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      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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      83 views

      A probability density function for pink noise

      I would like to understand pink (or $1/f$) noise better. However, clearly written resources are difficult to come by, and are usually concerned with its Fourier spectrum, or qualitative descriptions ...
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      27 views

      Cross product of multi-variate Gaussians and their expectations

      Let $a, b \in \mathbb{R}^3$ be two vectors, chosen independently from multi-variate Gaussian distributions ($a \sim N(\mu_a, \Sigma_a), b \sim N(\mu_b, \Sigma_b)$). I'm trying to find a closed-form ...
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      distribution of interval time of a mixture of random arrivals

      suppose there is a random arrival process with inter-arrival time following a distribution $f(x)$. I am thinking about a mixture of $N$ such processes and what the distribution of inter-arrival time ...
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      57 views

      Convexity of exponential family

      It is known that (given a $\sigma$-finite Borel reference measure $\nu$ on $\mathbb{R}$) the parameter space of an exponential family is convex in Euclidean space. However, my question is, for an the ...
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      91 views

      Central limit type theorems for compact Hausdorff topological groups?

      Given a compact Hausdorff topological group $G$ and probability measures $\mu$ and $\tau$ on the Borel sets of $G$, their convolution is the probability measure $(\tau*\mu)(A)=\int\int1_A(xy)d\tau(x)...
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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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      52 views

      Marginal probability mass function

      I have the joint PMF $\exp(y_1 \ln(\lambda)+y_2 \ln(c)+y_2\ln(\lambda)-\ln(y_1!y_2!)-\lambda(1+c))$ for a constant $c>0$. In canonical representation and mixed parameterization I have $\mathbf{\...
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      188 views

      Expected value of square[X/sigmaX] = 1/n^2(1+1/pi)?

      Please see the below link for the complete description. I already have an answer shown in the link, based on many Excel simulations ($n=4$ to $100$, $x_i$ generated by RAND() function of Excel). I ...
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      28 views

      Expected Euclidean norm of vector with i.i.d. Levy distributed entries

      let $X\in\mathbb{R}^n$ be a random vector with i.i.d. entries $X_i=Y_i-\mu$, where the $X_i$ are distributed according to a Levy distribution with stability index $\mu\in(1,2)$ and arbitrary skewness ...
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      Physical meaning of dividing the mean square by variance of a distribution

      In the field of chromatography, the so-called "efficiency" a Gaussian peak or at times an exponentially modified Gaussian peak is expressed as the mean squared divided by the variance of the peak. The ...
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      “Сross сubic variation” of two Brownian motions and interpretation of the simulation result

      Consider two independent 1-dimensional Brownian motions $W_{t},B_{t}$, with an equidistant partition of the interval $[0,T]$, and $n\Delta≡T$. How to calculate the expression below? Can we rewrite ...
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      168 views

      Find $\inf_{P_{X_1,X_2}}P_{X_1,X_2}(\|X_1-X_2\| > 2\alpha)$ , where $\alpha > 0$ and inf is over couplings

      Let $\mathcal X$ be a seperable Banach space with norm $\|\cdot\|$, and let $X_1$ and $X_2$ be random vectors on $\mathcal X$ with finite means. Question. Given $\alpha > 0$, what is value of, ...
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      65 views

      Semi-discrete Wasserstein distance to uniform

      Does the $p$-Wasserstein distance have a simpler expression when applied to these two distributions : A uniform distribution on $[0,1]^d$ A discrete distribution with $N$ equally-weighted point mass ...
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      93 views

      Entropy of probability distributions on a sphere

      I have a family of probability distributions on the $n$-dimensional sphere $\mathbb S^n \subset \mathbb R^{n+1}$ defined in the following way: $D_0$ is the uniform distribution, which is constructed ...
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      31 views

      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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