 ...
0
votes
1answer
604 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 ...
0
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2answers
844 views

Multivariate Power Law distributions?

Is there a text books or publications that describes multivariate power law/pareto distributions?
1
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0answers
455 views

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 ...
2
votes
1answer
664 views

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, ...
6
votes
2answers
287 views

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 ...