All Questions
Tagged with pr.probability probability-distributions
1,384 questions
0
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46
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Prove lower collinearity on the tails of Gaussian blob
Let us consider a $n$-dimensional Gaussian blob, i.e. a set of $N$ random vectors $\{\boldsymbol{X}^{(j)}\}_{j=1}^N$, with $n$ independent components, $X_i^{(j)}$, and such that $X_i^{(j)} \sim \...
- 305
3
votes
1
answer
269
views
Trace of product of two Wishart matrices
Let $A,B$ be two independent complex Wishart matrices, $A,B\sim CW_p(\mathbf{I},n)$, that is $A=\frac1n GG^\dagger$& $B=\frac1n QQ^\dagger$ where $G$ and $Q$ are independent $p\times n$ complex ...
- 188k
2
votes
1
answer
164
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Maximal entropy distribution on three variables knowing its marginals on any two
Observation 0: Given a finite set $X$, the probability distribution on $X$ with highest entropy is the uniform one. This is well known.
Observation 1: Given two finite sets $X,Y$ and two probability ...
- 301
2
votes
1
answer
213
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Gaussian expectation restricted to a convex polytope
Let $X$ be a Gaussian vector in $\mathbb{R}^n$ with $\mathbb{E}[X]=0$ and $\mathbb{E}[X X^\intercal]=I_n$. Let $\mathbf{S}$ be a convex polytope in $\mathbb{R}^n$ defined as the intersection of $m$ $(...
- 128k
222
votes
0
answers
18k
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Why do polynomials with coefficients $0,1$ like to have only factors with $0,1$ coefficients?
Conjecture. Let $P(x),Q(x) \in \mathbb{R}[x]$ be two monic polynomials with non-negative coefficients. If $R(x)=P(x)Q(x)$ is $0,1$ polynomial (coefficients only from $\{0,1\}$), then $P(x)$ and $Q(x)$ ...
- 391
2
votes
0
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306
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Why is it impossible to create a numerically balanced die with more than 120 sides?
I allow myself to contact you as a mathematics enthusiast. I have recently been intrigued by the concept of balance in dice and the assertion that it would be impossible to create a numerically ...
- 5,429
2
votes
2
answers
297
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Convergence of the row sums in a triangular null array with zero mean
Let $(X_{jn})_{1\leq j \leq n}$, $n\in \mathbb N$, be a triangular array of random vectors in $\mathbb R^d$ (the $X_{jn}$ are understood to be independent in $j$ for fixed $n$.). We say that the ...
- 128k
2
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0
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56
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Dirichlet series solution to Poisson Point Process question (repost from math.SE)
Reposting here because the bounty on the original math.SE post expired, with no solutions or comments received.
For any discrete subset $S$ of $\mathbb{R}^d$, consider a digraph formed by placing an ...
- 121
3
votes
1
answer
407
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Relative entropy equality for a sequence of Bernoulli random variables
We are given two joint probability distributions, $p$ and $q$, of $n$ Bernoulli random variables $X_1, X_2, \ldots, X_n$.
We denote by $p(x_k\mid x^{k-1})$ the probability $\mathbb{P}_p(X_k=x_k\mid ...
- 1,677
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0
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86
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Show that $\max_{P_X : X\in (0,1) } \left| \frac{\mathbb{E} [ f'(X) ]}{ \mathbb{E} [ f(X) ] } \right|$ is maximized by at most two mass points
Let $f$ be some given well-behaved function. Consider the following optimization problem overall probability distribution on $[0,1]$
\begin{align}
\max_{P_X : X\in [0,1] } \left| \frac{\mathbb{E} [ ...
- 671
4
votes
0
answers
129
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Maximum Likelihood and De Finetti's Theorem
I have a question about whether it is possible to use De Finetti's representation theorem for maximum likelihood estimation.
De Finnetti's theorem states that for any exchangable infinite sequence of ...
2
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0
answers
115
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Equivalence of score function expressions in SDE-based generative modeling
I am studying the paper "Score-Based Generative Modeling through Stochastic Differential Equations" (arXiv:2011.13456) by Yang et al. The authors use the following loss function (Equation 7 ...
- 63.9k
2
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0
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185
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An attempt to define expected value of a Riemannian manifold valued random variable - what'll go wrong?
Let $X:\Omega\to (M,g)$ be a random variable taking values in a Riemannian manifold $(M,g)$ with the Riemannian volume form denoted by $dvol_g(x).$ We know that there's no standard way to generalize ...
- 1,467
1
vote
2
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137
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Sum and alternating sum of a series of Bernoullian variates
Consider the random variables
$a_i,i=0,1,\ldots,n$ be random variables which take values from $\{-1,1\}$ independently and randomly with equal probability. Let
\begin{align}
S &= a_1+\cdots+a_n , ...
- 826
12
votes
4
answers
3k
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What structure is needed to define a Gaussian distribution on a given space?
In most textbooks, the normal distribution is defined on $\mathbb{R}^n$ by specifying its probability density function. This works perfectly well, but it isn't really amenable to generalisation.
I'm ...
4
votes
1
answer
197
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On a double sum involving binomial coefficients
For natural $n$, let
\begin{equation}
p_n:=2^{1-n}\sum_{v=1}^l \binom l{(v+l)/2}1(v\equiv l)
\sum_{u=1-v}^{v-1}\binom k{(u+k)/2}1(u\equiv k), \tag{1}\label{1}
\end{equation}
where $k:=\lfloor(n+1)/...
- 311
1
vote
1
answer
126
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What is this distributional closeness?
Let $P$ and $Q$ be two distributions over a sample space $\Omega$ which I would like to show are close under some choice of distance function. So far I have managed to show that there exists a subset $...
- 128k
4
votes
1
answer
250
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Does a subset with small cardinality represent the whole set?
Assume that we have heavy-tailed distribution $F(x)$ such that
\begin{align}
F(x)=\mathbb{P}[X\geq x]=x^{-0.5}.
\end{align}
Then, we produce $N$ independent samples $X_1,X_2,\ldots,X_N$ from this ...
- 109k
3
votes
2
answers
226
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Recovering measure support from the sequence of I.I.D random variables
Assume we have a Borel probability measure $\mu$ in $\mathbb{R}^n$ and a sequence of $\mu$ distributed I.I.D. random variables $x_n$. Is there a limit formula for $supp(\mu)$, something like closure ...
- 2,101
5
votes
1
answer
516
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Bounding the variance of a truncated Gaussian random variable
Suppose $X_1, X_2, X_3 \sim N(0, 1)$ are three independent standard normal random variables. I am interested in showing that:
$$\text{Var}[X_2\mid X_2 \geq X_1 - a, X_1 \leq X_3 + b] < 1,$$
where ...
-1
votes
1
answer
169
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joint density of two relevant random variables
It seems that for most of the examples to derive the joint density of two or more random variables, the random variables themselves need to be independent. Is it possible to get the joint density of ...
- 5,474
3
votes
1
answer
355
views
Quantitative results (with formal proof) on the median approximation of Chi-squared distribution
It is well-known that Chi-squared distribution $X_n$ with degree-$n$ freedom has an approximate formula for its median as $n\left(1-\frac{2}{9n}\right)^3$. Or $(X_n/n)^{\frac{1}{3}}$ is approximately ...
- 128k
4
votes
3
answers
553
views
Gaussian approximation of the characteristic function of Rademacher sum
I am searching an uniform bound for the characteristic function of some Rademacher sum.
Specificaly I want to estimate how much the characteristic function is close to a Gaussian.
We have in general ...
- 2,183
1
vote
0
answers
68
views
A one-sided/monotone version of min/max-stable distributions -- does this have a name?
In a couple of papers I am working on (in random graph theory) I have encountered the following property of certain probability distributions, which I will describe shortly, and I am wondering if this ...
- 5,429
5
votes
2
answers
139
views
Number of resampling until obtaining a uniform list
Let $A_0$ be a list of $ n$ distinct elements. By sampling with replacement the elements of $A_0$, we obtain a new list $A_1$ of $n$ elements that are not necessarily distinct. Repeat the same process ...
3
votes
1
answer
171
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Variance lower bound for natural exponential family
Let $Q$ be a probability measure on $\mathbb{R}$. Let $$Q_h(dy) = e^{y \cdot h} Q(dy) / M(h) \quad \text{where} \quad M(h) = \int e^{y \cdot h} Q(dy)$$ defined for $h \in (-c,\infty)$ with some $c &...
- 143
-2
votes
1
answer
152
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Branching process with varying offspring distribution at each step
Consider a simple branching process $Z_0,Z_1,Z_2...$ such that at every discrete step, a particle splits into $k\geq1$ particles where $k$ follows a discrete distribution with probability mass $p(k)$.
...
- 3,937
-1
votes
1
answer
77
views
Variance of the logarithm of the mixed Rademacher and complex Gaussian distribution
Consider the scenario where $X$ is a Rademacher random variable taking values $\{−1,+1\}$ with equal probability, and $Z$ is a complex Gaussian random variable with a mean of $0$ and a variance of $\...
- 287
4
votes
0
answers
143
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Projection of log-concave distribution on unit sphere surface
Let $\mathbf X : \Omega \to \mathbb R^d$ be a random vector following a zero mean, identity covariance log-concave distribution.
Is there any known upper bound for the probability density function of $...
- 141
2
votes
1
answer
112
views
Sample integral points in m-Ball
The problem I have is pretty simple, however I cannot find an answer.
I need an efficient algorithm to sample integral points within an m-dimensional ball of radius r around the origin (Euclidean norm)...
- 558
0
votes
1
answer
103
views
Is it reasonable to consider the subgaussian property of the logarithm of the Gaussian pdf?
Let $Y$ denote a Gaussian random variable characterized by a mean $\mu$ and a variance $\sigma^2$. Consider $N$ independent and identically distributed (i.i.d.) copies of $Y$, denoted as $Y_1, Y_2, \...
- 128k
2
votes
1
answer
239
views
Hoeffding's Lemma for bounded complex random variables?
If we have a real random variable $X$ such that $a\leq X\leq b$ almost surely, we can establish the following inequality:
\begin{align}
\mathbb{E}\left[\exp\Big(t(X-\mathbb{E}[X])\Big)\right]\leq\exp\...
- 37.7k
5
votes
1
answer
896
views
How can I sample uniformly from a citrus surface?
I want to sample from a Lemon surface uniformly. The equation of this surface is
$$16(x^2+z^2)+(y-2)^3 y^3=0.$$
I have read the paper
Stratified Sampling of 2-Manifolds
. The method described in this ...
- 128k
4
votes
3
answers
910
views
Solution to the fractional differential equation
What is the solution of the fractional differential equation
$$
f^{(\alpha-1)}(t) = tf(t)
$$
where $(\alpha)$ denotes the fractional derivative of order $\alpha$
EDIT: Background behind this ...
- 101
14
votes
1
answer
417
views
Lipschitz property of the determinant
$\newcommand{\A}{\mathcal A}\newcommand{\Tr}{\operatorname{tr}}$For $c$ and $C$ such that $0<c<C<\infty$, let $\A_{d;c,C}$ denote the set of all symmetric positive-definite real $d\times d$ ...
- 24.2k
3
votes
1
answer
346
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Simple anticoncentration bound for binomially distributed variable
The following question, which arose during my research, seems deceivingly simple to me, but I could not find any elegant and formal argument.
For a binomially distributed variable $X \sim \text{Bin} \...
- 128k
1
vote
1
answer
2k
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First hitting time for a drifted Brownian motion
While the solution for a first hitting time for a drifted Brownian Motion is well known, I want to post a different question.
Take a continuous-time stochastic process $X_t$ and define the the ...
0
votes
0
answers
74
views
Probability distribution for a Bayesian Update
I am struggling with a process like this:
$$X_t=\begin{cases}
\frac{\alpha\omega_t}{\alpha\omega_t+\beta(1-\omega_t)} & \text{with prob } p\\
\frac{(1-\alpha)\omega_t}{(1-\alpha)\omega_t+(1-\beta)(...
1
vote
1
answer
308
views
$L_1$ norm concentration of an empirical distribution
Suppose we have one random variable $X$, whose sample space is $\mathbb{X}=\{x_1,x_2,\dots,x_m\}$, and the size of the sample space is $m$. We have $N$ i.i.d. samples from this distribution, and $x_i$ ...
- 4,529
1
vote
0
answers
83
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Closed form volumes for intersecting modified cylinders
This question is somewhat related to the question Intersecting cylinders, but where the cylinders are now modified to orbifolds in the hypercube with singularities occurring at the vertices of the ...
- 383
4
votes
1
answer
425
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An exercise on log-concave random variable on the real line
Let $X$ be a real random variable with log-concave density $f$. Assume that $E(X) =0$ and $E(X^2)=1$.
Show that there is a universal (independent of $X$) constant $c>0$ such that:
$$P(X\in[-1/2;0])\...
- 128k
4
votes
2
answers
297
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Effect of small change in probability distribution on error probability
Let $X$ be a random variable and $Y=f(X)$ where $f$ is a deterministic function. Moreover, assume that there exists a deterministic function $g(.)$ such that the following probability is small.
\begin{...
- 128k
0
votes
1
answer
61
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What can we say about the order of convergence of a critical point of Gaussian mixture density to its limit when the parameter $h$ goes to $0?$
Density of Gaussian mixture with $n$ components is given by:
$$f(x):=C \sum_{i=1}^{n}e^{-\frac{1}{2}||\frac{x-x_i}{h}||^2}, x_i \in \mathbb{R}^d, h > 0$$
where $C$ is a normalization constant ...
- 128k
2
votes
0
answers
87
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A complex problem involving densities (likelihood functions) and optimization
Consider the following autoregressive process with normal errors:
\begin{equation}\label{7YlUV4i8nuO}\tag{I}
y_t = \phi y_{t-1}+ u_t, \quad u_t \overset{iid}{\sim} N(0,\sigma^2)
\end{equation}
We ...
- 13
1
vote
1
answer
115
views
The effect of a small change of the probability distribution on the output of the function
Suppose $X$, $Y$, $X'$ and $Y'$ are random variables whose probability density follows the following relations.
\begin{align}
\|p_X-p_{X'}\|_{\mathrm{TV}}&\leq\epsilon_1,\\
\|p_Y-p_{Y'}\|_{\mathrm{...
- 3,937
0
votes
1
answer
350
views
The expected value of a double exponential function of normal random variable
Let $X$ be a random variable from a normal distribution $N(\mu, \sigma)$. How do we calculate the expectation $E[e^{k\cdot e^{-X}}]$, where $k<0$?
I think we can use the moment generating function
...
- 188k
2
votes
2
answers
2k
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What is a two-sided geometric distribution?
I found in some articles (such as this) references to two-sided geometric distribution. But I went through texts of probability and did not find anything called "two-sided geometric distribution". ...
- 103
3
votes
3
answers
410
views
Statistical moments of $\frac X{X + Y}$ when $X$ and $Y$ are two independent random variables with a Beta distribution
I'm trying to find the moments (or the pdf but I'm less confident there's a closed form) of $\frac X{X + Y}$ where $X$ and $Y$ are two independent random variables with a Beta distribution. There's a ...
1
vote
1
answer
879
views
Does the (normalized) product of two independent binomial variables converge in distribution to a normal variable?
(I asked this question on MSE 10 days ago, but I got no answer.)
Let $X$ and $Y$ be two independent identically distributed binomial random variables with parameters $n \in \mathbb{N}$ and $p \in (0,1)...
1
vote
1
answer
204
views
Expected (maximum minus minimum) of Laplacian random variables
Suppose there are $n$ IID random variables denoted as $X=(X_1,\dots, X_n)$, they follow Laplace distribution with parameter $\lambda$, denoted as $Lap(\lambda)$. That is,
$$f(x)=\frac{1}{2\lambda}\exp ...
- 128k