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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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        36 views

        How to compute means μ1 and μ2 knowing sum of Skellam distributions f(k;μ1,μ2) and sum μ1+μ2, where k is from 2 to n?

        The probability mass function for the Skellam distribution for a count difference $ k=n_1-n_2 $ from two Poisson-distributed variables with means $\mu_1$ and $\mu_1$ is given by: $$ f(k;\mu_1,\mu_2)=...
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        26 views

        Statistics of perfect matching and incremental perfect matchings in bipartite planar graphs?

        Planar graph permanent can be reduced to determinants and so statistics should be amenable. Pick a uniformly random bipartite planar graph $G$ with $n$ vertices of each color and choose new ...
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        votes
        1answer
        134 views

        Maxima of Brownian motion

        It is well-known that Brownian motion attains infinitely many maxima in each time interval $[0,T]$ a.s.. From a physics perspective it seems reasonable that when the disorder of the path of a ...
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        votes
        1answer
        61 views

        Intersection of quadratic equations with planted solutions?

        Suppose we have two quadratic equations in $\mathbb R[x_1,\dots, x_n]$. What is the expected dimension of their intersection? In general what can we say about intersection of $k$ quadratics? How many ...
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        66 views

        Concentration inequality for max component of a multivariate Gaussian in the general case

        I am looking to bound the variance of the maximum component of a vector distributed multivariate Gaussian in the general case where the Gaussian distribution has arbitrary mean and full covariance ...
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        51 views

        Tail decay of the norm of spherical distribution

        Let the distribution of the random vector $X\in\mathbb{R}^n$ be orthogonally invariant. I'm considering the following expectation for any vector $\mu\in\mathbb{R}^n$: $$f(\|\mu\|)=\mathbb{E}\left[\exp(...
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        vote
        1answer
        99 views

        Moment generating function of random unit vector

        Let $X$ be uniformly distributed on the unit sphere $S^{n-1}$. Is there any result concerning the calculation or bound (particularly lower bound) of $$\mathbb{E}[\exp(X^Tv)]$$ for any $v$?
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        Tight bounds for finite de Finetti's theorem

        de Finetti's theorem roughly states that infinite sequence of exchangeable random variables are conditionally independent. I am looking for tight bounds for de Finetti's theorem in the following ...
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        1answer
        73 views

        Expected value of a random variable conditioned on a positively correlated event

        I have a random variable $x \in [a, b]$ with PDF $f(x)$ and an event $E$ which satisfies the following property for any $x'<b$. $$\Pr[E\mid x > x'] \geq \Pr[E]$$ My question is whether or not ...
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        1answer
        139 views

        Is there a 1/poly(n) or 1/polylogn upper-bound for this tail bound?

        Is there a good tail bound for $\operatorname{P}\!\Bigg[\bigg\vert\dfrac{\sum_{j=1}^n(\sum_{i=1}^n a_{i,j})^2}{n^2} -1\bigg\vert > \epsilon\Bigg]\,,$ where all $a_{i,j}$'s are iid, with $\...
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        45 views

        Concentration vs. Anti-Concentration

        Suppose $X_1, \dots, X_n$ are $n$ independent copies of a real random variable $X$. Then, what is the largest value of: $$\alpha_X = \min_{a_1,\dots, a_n \in [1,R]} \Pr\left[\sum_i a_i X_i <1\...
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        15 views

        Distribution Functions in Poisson Process

        We consider definition of Poisson processes that satisfy Condition 0 and 1 according to Billingsley section 23 . How find out the densities of $A_t , B_t , L_t$ defined as in problems section of ...
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        1answer
        81 views

        Total offspring of Poisson multitype branching process

        A normal branching process $Z_n$ initialized with $Z_0=1$ and offspring generated from $Pois(p),p<1,$ has a total progeny / total off spring distribution $$X=\sum_{n=0}^\infty Z_n$$ $X\in \mathbb{...
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        70 views

        Differentiability (Hessian) of $\int \log F$ when $\int \log f$ is differentiable?

        For a specific probability density function $f$ with support on ${\mathbb R}$, which is not differentiable everywhere, I have proven that the Hessian matrix of $$g(\theta) = \int \log f(x;\theta)d H(...
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        Probability of one event happening to two different individuals within a time frame [migrated]

        What is the probability of the same event happening to two (or more) different independent people within a specified time frame? For example, if I wave my hands to my friend, there is a probability ...

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