# 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**

vote

**2**answers

93 views

### 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 ...

**3**

votes

**2**answers

73 views

### Is there a name for “splitting a probability distribution into independent components”?

Suppose I have a random variable $\theta=(\theta_1,\dotsc,\theta_n)$; where the $\theta_i$ might have pairwise correlations. I decompose it into $\theta=\hat\theta(\phi_1,\dotsc,\phi_k)$, where $\hat\...

**-2**

votes

**0**answers

29 views

### Distribution of a post-selected random variable with power-law distribution

Background
Assume $X \sim \mathcal D$ is a random variable, distributed according to some distribution $\mathcal D$. Then postselection with respect to some set $A$ is defined as the conditional ...

**1**

vote

**0**answers

23 views

### Reference Request: Total Variation Between Dependent and Independent Bernoulli Processes

Let $X$ be a random variable taking values in $\{0,1\}^n$ with the following distribution. For each coordinate $i$, we have $p_i = P(X_i = 1) = c/\sqrt n$, where $c$ is a (very small) constant. ...

**2**

votes

**0**answers

50 views

### 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 ...

**0**

votes

**1**answer

68 views

### Strictly Proper Scoring Rules and f-Divergences

Let $S$ be a scoring rule for probability functions. Define
$EXP_{S}(Q|P) = \sum \limits_{w} P(w)S(Q, w)$.
Say that $S$ is striclty proper if and only if $P$ always minimises $EXP_{S}(Q|P)$ as a ...

**4**

votes

**3**answers

91 views

### A clean upper bound for the expectation of a function of a binomial random variable

I wonder if there is a closed-form, or clean upper bound of this quantity: $\mathbb{E}[|X/n-p|]$, where $X\sim B(n,p)$.

**0**

votes

**1**answer

80 views

### How to find a special random variable? [closed]

Suppose random variables $X_1$ and $X_2$ have the same distribution under P, $Y_1$ is an arbitrary random variable,let $Z_1:=X_1+Y_1$.Can we find a r.v. $Y_2$ which has same distribution as $Y_1$,such ...

**2**

votes

**1**answer

64 views

### Asymptotic bound on the total variation distance between a standard multivariate normal and a simple mixture

Let $P = N(\vec{0}, I^d)$ be a standard multivariate Gaussian distribution in $d$ dimensions. Let $Q$ be distributed the same as $P$, except that samples from $Q$ have one of their coordinates, chosen ...

**1**

vote

**0**answers

47 views

### About a class of expectations

Consider being given a $n-$dimensional random vector with a distribution ${\cal D}$, vectors $a \in \mathbb{R}^k$, $\{ b_i \in \mathbb{R}^n \}_{i=1}^k$ and non-linear Lipschitz functions, $f_1,f_2 : \...

**-2**

votes

**0**answers

30 views

### 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 ...

**1**

vote

**1**answer

52 views

### 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 ...

**4**

votes

**1**answer

51 views

### Rate of decay in the multivariate Central Limit Theorem

The celebrated Berry-Esseen inequality tells us that the rate of convergence in the univariate CLT is of magnitude $\frac{1}{\sqrt{n}}$ for sums $S_n=X_1+\cdots+X_n$ of independent random variables $...

**5**

votes

**1**answer

75 views

### Variance of sum of $m$ dependent random variables

I originally posted this question in Mathstackexchange, but since I got no answer I'm posting it also here.
Let $X_1,X_2,...$ be a sequence of identically distributed and $m$-dependent random ...

**1**

vote

**0**answers

92 views

### Different balls in bins: What is the probability distribution of the sum of the minimum of the two types of balls over all bins?

Assume that there are $N$ different bins and two different kinds of balls, $R$ red balls and $W$ white balls.
The red balls and the white balls are randomly distributed across the bins (that is, for ...