# 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.

981 questions
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### The norm of isotropic sub-Gaussian random vector may not be sub-Gaussian

Suppose $X$ is a isotropic sub-Gaussian $n$-dimensional random vector (i.e. $EXX^T=I_n$, and for any unit vector $u$,$\|\left<X,u\right>\|_{\psi_2}\le K$). It is said that $\|X\|_2-\sqrt n$ may ...
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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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### 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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### 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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### 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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### 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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### Solution of a 2D Recurrence sequence

Can we solve the following recurrence relation: $$a_{m,n} = 1 + \frac{a_{m,n-1}+a_{m-1,n}}{2}$$ with $a_{0,n}=a_{m,0}=0$? If not, can we get an estimate of the growth of $a_{m,n}?$ I encountered this ...
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### Independence of r.v.'s following a distribution that is the ratio between complex Gaussian and Chi-square r.v.'s

Given the following two R.V.s $$z_1 = \frac{x_1}{|x_1|^2 + |x_2|^2 + \cdots + |x_M|^2}$$ and $$z_2 = \frac{x_2}{|x_1|^2 + |x_2|^2 + \cdots + |x_M|^2}$$ where $x_i \sim \mathcal{CN}(0,a), \forall i$...