# 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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### Order statistics of correlated bivariate Gaussian

Suppose $(X_1,Y_1),...,(X_n,Y_n)$ are i.i.d. bivariate Gaussian with mean zero. Each coordinate has variance 1 and correlation between coordinates is $\rho\in[-1,1]$. I'm interested in the following ...
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### On the sum of uniform independent random variables

Let $X_1,...,X_n$ be independent uniform random variables in [0,1] and assume $c>1/2$. Is it true that $$\mathbb{P}\left[\sum_{i=1}^n X_i \leq n \cdot c\right]$$ is increasing with respect to $n$? ...
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### Looking for a generalization of Binomial distribution and it's properties

In my research (coming from computer science), I have encountered a family of discrete probability distributions that seems to be some sort of generalization of the binomial distribution. A ...
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### Concentration inequality for quadratic form of Gaussian variables with non-idempotent matrix

Given $y \sim N(0,\sigma^2 I)$, and $M$ that is a symmetric matrix (not necessarily idempotent) what is the distribution of ${y^T M y}$? is there a high probability bound on $|{y^T M y}|$? Most ...
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### What is the expected minimum total matching distance between two partitions of identically and independently distributed points?

Suppose a square $[0,1]\times [0,1]$ in which $N$ vehicles $V_i$ and $N$ riders $R_i$ are distributed identically and independently (say, uniform distribution), a bipartite matching (or a permutation, ...
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### Probability number A greater number B [migrated]

Given $a \in \{1,2,...,250\}$ and $b \in \{0,1,...,1000\}$ $a$ and $b$ are chosen randomly, how does one calculate the probability of $a > b$?
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### Fundamental difference between Poisson Point Process and Binomial Point Process

What is the fundamental difference between Poisson Point Process and Binomial Point Process? I am evaluating a solution in a Binomial Point Process setup. If I want to evaluate that in a Poisson ...
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### Couplings on empirical distributions

For a problem I've been working on, I'm thinking about couplings between true and empirical distrubutions. I have two datasets $S$ and $T$ with underlying measures $\mu_S,\,\mu_T$. And then I have ...
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### Resultant probability distribution when taking the cosine of gaussian distributed variable

I am trying to do a measurement uncertainty calculation. I have a gaussian distributed phase angle (theta) with a mean of 0 and standard deviation of 16.6666 micro radians. The variance is the ...
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### Maximizing the expectation of a polynomial function of iid random variables

Let $f \colon \mathbb R^N \to \mathbb R$ be a smooth function. Let $\mu$ be a probability measure on $[0,1]$ and $X_1, \ldots , X_N$ be i.i.d. random variables on $\mathbb R$. Question 1. What is ...
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### Concentration properties of inner-products in high-dimension

Let $S^K$ be the unit sphere embedded in $R^{K+1}$. $v \in S^K$ is randomly chosen from a uniform distribution over $S^K$. $A \subseteq S^K$ is a $d$-dimensional sub-manifold ($d \leq K$). Think of ...
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### Expectation of balls to reach capacity C with two bins of unequal probability

Let there be two bins $b_1$ and $b_2$. We denote the number of balls in $b_1$ as $X_1$ and $b_2$ as $X_2$. The probability a particular ball lands in $b_1$ is given by $p$, and $b_2$ given by $1-p$. ...
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### CLT for Martingales

I posted this question originally in math stack exchange, but I got no answer. (https://math.stackexchange.com/questions/2604591/clt-for-martingales) In wikipedia, there is a version of a CLT for ...
We consider the two distributions $$p_t = p_0 * N(0, tI),\quad q_t = q_0 * N(0, t I),$$ where $*$ denotes the convolution between two densities, while $p_0$ and $q_0$ have the same mean and ...