# Tagged Questions

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### Is it possible to construct any random variable on the Euclidean Probability space?

Let $(\Omega,\mathscr A,P)$ be an arbitrary probability space, and let $X:\Omega\to\mathbb R$ be a random variable. Then, one can generate a random variable $Y$ from the probability space ...
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### hi, I have a question about probability density function [migrated]

I've just read about probability density function from wiki( http://en.wikipedia.org/wiki/Probability_density_function ). In that article, there is some wired concept that I can't understand, please ...
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### Entropy on a draw from a random distribution.

Suppose I am attempting to calculate the entropy of a continuous, normally distributed random variable $X$, from the distribution $\mathcal{N}(\mu, \sigma)$. This is easy to to do - I just calculate ...
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### Beta distribution - changes in multiple time points

Let's say I have a set of daily data (assume iid) that I know is beta distributed (between 0 and 1). I can estimate the parameters of the distribution and calculate the tails etc. This would tell me ...
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### Two matrix Fisher distributions on SO(3)?

After the uniform distribution (normalized Haar measure), the matrix Fisher distribution seems to be the most popular probability distribution on the Lie group SO(3). The density is proportional to ...
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Assume $z \in \mathbb{R}^m$ and $x \in \mathbb{R}^n$. Assume we have proper density function $P(z)$ and proper conditional density function $P(x|z)$. We give the definition $$T(x_1,\ldots,x_n) := ... 0answers 83 views ### An inequality for moments of a random variable I'm interested in a class C of R^1-valued random variables \xi which satisfy an inequality of the type$$ (1) \qquad E|\xi|^p \leq F(E|\xi|^2), $$where p>2, F is a certain ... 1answer 81 views ### A special class of random variables I'm interested in classes C of R^1-valued random variables which possess the following properties: 1) the sum of two independent random variables from class C belongs to class C; 2) for any ... 0answers 58 views ### Maximizing the “uniformity” of a probability measure, with constraints, via path length minimization Background I want to find a probability measure for a continuous random variable, subject to moment constraints, that is maximally "uniform", as defined below: Definition: Maximally Uniform ... 1answer 157 views ### How to check if a symmetric random variables is the difference of two iid symmetric random variables I have the continuous symmetric random variable X in \mathbb{R}. If I know its distribution function F(x) what are the conditions on F(x) so that X=Y_1 - Y_2 where Y_i are also iid ... 4answers 201 views ### Estimating the Variance of a Discrete Normal Distribution Let f(x; \sigma) = \frac{1}{\sigma\sqrt{2\pi}}\cdot e^{-\frac{x^2}{2\sigma^2}} be the probability density function of a normal distribution \mathcal{N}(0, \sigma^2). We consider a discrete normal ... 1answer 200 views ### Measure concentration for law of large numbers The classical law of large numbers states that$$\frac1k\sum_{i=1}^k X_i \rightarrow \mathbb{E} X_1$$for i.i.d. X_1, X_2, \ldots with finite L^1 norm. I was wondering whether is it possible to ... 1answer 98 views ### Residual lifetime of heavy-tailed random variable The residual life time distribution of a random variable X with distribution function F is given by the formula R(t)=P[X_\text{res}\leq t] = ... 0answers 61 views ### Dominating Poisson with parameter depending on a Bernoulli Fix \mu >0 and take \lambda \geq 0. Let B_p \sim \text{Ber}(p) with p = \exp(-\mu - \frac{\lambda}2) . Define the random variable Y which is Poisson with parameter depending on the value ... 1answer 183 views ### Double Markovity Suppose we have a double Markov relation for three random variables X, Y and W as follows$$X\to W\to Y,$$and$$X\to Y\to W.$$How to prove that there exist functions f and g such that ... 2answers 109 views ### Approximate Moment Conditions It is known in classical probability that if two random variables X and Y obeys$$\mathbb{E} X^k = \mathbb{E}Y^k, \ \forall \ k \geq 1$$with additional condition that \mathbb{E}X^k does not ... 1answer 104 views ### Laplace transform of : t^{\gamma-1} F(\alpha,\beta,\delta,t), where F is the Gauss' hypergeometric function What is the Laplace transform of : t^{\gamma-1} F(\alpha,\beta,\delta,t), where \gamma >0  and F is the Gauss' hypergeometric function. Thanks! 1answer 180 views ### Using a probability measure, P, defined on uncountable sets to construct a probability measure, P' on singleton P-null sets Let \Omega be an uncountable set and (\Omega, \mathcal{F},P) be a probability space built on \Omega. Let S \subset \{A \in \mathcal{F}: P(A)=0,\;|A|=1\}:|S|<\infty be a finite subset of ... 1answer 209 views ### Probability distribution of uAv… Consider the complex domain ℂ. If U and V are 2 unitary random matrices and A is a deterministic matrix. What is the distribution of u^HAv ( or ||u^HAv||^2) where : u is a column vector of U. v ... 2answers 169 views ### How to calculate P(\sum_{i=1}^{m}(A_i+S_i)\le L) with A_i,L\sim\text{exp}(\lambda),S_i\sim\text{exp}(\mu) and positive integers \lambda\neq\mu? Recently I was stumped by the calculation of the probability$$\mathbb{P} \big(\sum_{i=1}^{m} (A_i + S_i) \le L < \sum_{i=1}^{m+1} (A_i + S_i) \big)$$where A_i \sim \text{exp}(\lambda), S_i \sim ... 1answer 123 views ### Push-forward density as surface integral [closed] Let X be a random variable taking values in \mathbb R^n with a probability distribution \mathbb P that has a density p. Consider further a linear mapping \pi: \mathbb R^n \to \mathbb R^m, ... 1answer 139 views ### Distribution of entries of a doubly-sorted random matrix Take an n \times n random matrix whose entries are i.i.d. with uniform distribution in [0,1]. Look at the sums of the elements of each row and then permute the rows so that these sums form an ... 1answer 185 views ### Concentration of sum of powers of normals Let Z_1,Z_2,\ldots,Z_n be i.i.d. copies of a random variable Z distributed as \frac{1}{\sqrt{2}}X+i\frac{1}{\sqrt{2}}Y with X and Y independent standard Normal random variables ... 1answer 138 views ### Singular distributions: Applications and Instances Singular distributions are special mathematical objects. They have an interesting property of not having a density function, defined on a set with Lebesgue measure zero. Cantor distribution is the ... 1answer 174 views ### Quantiles moments and Convergence QUESTION: Let F be an absolutely continuous distribution function with density f, and F_{n} be its nth empirical distribution. Suppose that t\in (0,1) is constant. Is true the convergence ... 1answer 142 views ### Derive concentration bound for the derivative It that true to conclude that if a random f(z) is a sub-Gaussian random variable for a constant value of z, its derivative f'(z)|_{z=k} with respect to variable z is also sub-Gaussian? In ... 1answer 146 views ### A calculation involving a uniform random variable quantile THE PROBLEM: Let U be a uniform distribution and U_{n} be its nth empirical distribution. Suppose t\in (0,1) and n\in \mathbb{N} are constants. What's the explicit expression to ... 1answer 71 views ### Running supremmum of a Levy process Let X be a cadlag Lévy process with X_0=0 and let p be a real number in [1,\infty). Then, the following are equivalent. 1): X is L^p-integrable. 2): X^*_t= \mathop{\sup}_{0\leq s\leq t} ... 2answers 161 views ### Empirical estimator for total variation distance between two product distributions Let X = (X_1, X_2, \ldots , X_n) be an n-dimensional random variable, where each X_i is a random variable on finite discrete set S. In addition, X_i are independent of each other (but not ... 1answer 124 views ### Question about characteristic function with independence assumption Let X be a random vector taking values in \mathbb R^2 with probability density p(x) = p_1(x_1)p_2(x_2), i.e. the components of X are independent. Let V be an open set in \mathbb S^1, the ... 2answers 179 views ### Joint probability distribution as functions Suppose X and Y are correlated random variables in a finite set {\mathcal A}, and let f, g be functions that map elements from {\mathcal A} to {\mathcal B} for some finite set {\mathcal ... 3answers 343 views ### Maximum of the expectation of maximum of Gaussian variables Suppose X=(X_1,\ldots,X_n) is a Gaussian vector with each entry X_i marginally distributed as \mathcal{N}(0,1). Want to find out the possible maximum of$$\mathbb{E}\max_{1\le i\le n}|X_i|$$and ... 0answers 79 views ### Learning resources for Probability Distributions/Models [closed] I've a good background in basic probability. I need to learn and get a good grip on the probability distributions and stochastic processes, counting processes, and other related topics. I am already ... 1answer 88 views ### Random weighted selection without replacement I am using the following procedure to select m different numbers \{i_1,\ldots,i_m\} from the set \Omega = \{1,\ldots,N\}, with m,N\in\mathbb{N} such that m< N. Selection procedure ... 0answers 56 views ### Bounds or approximations for the conditional probability of an event involving correlated random variables Let \tilde{\gamma_1}, \tilde{\gamma_2}, \ldots, \tilde{\gamma_N} be exponential random variables (RVs) that are correlated with each other. Let \gamma_n be another exponential RV that is ... 1answer 101 views ### General version of Skorokhod representation of random variables Let F: \mathbb{R} \to [0,1] be cumulative distribution function (cdf). The standard way to build a random variable \tau on ([0,1],\mathcal{B},\text{Leb}) with F as its cdf is using the ... 3answers 206 views ### Lipschitz continuous maps from \mathbb R^n to \mathbb R^n that preserve Gaussian measure? The only ones I can think of are linear maps like rotations and permutations. Is there a more general characterization? 1answer 153 views ### Characterizations of the GOE/GUE family of distributions This question is somewhat related to this one. Loosely speaking, when should I expect a GOE/GUE distribution? The angle of my approach to this is not through statements such as "there is a natural ... 1answer 431 views ### Mean of i.i.d Random Variables With No Expected Value Let X be an integer-valued random variable and let X_n be the sum of n independent realizations of X. I would like to understand the behavior of X_n/n for large n in some cases where X ... 1answer 107 views ### Variance of maximum of mixture of gaussians Let \{X_i\} be an iid collection of standard normal (N(0,1)) random variables . Let X = (X_1,\ldots,X_n), and consider a function of the form f(X) = \max(A\cdot X), where A is some ... 0answers 281 views ### 1-Wasserstein distance between two multivariate normal The p-Wasserstein between two measures \nu_1 and \nu_2 on X is given by ... 1answer 290 views ### Convergence rate of the central limit theorem near the center of the distribution I'm looking for fast convergence rates for the central limit theorem - when we are not near the tails of the distribution. Specifically, from the general convergence rates stated in the Berry–Esseen ... 1answer 111 views ### explicit expressions of the distribution of sums of i.i.d. logistic random variables Where can I find the explicit expression of the distribution of the sum of n i.i.d. logistic random variables, for n=2,3,4... The expressions given in "On the convolution of logistic random ... 3answers 175 views ### Expected value of swaps Suppose you have a list of non negative numbers of size N. Now you calculate the maximum element in the list by scanning the list linearly and constantly updating a variable which has initial value of ... 3answers 386 views ### Are there known expressions for total variation distance between N(0,\sigma_1^2) and N(0,\sigma^2) Are known expressions for total variation distance between N(0,\sigma^2) and N(0,\sigma^2+\epsilon) for small \epsilon? The only thing I seem to find is things are expression about the mean but ... 0answers 77 views ### Fitting distribution to spatial data I am studying a physical process generating data which projects nicely into two dimensions with non-negative values. Each process has a (projected) track of x-y points -- see the image below. ... 0answers 61 views ### Can truncated/non-smooth distributions be used as priors/posteriors in Variational Bayesian methods? Variational Bayesian methods can sometimes be a good alternative to Markov Chain Monte Carlo numerical evaluation of probability distributions. They do this, as I understand it, by approximating the ... 0answers 104 views ### approximation of probability distribution I have a question: Let \mu be a probability distribution defined on (\mathbb{R}, \mathcal{B}(\mathbb{R})) satisfying$$\int_{\mathbb{R}}|x|d\mu<+\infty$$Set$$A_n=\Big\{\frac{i}{n}:~ ...
Let $X$ be a random vector in $\mathbb R^n$ with probability distribution $\mathbb P_X$. Now when given only the family of distributions \begin{align*} \left\{ \mathbb P_{v_1 X_1 + \dots + v_n ...