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
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2answers
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 ...
3
votes
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\...
-2
votes
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 ...
1
vote
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. ...
2
votes
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 ...
0
votes
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 ...
4
votes
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 $X\sim B(n,p)$.
0
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0answers
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 ...
0
votes
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 ...
2
votes
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 ...
1
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0answers
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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votes
0answers
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
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 ...
4
votes
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 $...
5
votes
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 ...
1
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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 ...
2
votes
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 $\...
-1
votes
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 $a > b$?
0
votes
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 ...
0
votes
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 $1-p$. ...
1
vote
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 ...
-1
votes
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\...
1
vote
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 ...
0
votes
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 ...
2
votes
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 ...
22
votes
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 ...
1
vote
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 ...
1
vote
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 \...
3
votes
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
votes
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 $x_i \sim \mathcal{CN}(0,a), \forall i$...
1
vote
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}(...
2
votes
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
vote
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 ...
0
votes
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 ...
0
votes
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 (...
0
votes
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]- \...
0
votes
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{...
2
votes
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 ...
1
vote
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\...
-1
votes
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 ...
-1
votes
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 ...
0
votes
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 ...
0
votes
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
vote
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 > ...
0
votes
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
votes
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
vote
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
vote
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 ...
