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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questions with no upvoted or accepted answers
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An Inequality of Expected Value of Random Variables
I encountered the following problem in my research:
Suppose there are $N$ random variables that are independent and identically distributed (IID). The probability density function (PDF) of these ...
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201
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Distribution and expectation of inverse of a random Bernoulli matrix
This question cropped up as a part of my research. Let us assume a $n\times n$ random matrix $\mathbf{M}$ with elements iid distributed to a Bernoulli distribution that takes values $\{0,1\}$ with ...
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43
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Small parameter expansion of probability density
I am trying to describe the motion of a particle that moves according to the Langevin equations
\begin{align}
\dot{x}&(t)=v_0\cos{\beta(t)},\tag{1}\\
\dot{y}&(t)=v_0\cos{\beta(t)},\tag{2}
\end{...
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71
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Why does Y. Moshe Vardi use this specific matrix when estimating source-destination traffic intensities with EM algorithm?
Sorry for the verbose title, but the question is super specific. If you happen to know a site better suited for these types of question, feel free to direct me.
The article to which I am referring to ...
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91
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Distribution of Euclidean distance given two dependent normal distributions
I'm stuck on identifying the distribution of
$$
r = \sqrt{x^2 + y^2}
$$
when $\langle x,y\rangle \sim \mathcal{N}(\boldsymbol{\mu}, \boldsymbol{\Sigma})$ are jointly normally distributed.
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0
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62
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(Anti-)concentration of gap between largest and second largest component of multivariate random gaussian vector
Let $n$ be a large positive integer and let $Y=(Y_1,\ldots,Y_n)$ be a zero-centered random $n$-dmensional real vector with covariance matrix $\Sigma$, an $n$-by-$n$ positive definite matrix with ...
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36
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Grid of mostly independent variables
We're given an finite grid of random variables like so:
$$
\begin{bmatrix} A & B &... \\C & \ddots \\ \vdots&&X\\
\end{bmatrix}
$$
a subset of variables on the gird is independent ...
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98
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Cartesian product of Poisson processes
Consider $n$ smooth, compactly supported functions $\phi_1,\dots, \phi_n \in C_c^\infty(\mathbf{R})$, and generate $n$ independent Poisson spatial processes $N_1,\dots,N_n$ on $\mathbf{R}$, each with ...
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145
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Is there a name for a random variable that is the absolute value of the difference between two iid discrete uniform variables?
I'm working on a project and I needed to calculate the distribution of the difference between two iid discrete uniform variables (sorry for the long title).
That is, let $I, J$ be two iid discrete ...
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44
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How to use the mixed normal distribution to construct a proper statistics?
For a random vector $\xi_n \in \mathbb{R}^p$, if $\xi_n \rightarrow_d N(\mu, \Sigma)$, we can construct
\begin{equation*}
\Psi := \xi_n^{\top} \widehat{\Sigma}^{-1} \xi_n
\end{equation*}
for ...
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0
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32
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Condtions for a stochastic process to be locally non-factorizable
Given a stochastic process $X=(X_t)_{t\in I}$ on $\mathbb{R}^d$ with continuous sample paths supported on a prob. space $(\Omega, \mathscr{F}, \mathbb{P})$ and such that each pair $(X_s, X_t)$, with $(...
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194
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A new notion of probability coupling
Let $X$ and $Y$ be two discrete random variables distributed according to $\mu$ and $\nu$, respectively. Consider the following optimization problems
$$\inf_{\pi\in \Pi(\mu, \nu)}\Pr(X\neq Y),$$
...
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0
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55
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Quantitative bounds on convergence of Bayesian posterior
Let $Y$ be a random variable in $[0,1]$, and let $X_1, X_2, \ldots$ be a sequence of random variables in $[0,1]$. Suppose that the $X_i$'s are conditionally i.i.d given $Y$ ; in other words, I'd like ...
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1
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294
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Finding a connection between two types of convergence
Please, help me find connections between two types of convergence:
Let $\{X_n\}_{n\ge1}: (\Omega,F,P) \rightarrow (\mathbb{R},Bor)$ be a sequence of r.v., there are two convergences:
1) $X_n \...
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111
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Metrics on the space of distributions in terms of p.d.fs
If two probability distributions (on the same measure space) are s.t they have p.d.fs and the $L^1$ distance between the p.d.f.s is large, then is there a choice of a ``nice" metric $d_{\rm ...
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64
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Dependence rank: what is the size of the largest subcollection of random variables which is statistically independent?
Let $X_1,\ldots,X_p$ be random variables on the same space. Define their dependence rank, denoted $rank(X_1,\ldots,X_p)$ as the largest nonnegative integer $k$ such that there is a subcollection of $k$...
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115
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Law of large numbers and Central Limit Theorem for eigenvalues of perturbed matrices
I'm looking for results where perturbation by iid random entries to a matrix will result in convergence of the eigenvalues to the original eigenvalues. More precisely,
Let $ \forall n \in \mathbb{N},...
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0
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74
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Tracy Widom type results for asymptotic distribution of the $k$-th largest eigenvalue of the sample covariance when $n, p \to \infty$?
Earlier I asked a question: Distribution of the $k$-th largest eigenvalue of in the sample covariance matrix?, but I forgot to mention that I'd like results for asymtotic regime. So, I'm posting here ...
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105
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Measure on a set and its value on $\emptyset$
After my first post here, I have one more doubt which is bothering me. It concerns Minlos's book Introduction to mathematical statistical physics again. To fix the notation, we have $\Lambda \subset \...
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435
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Laplace transform of a random variable: Inversion formula from an interval
Let $X$ be a non-negative random variable with a CDF $F$. Let $L_X(t)$ denote the Laplace transform of $F$, i.e.,
\begin{align}
L_X(t)=E[ e^{-tX}], \quad t \ge 0
\end{align}
It is known that $L_X(...
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0
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162
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Sampling uniformly from a ball in SPD manifold
I'm trying to sample uniformly from a ball around the identity matrix in the manifold of symmetric positive-definite matrices (SPD), i.e., $\mathcal{B}(R) = \{X \in \mathcal{P}(n) : d_\mathcal{P}(I_n, ...
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78
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Empirical estimation of $\inf_{\gamma \in \Pi(\mu,\nu)}\gamma(\Omega)$, given i.i.d samples from $\mu$ and $\nu$
Let $\mathcal X$ be a Polish space and $\Omega \subseteq \mathcal X^2$ be open. Let $\mu$ and $\nu$ be probability measures, and consider the quantity $c_\Omega(\mu,\nu)$ defined by
$$
c_\Omega(\mu,\...
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0
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294
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Wasserstein distance between rotated conditional distributions
Suppose we have a probability distribution $\rho$ on $\mathbb{R}^d$. Let $ E \subset \operatorname{supp}(\rho) $, and $R_\theta$ a rotation of angle $\theta$ such that $ R_\theta E \subset \...
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0
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49
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How to get the joint distribution of multimodal deep Boltzmann machine?
Here is the graph model of the multimodal dbm. I want to know how to inference the joint distribution of this probability graph model.
Some example equations are here. For the expression, I have two ...
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0
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127
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Who invented the gamma distribution?
Who was the first person who invented the gamma distribution? Can you also give the reference of the article in which he/she did that.
1
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0
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104
views
Hashed coupon collector
The story:
A sport card store manager has $r$ customers, that together wish to assemble a $n$-cards collection.
Every day, a random customer arrives and buys his favorite card (that is, each customer ...
1
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0
answers
72
views
Large Deviation of Triple Poisson Product
Let $X_i$ with $i=1,\ldots,n$ be independent Poisson variables, $X_i$ with parameter $\lambda_i.$
Let $\circ$ be a group operation on a group of size $n.$
I would like to obtain a large deviation ...
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0
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96
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Joint distribution of two weighted sums of IID random variables
Let $X_1, X_2, \dots$ be independently uniformly distributed random variables in $\{-1, +1\}$ and let $a_1, b_1,a_2,b_2, \ldots \in \mathbb{R}$ be fixed, bounded and of non-zero average. Let $Y_n=...
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101
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A question about pdfs with likelihood ratio order
Suppose $f_1,f_2,\dots$ are pdfs of absolutely continuous random variables with the same support (say an interval). Assume that $\{f_i\}$ are strictly positive in their support. Furthermore, $\frac{...
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0
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56
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About a class of expectations
Consider being given a $n-$dimensional random vector with a distribution ${\cal D}$, vectors $a \in \mathbb{R}^k$, $\{ b_i \in \mathbb{R}^n \}_{i=1}^k$ and non-linear Lipschitz functions, $f_1,f_2 : \...
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217
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Different balls in bins: What is the probability distribution of the sum of the minimum of the two types of balls over all bins?
Assume that there are $N$ different bins and two different kinds of balls, $R$ red balls and $W$ white balls.
The red balls and the white balls are randomly distributed across the bins (that is, for ...
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0
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49
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Stochastic Control: Markovian restriction
Consider a stochastic control problem,
$$v^C(0,x) = \mathbb{E} \Big[\int_0^\tau f(X_t,C_t) d t + (T-\tau)|X_\tau|\Big] $$
where $X_t$ is a weak solution to the SDE
$$dX_t = C_t dB_t, \quad X_0 = x \...
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1
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394
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How to calculate the probability of 2 events happening in time series under only cdf information?
In time domain $0\rightarrow T$, there are two independent events $A$ and $B$.
$B$ follows Poisson Process with density $\lambda$. It's easy to get $P_B(t)$ which denotes $P_B(N(\tau+t)-N(\tau)\geq 1)...
1
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0
answers
44
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Show a convolution of distributions ε-close to min-entropy k is ε^t-close to min-entropy k
Assume $X_1,...,X_t$ are independent distributions on $\mathbb{Z}_2^n$ s.t. each $X_i$ is $\epsilon$-close to min-entropy $k$; i.e. there exist distributions $Y_1,...,Y_t$ on $\mathbb{Z}_2^n$ s.t:
$$ \...
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0
answers
66
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Probability distribution from standard domain (two primes) - IV
Pick a random pair $(a,b)\in\mathbb Z_n^2\setminus\{0,0\}$. Denote $N_2(a,b,n)$ to be minimum $\ell_2$ norm of vector $(x,y)$ as $(x,y)$ ranges over all non-zero integral solutions to $(x,y)\equiv t(a,...
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0
answers
186
views
Calculating the expectation of a sum of dependent random variables
Let $(X_i)_{i=1}^m$ be a sequence of i.i.d. Bernoulli random variables such that $\Pr(X_i=1)=p<0.5$ and $\Pr(X_i=0)=1-p$. Let $(Y_i)_{i=1}^m$ be defined as follows: $Y_1=X_1$, and for $2\leq i\leq ...
1
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0
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135
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Choice of residual function for least squares error minimization
Good morning,
I have the a set of data $(\sigma,D,\alpha_0)_i$, $i=1...n$ data.
I want to determine two parameters $K_{IC}$, $C_f$ in the basic equation given as
$K_{IC} = \sigma \sqrt{D} k_0(\...
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0
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264
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Expected value of inverse of complex non-central Wishart matrix
I have a matrix $W$ that abides a complex non-central Wishart distribution.
My question is what the expectation of the inverse is, i.e., how to compute
$$\mathbb{E}(W^{-1}).$$
I have tried to read up ...
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0
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60
views
Families of distributions with a certain symmetry property?
Consider the probability distribution $\mathcal{N}_n$ on $\mathbb{R}^n$ whose density is $$(2\pi)^{-n/2}e^{-\frac{1}{2}||\vec{x}||^2} = \prod_{i=1}^n \frac{1}{\sqrt{2\pi}} e^{-\frac{1}{2}x_i^2}$$
...
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2
answers
324
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Use Importance sampling for multimodal and multivariate distribution draws, how to choose proposal distribution?
I'm in trouble trying to generate samples following a particular distribution which is not numerically known perfectly. Let us consider a $R^n$ space provided with an orthonormal base $( e_{1},...,e_{...
1
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0
answers
165
views
Upper bound for $\sup_{W_d(Q, P) \le \epsilon} Q(A)$, where $W_d$ is the Wasserstein metric
Let $X=(X,d)$ be a metric space and let $W_d$ denote the Wasserstein metric induced by this metric, on the space of probability distributions on $X$. Let $\epsilon \ge 0$, $A$ be a Borel subset of $X$...
1
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0
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68
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Reference request on numerical integration
Let $\rho:\mathbb R^d\to\mathbb R_+$ be a density function with finite first moment, i.e.
$$\int_{\mathbb R^d}~ \rho(x)dx~=~1 \quad \mbox{ and }\quad \int_{\mathbb R^d}~ |x|\rho(x)dx<+\infty.$$
...
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0
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85
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Conditonal convergence implies convergence?
Note : All measures below are probability measures.
Let $\mu_n(X,Y)$ be a random probability measure on $\mathbb C$ depending on two random variables X and Y with values in $\mathbb{R}^N$.
Actually,...
1
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0
answers
218
views
CLT for random sums: Anscombe's Theorem vs. "classical" version
Given a compound Poisson distribution
$$S(t):=\sum_{k=1}^{N(t)} X_{k}$$ with
$N(t)\in\mathbb{N},\,t\geq0$ a Poisson process with rate $\lambda.$
$X_{k}\in L^{2}$ are iid random variables, i.e. $\...
1
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0
answers
105
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Bounding quantiles of the noncentral chi distribution
I need to bound the empirical quantiles for a noncentral chi distribution (not chi-squared) $\chi_\nu(\lambda)$, where $\nu$ is the number of degrees of freedom and $\lambda$ the non-centrality ...
1
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0
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66
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Showing that $b$ is a inner point of $\mathcal{G}$ where $\mathcal{G}$ is a subset of $\mathbb{R}^{N+3}$ determined by $\mathcal{M}^{+}$
Let $(\Xi,\mathscr{E})$ be a measurable space, $(\mathbb{R_{+}},\mathfrak{B})$ other measurable space where $\mathfrak{B}$ a $\sigma$-algebra. We consider the measurable space $(\Xi\times\Xi\times\...
1
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0
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188
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References for the theory of Hadamard functions and compositions of random vectors
Recently, I fell in love with the pointwise/elementwise/componentwise/Hadamard/Schur functions and compositions of random vectors such as Hadamard squares and products of random vectors. Here is one ...
1
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0
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98
views
Higher order derivative of negative power of cosine function
This is a question I encountered in my own research on Generalized
Hyperbolic Secant (GHS) distributions. It is known that the Laplace transform of the
basis measure for this family is
$$L\left( \...
1
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0
answers
301
views
Distribution of top singular vector of Bernoulli ensemble
Consider the distribution given by taking the top singular vector of a matrix whose entries are equally likely to be +1 or -1.
This is not well-defined for matrices where the subspace achieving the ...
1
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0
answers
414
views
What is the maximum entropy distribution over the integers
Let $μ=0,σ>0$. What is the maximum entropy distribution over the integers with mean $μ$ and variance $σ^2$?
Is Skellam distribution a maximum entropy distribution? Is there a closed-form ...