# Questions tagged [probability-distributions]

In probability and statistics, a probability distribution assigns a probability to each measurable subset of the possible outcomes of a random experiment, survey, or procedure of statistical inference.

1,937 questions
Filter by
Sorted by
Tagged with
45 views

• 25
87 views

### Concentration bound for a increasingly weighted sum of bernoulli random variables

Given $x_1,x_2,\ldots,x_n$ i.i.d. bernoulli random variables with $P(x_i=1)=\frac1n$. Given a constant $c=1+\frac{1}{m}, m\geq n$. Is there an explicit theorem that can derive a concentration argument ...
• 25
85 views

### When are increasing functions on posets (specifically, lattices) the CDF of a probability measure?

This is perhaps a basic question, but I couldn't find a reference. Let $P = (X,\leq)$ be a poset. Given a probability measure $\mu$ on $P$ (with respect to the Borel $\sigma$-algebra generated by sets ...
1 vote
33 views

• 1,298
71 views

Let $X_p$ for $p\in \mathbb{Z}$ be a collection random variables that satisfy for all $k>0$, $p\in \mathbb{Z}$: $$\sum_{p_1+\dots+p_k=p} \mathbb{E}[X_{p_1} \dots X_{p_k}]=\begin{cases} 0 &... 0 votes 0 answers 36 views ### What is the direct role of exchangeability in ensuring coverage in conformal prediction? I was wondering how exchangeability directly relates to the proof of the coverage guarantee in conformal prediction. In most papers I have seen, usually they say that by exchangeability the order of ... 0 votes 0 answers 15 views ### Position dependent service time in queue Is there any literature for queuing analysis (waiting time, capacity etc.) of a queue with service time that depends on the position of the customer in the queue? I have encountered a problem where a ... • 13 1 vote 1 answer 87 views ### Maximum column norm of random A^{-1}B Suppose that A is an n by n Gaussian matrix (each component i.i.d. normal distributed with mean 0 and variance 1). Let b be a n-Gaussian vector. Then it could be easily proven that the ... • 33 1 vote 1 answer 64 views ### Queues wait for other queues- A communication problem I am working on a problem which involves a single server that requires multiple inputs to do a computation. Each of these inputs arrive as a Poisson process with rate \lambda. Hence, a situation ... • 13 0 votes 0 answers 42 views ### Are there probability densities \rho, f_n such that \lim_n \frac{[\rho * f_n]_\alpha}{\|\rho * f_n\|_\infty} = \infty? We fix \alpha \in (0, 1). Let [f]_\alpha be the best \alpha-Hölder constant of f: \mathbb R^d \to \mathbb R^k \otimes \mathbb R^m, i.e., [f]_\alpha := \sup_{x \neq y} \frac{|f(x) - f(y)|}{|x-... • 939 1 vote 1 answer 50 views ### Lower bound the best \alpha-Hölder constant of a convolution Let \mathcal D_1 be the set of bounded probability density functions on \mathbb R^d. This means f \in \mathcal D_1 if and only if f is non-negative measurable such that \int_{\mathbb R^d} f (... • 939 1 vote 2 answers 81 views ### Is the difference between \alpha-Hölder constants of f*\rho and g*\rho controlled by \|f-g\|_\infty? Let \mathcal D_1 be the set of bounded probability density functions on \mathbb R^d. This means f \in \mathcal D_1 if and only if f is non-negative measurable such that \int_{\mathbb R^d} f (... • 939 0 votes 0 answers 84 views ### Comparison between the expected values of the inverse of the CDF of binomial-distributed random variables Let us denote with F(x;j,\mu) the cdf of a Binomial distributed random variable with j trial with success probability \mu considered in x, and let f(x;j,\mu) be the pmf. Defining 0\leq \... 0 votes 1 answer 87 views ### Reverse Pinsker's inequality for smooth density classes Suppose we are given a class of probability density functions \mathcal{F} so that for every f \in \mathcal{F} we have \alpha \leq f \leq \beta for some positive \alpha, \beta \in \mathbb{R}_+ ... 2 votes 1 answer 167 views ### Reference request: Best book to cite on a property of the family of Cauchy distributions Kai Lai Chung once began a section of a textbook on probability by writing "Everybody knows" that$$ e^x = \sum_{n\,=\,0}^\infty \frac{x^n}{n!}. $$(with those quotation marks). Other ... • 12.2k 0 votes 1 answer 46 views ### Everywhere existence of marginals Let f\in L^1(\mathbb{R}^2) be a (joint) probability density function which satisfies f(x,y)>0 for all (x,y)\in \mathbb{R^2}. What is a necessary and sufficient condition under which the ... • 3,554 0 votes 0 answers 29 views ### Moment generating function for product states In the sequel B=M_\ell(\mathbb{C}). For M\in\mathbb{N} fixed and N\geq M I consider the symmetrizer \pi_{M,N}(x_M)\in B^{\otimes N}, which is the symmetrized tensor product of a_1,...,a_M ... • 63 1 vote 1 answer 172 views ### Chebyshev's inequality for Poisson distribution Reading an old Richard Karp paper, in which he mentions this argument "Application of Chebyshev's inequality yields the result that, if X is Poisson distributed with mean \lambda, then E(X\... • 67 0 votes 0 answers 21 views ### reference request: product measures defined by a subsequence of measures Suppose \{\mu_n\}_{n\in\mathbb{N}} is a sequence of pairwise equivalent probability measures, each of which is defined on \mathbb{R}. Let \bigotimes_n\mu_n be the product measure defined on \... 2 votes 1 answer 67 views ### From convergence of sequences to uniform convergence in probability For n=1, 2,\ldots consider a sequence of sets of ascending integers I_n=\{\underline{i}_n,\underline{i}_n+1, \ldots, \overline{i}_n\}, with \underline{i}_n \to \infty and \underline{i}_n=o(\... • 157 2 votes 0 answers 41 views ### If a probability measure is a mixture of products of its marginals, does it have finite moments? Let \mu be a Borel probability measure on \mathbb{R}^n. For a linear subspace E\subset \mathbb{R}^n, let \mu_E denote the marginal of \mu on E. The usual orthogonal complement of E is ... • 716 1 vote 1 answer 111 views ### A property of the distribution related to stochastic ordering Let X be a random variable with a symmetric support S\subset[-M,M] for some M>0. (i.e., if x is a point of increase of CDF F_X(\cdot), so is -x.) Has the infimum value of c such that \... • 19 0 votes 0 answers 62 views ### Asymptotic stochastic ordering for weighted sum of i.i.d. random variables Are you aware of any literature focusing on the conditions such that for two i.i.d. sequences of discrete r.v.'s \{X_n\} and \{Y_n\}, a_1X_1+a_2X_2+\ldots+a_nX_n\geq_1 a_1Y_1+... • 19 0 votes 0 answers 69 views ### Maximizing the trace of the resolvent of a Wishart matrix over positive unit trace matrices? Let G be a standard d \times d Wishart random matrix and consider the problem of maximizing the function$$ f(M) = \mathbb{E}\Big[\mathrm{tr}((G + M^{-1})^{-1})\Big], $$over the class of real ... • 384 0 votes 1 answer 38 views ### Gumbel-Softmax like function I am trying to train my DNN models and face some mathematical problems. Let me explain my goal. Consider an input tensor like [1,2,3,4,5]. I aim to obtain a one-hot encoded vector of the argmax of ... 1 vote 0 answers 67 views ### Gibbs Priors form a Martingale I am working on adapting variational inference to the recently developed Martingale posterior distributions. The first case, which reduces the VI framework to Gibbs priors, is proving hard to show as ... 0 votes 1 answer 110 views ### Limit distribution of the self-normalized sum of Cauchy random variables This is something that has come up in my research. I originally posted this question on CrossValidated but realized it might be better suited for this site. I have deleted the question there (in case ... 1 vote 1 answer 212 views ### Upper-bound of the tail of a weighted sum of iid random variables I have a question related to this one. X_i are n iid random variables with CDF 1_{[0,+\infty[}(x) \Phi(x), i.e. it is a mixture between a folded Gaussian and a delta in 0, both with weight 1/2.... • 65 0 votes 0 answers 74 views ### Random walks on groups I recently started reading Wolfgang Woess' book titled "Random Walks on Infinite Groups". In the section where he introduces Markov chains and random walks on a set X, he has defined a ... 1 vote 1 answer 124 views ### Stochastic order on weighted sum of iid random variables X_i are n iid random variables with CDF 1_{[0,+\infty[}(x) \Phi(x), i.e. it is a mixture between a half Gaussian and a delta in 0, both with weight 1/2. I would like to show that, \forall a \... • 65 2 votes 1 answer 87 views ### Deriving the distribution of standardized variables with empirical mean and standard deviation I'm working with a set of independent and identically distributed random variables \{ x_i \}_{i=1}^N, where each x_i follows a Gaussian distribution P_X(x) = \mathcal{N}(x; \mu, \sigma^2). This ... • 185 9 votes 1 answer 351 views ### Characterize algebras of the "topological simplices" operad The operad of topological simplices, which I'll denote \Delta, has as n-ary operations the set$$ \Delta_n:=\left\{P\colon\{1,\ldots,n\}\to[0,1]\;\middle|\;1=\sum_{i=1}^n P(i)\right\} $$of ... • 8,589 0 votes 0 answers 46 views ### Existence of derivative of distribution of exponential family? Suppose (X, \mathcal{F}) is a measurable space and \left\{F_\theta, \theta \in \Theta\right\} is a distribution family on (X, \mathcal{F}). When \left\{F_\theta, \theta \in \Theta\right\} is ... 1 vote 2 answers 227 views ### Relationship between fixed points and inversions in permutations Inversions in a permutation Y are defined as pairs where Y_a < Y_b but a > b, while fixed points in Y are defined as elements where Y_a = a (i.e., 1-cycles). Let S_\alpha be the set ... • 173 0 votes 0 answers 27 views ### Probability density function estimation for rare events My goal is to numerically estimate the probability density function (pdf) P(f) for the function f(x_1,x_2,\cdots,x_n). Here the random variables x_1,x_2,\cdots,x_n are drawn from the independent ... • 429 0 votes 0 answers 49 views ### Computing the Laplace transform of an expression I would like to find the Laplace transform of the following expression with respect to the Laplace parameter s  \int_{z=u}^{\infty} e^{-az/c} g^{'}(\dfrac{z-u}{c}) \int_{x=0}^{\infty} \varphi(z-x)dF(... • 1 7 votes 1 answer 533 views ### A variation on the Borel–Cantelli lemma theme Let X,X_0,X_1,\dots be nonnegative independent identically distributed (i.i.d.) random variables. Let \begin{equation*} E:=\bigcap_{n\ge0}B_n, \end{equation*} where \begin{equation*} B_n:=\... • 119k 2 votes 1 answer 121 views ### Convexity of a function Let: F_{j+1,y}(s) be the cumulative distribution function of a binomial distribution with mean y, j+1 independent trials considered for s successes. Is it possible to show in any way that: \... 1 vote 0 answers 79 views ### How can one build a min-2-wise independent small sample space from min-3-wise permutations? I have been studying a polynomial-size set of permutations from one of my lectures. The below image, taken from the lecture notes PDF, illustrates how to construct min-3-wise permutations. My ... • 15 1 vote 1 answer 59 views ### Upper bound I_R := \int_{B_R^c} |x| (P_t \ell_\nu) (x) \, \mathrm d x in terms of R, \nu, t? Let (p_t)_{t >0} be the Gaussian heat kernel on \mathbb R^d and (P_t)_{t >0} its induced semi-group, i.e.,$$ \begin{align} p_t (x) &:= (4\pi t)^{-\frac{d}{2}} e^{-\frac{|x|^2}{4t}}, \...
• 939
Is anything known about the distribution of the algebraic area, à la Lévy's stochastic area, of a Brownian excursion in the half-plane? To be precise, letting $x>0$, we consider the path \$(X_t,Y_t)...