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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.

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Probability distribution of total time for a job, given a workflow graph

$$ \begin{array}{cccccccccccc} & & \text{A} \\ & \swarrow & & \searrow \\ \text{B} & & & & \text{C} \\ & \searrow & & \swarrow \\ \downarrow & &...
Michael Hardy's user avatar
2 votes
0 answers
94 views

the projection distribution induced by integral points on the sphere

Let $A=\{\mathbf{v} \in \mathbb{Z}^{n}: \|\mathbf{v}\|^2= m \}$ and a fixed $\mathbf{y}\in \mathbb{R}^n$, the norm here refers to the Euclidean norm. Suppose $\mathbf{x}$ is a uniform distribution on ...
constantine's user avatar
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1 answer
17 views

Prove that the regularized incomplete beta function monotone with each of its parameter

Consider the regularized incomplete beta function $I_x(a, b)$ with $x \in [0,1]$ and $a, b > 0$. I am hypothesizing that the function is monotone decreasing with respect to $a$ and monotone ...
Shiwen Yang's user avatar
1 vote
1 answer
115 views

trying to get intuition into why Cross Entropy will always be greater or equal to the Entropy

I understand what entropy measures and cross entropy is the same except it is uses another distribution $q$ to compare it against $p.$ Is it because the log function is concave down so the predictions ...
Chris Blodgett's user avatar
2 votes
1 answer
147 views

Does $X_t$ with $t>0$ admit a density?

$ \newcommand{\RR}{\mathbb{R}} \newcommand{\TT}{\mathbb{T}} \newcommand{\NN}{\mathbb{N}} \newcommand{\PP}{\mathbb{P}} \newcommand{\EE}{\mathbb{E}} \newcommand{\FF}{\mathbb{F}} \newcommand{\PPP}{\...
Akira's user avatar
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2 votes
1 answer
68 views

Proof of the monotonicity of a regularized incomplete beta function

I want to prove the monotonicity of $I_r(nr, 2+(1-r)n)$ on $n$ but has no clues. The $I$ is the regularized incomplete beta function, defined as follows: $$I_r(nr, 2+(1-r)n)=\frac{\int_0^r x^{nr-1}(1-...
J H's user avatar
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2 votes
1 answer
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 ...
Betty's user avatar
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3 votes
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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 ...
Roger Van Peski's user avatar
1 vote
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33 views

Definition of "interval of continuity" for function defined on sets

At the beginning of Chapter 8 of Kubilius's Probabilistic Methods in the Theory of Numbers, the author defines $Q=Q(E)$ to be a completely additive nonnegative function defined for all Borel subsets $...
Greg Martin's user avatar
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Calculate the distance between categorical distributions when different categories are somehow related [closed]

I want to calculate the distance between some categorical distributions, for example A:0.1, B:0.2, C:0.7. I know this can be done by simply using kl divergence or cross entropy. But in my case ...
Nicolas Fabre's user avatar
2 votes
0 answers
50 views

stochastic process and integral

Let $(X_n(t))_{t\in [1,+\infty], n\geqslant 1}$ be a sequence of nonnegative random variables and $(\mathcal{F}_s)$ a filtration ($\mathcal{F}_s \subset \mathcal{F}_r$ for $s\leqslant r$). We assume ...
20Xblog8x12's user avatar
1 vote
1 answer
86 views

Expected number of solutions of a random quadratic polynomial system over a finite field

Let $\mathbb{F}_q$ be a field of $q$ elements. Let $a_{i,j,k}$, $b_{i,j}$, $c_i$ ($1 \leq i \leq m$, $1 \leq j \leq k \leq n$) be independent uniformly distributed random variables in $\mathbb{F}_q$, ...
en-drix's user avatar
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1 answer
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Bounded density for determinant of GOE

Let $M$ a random GOE matrix, i.e. $M=(M_{i,j})$ is a symmetric matrix and the $M_{i,j},i\leq j$ are independent centred Gaussien entries with variance 1, except on the diagonal where the variance is $...
kaleidoscop's user avatar
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Finding a collection of random variables satisfying (exactly or numerically) a given set of moment identities

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 &...
Adrien Laurent's user avatar
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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 ...
medislamm123's user avatar
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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 ...
Ishan's user avatar
  • 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 ...
ZZZZZZ's user avatar
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1 vote
1 answer
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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 ...
Ishan's user avatar
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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-...
Akira's user avatar
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1 vote
1 answer
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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 (...
Akira's user avatar
  • 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 (...
Akira's user avatar
  • 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 \...
Marco Max Fiandri's user avatar
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 $\alpha, \beta \in \mathbb{R}_+$ ...
spacetimewarp's user avatar
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 ...
Michael Hardy's user avatar
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 ...
Amir Sagiv's user avatar
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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 $a_1$,...,$a_M$ ...
Kris's user avatar
  • 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\...
OmarR's user avatar
  • 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 $\...
Sanae Kochiya's user avatar
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(\...
Jack London's user avatar
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 ...
Tom's user avatar
  • 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 \...
Ben's user avatar
  • 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 $\{Y_n\}$, \begin{equation} a_1X_1+a_2X_2+\ldots+a_nX_n\geq_1 a_1Y_1+...
Ben's user avatar
  • 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 ...
Drew Brady's user avatar
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 ...
Woosung Kang's user avatar
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 ...
BayesRayes's user avatar
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 ...
Sanket Agrawal's user avatar
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 $1/2$....
odile's user avatar
  • 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 ...
Dimitri's user avatar
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 \...
odile's user avatar
  • 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 ...
user1172131's user avatar
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 ...
David Spivak's user avatar
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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 ...
Jaimin Shah's user avatar
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 ...
virtuolie's user avatar
  • 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 ...
Guoqing's user avatar
  • 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(...
Rosy's user avatar
  • 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:=\...
Iosif Pinelis's user avatar
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: $\...
Marco Max Fiandri's user avatar
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 ...
A. H.'s user avatar
  • 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}}, \...
Akira's user avatar
  • 939
4 votes
0 answers
131 views

Algebraic area of Brownian half-plane excursion

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)...
Timothy Budd's user avatar
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