# 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.

1,064 questions

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### 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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### 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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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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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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### 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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### 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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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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### 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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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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### 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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### 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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### 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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### 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 ...