# 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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### 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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72 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\...

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28 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 ...

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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. ...

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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 ...

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**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 ...

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89 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 $X\sim B(n,p)$.

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34 views

### Estimator example cube [closed]

We make subsequent throws of a fake cubic cube for which the probability of falling out six is 1/6 - epsilon, the probability of falling out of one is 1/6 + epsilon and the others eyes drop out with ...

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**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 ...

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63 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 ...

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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 : \...

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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 ...

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**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 ...

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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 $...

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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 ...

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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 ...

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**1**answer

71 views

### How to efficiently sample uniformly from the set of $p$-equipartitions of an $n$-set?

I have a question related to this one. For $n,p \in \mathbb{N}_+$ such that $p\mid n$, let $\mathcal{P}^{\rm eq}$ be the set of all equipartitions of $n$ in $p$ sets; i.e., in sets of equal size $\...

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12 views

### 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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45 views

### 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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75 views

### 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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94 views

### 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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59 views

### On probabilistic extension for Bernstein polynomials

Suppose $X_m\sim p_m(x)$ is a discrete distribution on $[0,1]$ where the value takes multipliers of $\frac{1}{m}$ (e.g., $p_m(x=\frac{k}{m})=\frac{1}{m+1})$. Suppose $p(x)=\lim\limits_{m\rightarrow\...

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61 views

### Probability of two Points being divided by an high-Dimensional Hyperplane

I have two points $x_1,x_2 \in \mathbb S^n $ which are distant $d$ from each other, where $d<<1$.
I also have a vector $v$ sampled uniformly at random from $\mathbb S^n$.
What is the ...

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67 views

### Taylor series expansion of quantile function

Suppose $Y$ and $Z$ two random variables, $\lambda \in \mathbb{R} $.
We note $F^{-1}_{Y}(\alpha)$ the quantile function of the variable $Y$ at the quantile level $\alpha \in (0,1)$.
Do you have any ...

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**1**answer

277 views

### Monotonicity, Convexity, and Smoothness of the KL-Divergence between Two Brownian Motions with Different Initializers

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 ...

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964 views

### 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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49 views

### Approximate $\log \mathbb E_P[\exp(th(x)]$ for a function $h$ which is lipschitz and has finite moments of order 1 and 2 w.r.t $P$

Let $P$ be a probability measure on a space $\mathcal X$ and $h: \mathcal X \rightarrow \mathbb R$ is measurable function with finite moments of order 1 and 2. I'm interested in approximating the ...

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67 views

### Expectation of random variables coincides

Let $Y_1:=(X_i)_{i \in \mathbb Z}$ be a family of random variables that are identically distributed but not necessarily independent.
We can then also define the shifted sequence $Y_2:=(X_{i+1})_{i \...

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163 views

### 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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69 views

### 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 $x_i \sim \mathcal{CN}(0,a), \forall i$...

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66 views

### Correlation between r.v.'s following a distribution that is the ration between complex Gaussian and Chi-square r.v.'s

Given the following two R.V.s
$$z_{1} = \frac{x_{1}}{|x_{1}|^2 + |x_{2}|^2 + ... + |x_{M}|^2}$$
and
$$z_{2} = \frac{x_{2}}{|x_{1}|^2 + |x_{2}|^2 + ... + |x_{M}|^2}$$
where $x_{i} \sim \mathcal{CN}(...

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50 views

### Distribution of weighted sum of non-central chi-squared r.v.'s: log-concave?

Let $n\geq 2$, and $X_1,\dots,X_n$ be independent non-central r.v.'s, where $X_i \sim \chi^2(\delta_i)$; and $w_1,\dots, w_n > 0$.
Letting $$X \stackrel{\rm def}{=} \sum_{i=1}^n w_i X_i$$ is it ...

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54 views

### Expected norm of linear maps

I want to compute the expected norm of a vector-matrix multiplication. I have a vector $x \in \mathbb{R}^n$ with norm one and a matrix $M \in \mathbb{R}^{n \times n}$, whose entries are iid taken from ...

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31 views

### For which vector norms $\|\cdot\|$ and information divergences $D_f$ do we have $|p^Tv-q^Tv| \le \min(D_f(p,q),D_f(q,p))\|v\|$?

By Cauchy-Schwarz, this holds for the total-variation distance, since
$$|p^Tv - q^Tv| \le \|p-q\|_1\|v\|_\infty = 2TV(p,q)\|v\|_\infty,
$$
for every vector $v \in \mathbb R^n$ and probability ...

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37 views

### Marginal Distribution of Partition Matrix

Suppose that $X\sim IW_{p}(n,I_p)$ has an inverse Wishart distribution, which probability density function is
$$f(X\mid n)\propto |X|^{-\frac{n+p+1}{2}}exp\Big(-\frac{1}{2}tr( X^{-1})\Big),~~\qquad (...

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65 views

### Bounding the total variation distance between two measures from a given set

I have a distance on the space of probability measures on $[0,2]$. It is defined as such for two probability measures $\mu_1$ and $\mu_2$ :
$d_p(\mu_1,\mu_2) := \sum_{k=0}^p ( \mathbb{E}[X_1 ^k]- \...

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39 views

### Absolute continuity of probability measures determined by dependence structure

We are on $\mathbb{R}^d$ with Borel $\sigma$-algebra. Let $\mu_1, ..., \mu_d$ be probability measures on $\mathbb{R}$ and $\Pi(\mu_1, \mu_2, ..., \mu_d)$ be the set of probability measures on $\mathbb{...

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**1**answer

56 views

### Mutual information between continuous and discrete variables from numerical data

I am looking for references/measures to estimate the mutual information between a continuous (C) and discrete (D) variable, given a real-world (i.e. finite sample) data set. C is uniformly distributed ...

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**1**answer

62 views

### MGF of a RV that is the ratio between a complex Gaussian and a Chi-squared RVs

Given the following p.d.f., which is the p.d.f. of the real and imaginary parts of a random variable that is the ratio between a complex Gaussian and a Chi-squared RVs:
\begin{equation*}
f_U(u)=\exp\...

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25 views

### IGARCH model property (conditional distribution) when used to model sum of log returns

I already asked this in the quant, crossvalidated, and math SEs, but no help there. I'm not sure many people are familiar with whatever I'm asking, and I tried rewording the question too, but seems ...

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29 views

### conditional distribution multivariate gaussian generalized inverse

given $\begin{bmatrix}X\\Y\end{bmatrix}\sim N\left(\begin{bmatrix}\mu_X\\\mu_Y\end{bmatrix};\begin{bmatrix}\Sigma_{XX}&\Sigma_{XY}\\\Sigma_{YX}&\Sigma_{YY}\end{bmatrix}\right)$ How can I prove ...

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62 views

### How can we show that the total variation distance of $X_s$ and $Y_s$ is bounded by the distance of $(X_t)_{t\ge s}$ and $(Y_t)_{t\ge s}$?

Let $(X_t)_{t\ge0}$ and $(Y_t)_{t\ge0}$ be real-valued time-homogeneous Markov processes with a common transition semigroup $(\kappa_t)_{t\ge0}$. Let $\mathcal L(Z)$ denote the distribution of a ...

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148 views

### How do we introduce a signed finite measure on the space of curves confined into the box $[0,1]^{n}$?

Given $\Omega_{n} = \{\alpha:[0,1]\rightarrow[0,1]^{n}\,|\,\alpha\,\,\text{is smooth}\}$, consider the equivalence relation:
\begin{align*}
& \alpha_{1} \sim \alpha_{2} \Leftrightarrow \int_{0}^{1}...

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99 views

### The minimum of the reciprocals of some Poisson random variables

Let $X_1,\dots,X_k$ denote a collection of independent samples of a Poisson random variable whose mean also happens to be equal to $k$. Does the quantity $$k\boldsymbol{E}\min\left\{ \frac{1}{1+X_{1}}...

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**1**answer

75 views

### Product of independent random variables and tail deconvolution

Suppose $X, Y$ are two independent non-negative random variables. The conditions
(i) $\mathbb{P}(X > t) = \frac{C}{t^p} + o(t^{-p})$
(ii) $\mathbb{P}(Y > t) = o(t^{-q})$ for any $q > ...

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**1**answer

58 views

### How to calculate the probability of 2 events happening in time series under only cdf information?

In time domain $0\rightarrow T$, there are two independent events $A$ and $B$.
$B$ follows Poisson Process with density $\lambda$. It's easy to get $P_B(t)$ which denotes $P_B(N(\tau+t)-N(\tau)\geq 1)...

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844 views

### Differentiating an integral that grows like log asymptotically

Suppose I have a continuous function $f(x)$ that is non-increasing and always stays between $0$ and $1$, and it is known that
$$ \int_0^t f(x) dx = \log t + o(\log t), \qquad t \to \infty.$$
...

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**2**answers

43 views

### Approximate the variance of multiple normal distributions with the same standard deviation

Given a number of normal distributions $N(\mu_1, \sigma^2), N(\mu_2, \sigma^2), ..., N(\mu_n, \sigma^2)$ with fixed variance $\sigma^2$, but not necessary equal means. My question is how to ...

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**0**answers

29 views

### Show a convolution of distributions ε-close to min-entropy k is ε^t-close to min-entropy k

Assume $X_1,...,X_t$ are independent distributions on $\mathbb{Z}_2^n$ s.t. each $X_i$ is $\epsilon$-close to min-entropy $k$; i.e. there exist distributions $Y_1,...,Y_t$ on $\mathbb{Z}_2^n$ s.t:
$$ \...

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**1**answer

148 views

### Integrating nasty gaussian over square root

TLDR: trying to solve,
$$\int_1^\infty \exp\left(-\frac{x^2}{2\omega^2}\right) \frac{1}{\sqrt{ax^2+bx-1}}dx$$
After doing some reading and looking at some other questions 1, 2 (and even going through ...