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.
1,919
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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 ...
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28
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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 $a_1$,...,$a_M$ ...
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139
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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\...
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17
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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 $\...
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28
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Modeling a population over time given certain parameters [closed]
I am trying to model a population over a span of time. The span of time is between the birth of the first person and the death of the last one, say the years 1867 and 2151. Total people born is a ...
2
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1
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58
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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(\...
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36
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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 ...
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What are the assumptions when dealing with the EM Algorithm in order to calculate $f_{\textbf{Y}|\textbf{X},\theta}(\textbf{Y}|\textbf{X},\theta)$?
Consider the EM Algorithm. In order to apply it, we are given the observed data $\textbf{X}$ (generated by some distribution depending on some parameters), which can be a vector, a matrix or a matrix ...
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63
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Inequality related with log-concave distributions
Fix any $n$-dimensional unit vector $\mathbf v$.
Let $\mathbf x$ be a random vector following the $n$-dimensional standard normal distribution. It has been shown (Analysis of Perceptron-Based Active ...
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108
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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
\...
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57
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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+...
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59
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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 ...
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80
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Stochastic dominance for (~)random harmonic series
$\DeclareMathOperator\Pr{Pr}$Consider the series $\sum_n^\prime a_nR_n$, where $a_n=\frac{(-1)^n}{n+c}$ for some constant $c\in(0,1)$ and $\{R_n\}$ denotes a sequence of i.i.d. Bernoulli random ...
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1
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30
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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 ...
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65
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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 ...
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1
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89
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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 ...
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39
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How to modeling continuous batching in large-scale inference with queuing theory approach?
I want to model continuous batching in large model inference problems, but my knowledge in data theory is insufficient, and I haven't found an appropriate queuing theory model to use for modeling. So, ...
1
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1
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182
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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 $1/2$....
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64
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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 ...
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1
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115
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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 \...
2
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1
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75
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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 ...
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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 ...
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0
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44
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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 ...
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2
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199
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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 ...
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27
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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 ...
0
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0
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45
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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(...
7
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1
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515
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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:=\...
2
votes
1
answer
105
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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:
$\...
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0
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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 ...
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1
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58
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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.,
$$
\begin{align}
p_t (x) &:= (4\pi t)^{-\frac{d}{2}} e^{-\frac{|x|^2}{4t}},
\...
4
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Algebraic area of Brownian half-plane excursion
Is anything known about the distribution of the algebraic area, à la Lévy's stochastic area, of a Brownian excursion in the half-plane? To be precise, letting $x>0$, we consider the path $(X_t,Y_t)...
1
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1
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53
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How does Chernoff-Hoeffding bound with limited independence reduce to the usual generic CH bound with complete independence
As the title might suggest, I am referring to this paper https://www.cs.umd.edu/~srin/PDF/ch-bounds.pdf , titled : Chernoff-Hoeffding Bounds for Application with Limited Independence.
The theorem in ...
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1
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71
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Optimization: Determine the categorical pmf that maximizes the objective function
Let $T$ denote a $J$-component categorical random variable with pmf
$$
\mathsf P(T=t_j)=w_j,\quad j=1,2,\dots,J,
$$
where $t_j\in[0,t_\max]$, $t_\max>0$.
I came across a problem that seeks to ...
1
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1
answer
80
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fourth-order multivariate Gaussian integral
I am struggling with an integral of form
$$ \int_{\mathbb R^n} y\otimes y~ \langle Ay,y\rangle \, \mathrm d N(0,\Sigma)(y). $$
I assume that it will involve the trace of some product of $R$ and $\...
2
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1
answer
91
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expectation of the product of Gaussian kernels and their input
I was wondering if anybody knows how to solve: $$\mathbb{E}{\mathbf{z} \sim \mathcal{N}(\mathbf{0}, \mathbf{I})}\left[ (\mathbf{x}{i} - \mathbf{z})(\mathbf{x}{j} - \mathbf{z})^\top \exp\left( - (\...
13
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1
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A disc contains many random points. Each point is connected to its nearest neighbor. What is the expectation of average cluster size?
A disc contains $n$ independent uniformly distributed points. Each point is connected by a line segment to its nearest neighbor, forming clusters of connected points.
For example, here are $20$ random ...
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3
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Proving the uniform distribution maximizes the expected value of the product of a random draw of $m$ elements from discrete distribution
Say I have a discrete probability distribution $p_i$, so $0 \le p_i \le 1$ and $\sum_i{p_i}=1$. We sample $m > 1$ draws $D$ from this distribution proportional to $p_i$ with replacement, and ...
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79
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Representation theory for symmetries of probability distribution functions
I would like to parameterize all the possible modifications to a probability density function. Is there a representation theory for this? Something along the lines of, these are all the operators $L$ ...
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0
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67
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Concentration bounds for sum of weighted sampling without replacement
Let $X$ be a collection of $2l$ non-negative numbers $X_1,X_2,\ldots,X_{2l}$. We draw $l$ weighted (proportional to values) samples without replacement from $X$. Let $S$ denote this set of $l$ samples....
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1
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117
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Central limit theorems involving nominal-valued random variables
Suppose $X$ is a random variable taking values in a finite set $\{a_1,\ldots, a_k \}$ and for $i=1,\ldots,k,$ $Y_i = \begin{cases} 1 & \text{if } X=a_i, \\ 0 & \text{otherwise.} \end{cases}$
\...
3
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0
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70
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Monotone Characteristic Function
Let $X$ be a continuous, symmetric random variable such that its characteristic function $\phi_X$ is real, symmetric and with $\lim_{t\to\infty}\phi_X(t)=0$.
What other properties must $X$ have in ...
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0
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20
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Asymptotic behavior of a truncated MGF of a quadratic form of a standard gaussian
Assume $X$ follows an $n$ dimensional standard Gaussian distribution and $A$ is an $n\times n$ semi-definite matrix. I am trying to analyze the asymptotic behavior of $\mathbb{E} e^{-X^TAX} \mathbf{1}...
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1
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62
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Conditioned on the expectation and covariance, is the total variation distance maximal for Gaussian distributions?
I want to find two distributions $p_1, p_2$, whose total variation distance is the largest between all pairs of distributions whose expectations $\mu_1, \mu_2\in \mathbb{R}^d$ and covariances $\...
0
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1
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56
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Correlation for a Sum of random vectors from the sphere multiplied by matrices
Let $A_1,\dots,A_n\in \mathbb{R}^{d\times d}$ be some matrices. Suppose we sample $x_1,\dots,x_n,y\sim \mathcal{U}(\mathbb{S}^{d-1})$, where $\mathcal{U}(\mathbb{S}^{d-1})$ is the uniform distribution ...
1
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2
answers
116
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Approximation of the Gaussian Mills' ratio
Let $R(t) = \frac{1 - \Phi(t)}{\varphi(t)}$ where $\Phi, \varphi$ represent the CDF and PDF of the standard Normal distribution, respectively.
I am interested in approximations of the function for $t &...
2
votes
0
answers
123
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How to sample uniformly over a polytope knowing its vertex presentation?
Say that a convex polytope $P$ is presented as $P = \mathrm{Conv}(v_1, \dots , v_m)$.
I would like to sample over $P$, without generating the facet presentation of the polytope.
How can I do that?
I ...
4
votes
2
answers
405
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Probabilty measures that are both discrete and continuous
Consider a measure space $\left(S,\Sigma\right)$ where each state $s\in S$ can be expressed as $s=\left(x,c\right)$, where $x\in\mathbb R$ and $c\in\mathbb N$. E.g., suppose $s$ denotes the state of a ...
-2
votes
1
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249
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On Impossible events
Let's consider a continuous random variable $X$ distributed according to a PDF $p(x):\mathbb{R}\mapsto \mathbb{R}_{\geq 0}$.
Is there a meaningful sense in which one could say that for any $x_0:p(x_0)=...
4
votes
1
answer
285
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Why should we expect this odd behavior of negative binomial distributions?
In independent Bernoulli trials with probability $p$ of success on each trial, let $X$ be the number of failures before the $n$th success. Then
$$
\Pr(X=x) = \binom{-n}{\phantom{+}x} (-q)^x p^n \text{ ...
1
vote
1
answer
112
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Kolmogorov inequality for Bernoulli random variables
This question is also asked on math stackexchange. The question is about one inequality which shows in Kolmogorov's paper (inequality (3.1)) but is not proved. The inequality says that, if we assume $...