All Questions
Tagged with pr.probability probability-distributions
1,384 questions
3
votes
2
answers
216
views
Approximating the probability that two Binomial variables are equal
Let $X,Y\sim Bin(n,p)$ be independent R.V.s and let $z\in[n]$ be integer.
My goal is to approximate the probability that $P[X-Y=2z]$.
What i need is a tight enough bound with error that is at most $o(\...
- 31
2
votes
1
answer
177
views
Optimization over Poisson-binomial distributions
I am studying the problem of how an expected utility maximizer should optimally form a portfolio of uncorrelated Bernoullis.
Fix an increasing sequence of $n$ numbers in $(0,1)$, $0<p_1<\dots<...
2
votes
1
answer
170
views
Law of large numbers for a continuum of Bernoullis
Suppose I have a family of $n$ independent Bernoulli random variables described by a vector of parameters $(p_i)_{i=1}^n$. As it is well known, the number of successes within this family is a random ...
4
votes
0
answers
112
views
MGFs of sum of (Rademacher) independent variables and (hyperbolic/spherical) Pythagorean theorem
Consider a set of iid random variables $X_1, X_2, \ldots$ (distribution to-be-specified later). For real numbers $a_1, a_2, \ldots$ (with $\sum_{k} a_k^2 < \infty$) define
$$X = a_1 X_1 + a_2 X_2 +...
- 637
3
votes
1
answer
70
views
A rearrangement majorant of two random variables
$\newcommand{\Om}{\Omega}\newcommand{\F}{\mathcal F} $Let $X$ and $Y$ be random variables (r.v.'s) defined on a non-atomic probability space $(\Om,\F,P)$ such that $P(X<0)>0$ and $P(Y<0)>0$...
- 128k
0
votes
1
answer
78
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Uniform concentration bound (function-valued random variable / continuous stochastic process)
I'm trying to consider a probability space $\Omega$ and
$f(x,\xi):\mathcal{X}\times\Omega\to\mathbb{R}$ (stochastic process over space? or function-valued random variable?), where $\mathcal{X}\subset\...
- 321
2
votes
0
answers
118
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 ...
- 253
2
votes
1
answer
246
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}{\...
- 835
2
votes
1
answer
136
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
1
vote
0
answers
43
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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 $...
- 12.8k
2
votes
0
answers
55
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 ...
- 29
0
votes
0
answers
74
views
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 &...
- 133
0
votes
0
answers
90
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 ...
- 21
1
vote
1
answer
99
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
0
votes
0
answers
87
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 \...
1
vote
1
answer
200
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
2
votes
1
answer
86
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(\...
- 195
2
votes
0
answers
54
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
115
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
73
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+...
- 19
1
vote
0
answers
68
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 ...
- 29
0
votes
1
answer
155
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 ...
- 103
1
vote
1
answer
301
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
99
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 ...
- 9
1
vote
1
answer
147
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
119
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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 ...
- 305
7
votes
1
answer
556
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:=\...
- 128k
1
vote
0
answers
84
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 ...
- 43
4
votes
0
answers
142
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)...
- 3,927
1
vote
1
answer
115
views
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 ...
- 121
2
votes
1
answer
138
views
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( - (\...
- 275
14
votes
1
answer
1k
views
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 ...
- 3,527
-1
votes
3
answers
215
views
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 ...
- 109
1
vote
0
answers
93
views
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$ ...
- 119
0
votes
0
answers
116
views
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....
- 11
3
votes
0
answers
83
views
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 ...
- 245
0
votes
1
answer
85
views
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 $\...
- 265
0
votes
1
answer
69
views
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 ...
- 155
5
votes
2
answers
730
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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 ...
- 101
-2
votes
1
answer
260
views
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)=...
- 181
1
vote
1
answer
186
views
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 $...
- 33
0
votes
0
answers
55
views
Sum of Skellam-distributed number of random variables
Suppose $X_i$ are i.i.d, and $N \sim \text{Skellam}(\mu_1$, $\mu_2$). Is it possible to find a closed form for the p.d.f of $S_N$, defined by $S_N = X_1 + \cdots X_N$ when $N \ge 0$, and $S_{-N} = -...
- 11
2
votes
1
answer
156
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Some identities from graph theory and probability
The other day I attended a seminar about probability. I took some notes and I am now revising it and trying to understand some steps that were omitted by the lecturer. To formulate my question, ...
- 1,305
0
votes
1
answer
231
views
Concentration inequalities for random sampling without replacement
Let a population $C$ consist of $N$ values $c_1, c_2, \cdots, c_N$, with $c_i\in \{0,1\}$. Let $X_1, X_2, \cdots, X_n$ denote a random sample without replacement from $C$ and let $Y_1, Y_2, \cdots, ...
- 105
4
votes
1
answer
277
views
Limit of distributions
Suppose that $X_1,X_2,\ldots, X_n$ are i.i.d random variables with continuous density $f(x)$, which is defined in the whole $\mathbb{R}$. Consider $$s(x)=\lim_{n\to\infty}\frac{1}{n}\log\mathbb{P}(\...
- 233
2
votes
1
answer
246
views
What's the lower bound of the correlation coefficient?
Suppose a random variable $X \in \mathbb{R}$ follows a discrete distribution $p$ and takes $n$ values. We assume $E[X]=0$ and $|X|\le M$, where $M$ is a constant. Given a smooth and monotonic ...
- 49
1
vote
0
answers
148
views
conjecture for general form of minimax estimator
I had previously posed an overly ambitious version of this conjecture here,
Form of minimax estimator,
which was quickly shot down by Václav Voráček (on twitter) and Iosif Pinelis (MO answer in the ...
- 6,541
3
votes
0
answers
77
views
Distribution of waiting time conditioned on a fixed time length
FYI, this question is a duplicate from math stack exchange
I ask here again because I got no response.
Suppose, I work in a factory production line. The time for me to finish wrapping product $A$ (or $...
1
vote
1
answer
341
views
Form of minimax estimator
Let $\Delta$ be the set of all probability distributions over $\mathbb{N}=\{1,2,\ldots\}$ and fix some $\mathcal{P}\subseteq\Delta$.
Suppose additionally that $\Delta$ is endowed with some norm $||\...
- 6,541
1
vote
1
answer
208
views
Extreme confusion with the exact meaning of Gaussian measure with "translation-invariant" covariance
In physics literature, the covariance of a Gaussian measure $\mu$ on a function space is denoted as $C(x,y)$. Moreover, they say that if the covariance is translation-invariant, then actually $C(x,y)=\...
- 3,477