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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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92 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 ...
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2answers
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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0answers
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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0answers
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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0answers
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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1answer
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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3answers
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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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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1answer
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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1answer
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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1answer
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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1answer
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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1answer
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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0answers
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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1answer
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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0answers
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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1answer
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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0answers
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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1answer
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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1answer
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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1answer
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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0answers
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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1answer
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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6answers
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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1answer
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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1answer
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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4answers
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 ...
2
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1answer
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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1answer
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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0answers
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 ...
1
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1answer
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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0answers
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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0answers
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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0answers
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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0answers
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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1answer
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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1answer
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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0answers
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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0answers
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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0answers
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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0answers
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}...
4
votes
2answers
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}}...
1
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1answer
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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1answer
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)...
8
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2answers
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 $$ \int_0^t f(x) dx = \log t + o(\log t), \qquad t \to \infty.$$ ...
1
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2answers
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 ...
1
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0answers
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: $$ \...
2
votes
1answer
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 ...