All Questions
Tagged with pr.probability st.statistics
317 questions with no upvoted or accepted answers
1
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101
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calculating how much to oversell given an acceptable risk (statistics)
I have a shared resource with a finite capacity (let's say 100), and I have usage data (2 hours average of samples taken each 20 seconds). I accept a risk of 10% per year to reach the capacity.
...
1
vote
0
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153
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Sampling without replacement: probability for total successes from successes in sample?
Consider drawing $n$ balls from an urn containing $N$ balls, of which $m$ are red. If i know $N$, $m$ and $n$ i can use the hypergeometric distribution to calculate the probability that my sample ...
1
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0
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177
views
Conditioning over Conditional probability? (also: $\phi$-mixing sequences)
For two sub $\sigma-$fields $\mathscr{F}$ and $\mathscr{G}$ of a probability space $(\Omega , \mathscr{A} , P)$ we define $\phi$ mixing as follows:
$$
\phi(\mathscr{F},\mathscr{G}) = \sup \{ |P(G|F) - ...
- 11
1
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0
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336
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Normalized correlation with a constant vector
I am confused how to interpret the result of preforming a normalized correlation with a constant vector. Since you have to divide by the standard devation of both vectors (reference: http://en....
- 171
1
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0
answers
1k
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Distribution of uniform-normed random vector
What is the pdf of $\vec{Y} = \frac{\vec{X} }{\lVert \vec{X} \rVert_\infty}$ with $\vec{X}$ a random vector following a multivariate standard normal distribution (zero-mean $\vec{\mu} = 0$ and ...
- 21
1
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0
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236
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density for Gaussian gram matrices
Let $Z \sim \mathcal{N}(0,\Sigma \otimes I)$ (so the columns of $Z$ are distributed $\mathcal{N}(0, \Sigma)$) and $A = Z'Z.$ Is there a name for the distribution on $A$? Is the density known?
- 922
1
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0
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554
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How to obtain tail bounds for a linear combination of dependent and bounded random variables?
Hi everyone,
Note: This question is a general case and edited version of my previous question ``How to obtain tail bounds for a sum of dependent and bounded random variables?''.
I am looking for ...
- 197
1
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0
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397
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Random Walk vs Branching process
1) Let us consider the set of all $N!$ permutations of the $N$ elements ${1, 2, . . . ,N}$. In the random state, each permutation of these elements occurs
with probability 1/N!. The probability $Pm(N)$...
1
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0
answers
466
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Bounding point-wise maximum of the absolute difference of two convex functions
Let $\Delta: R \times R \rightarrow R_{+}$ be a positive and convex function (convex in, say, both the arguments) called the loss function.
Let $x \in R^d$. Moreover, let $H_1,...,H_r$ be sets of ...
- 11
0
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0
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44
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Large Deviation Principle for an adaptive sampling rule for Multi Armed Bandits
Consider the following adaptive strategy for sampling from a Multi Armed Bandit with $K$ arms:
Split the $T$ rounds into $N (\in \mathbb{N})$ disjoint intervals. Each interval is indexed by $i=1,2,\...
- 9
0
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0
answers
85
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When is a family of distributions "closed" with respect to minimal sufficient statistics?
As in the title, I am interested in understanding how to express the idea that a parametric family of distribution is "closed" with respect to minimal sufficient statistics. Before giving ...
0
votes
1
answer
108
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RMT for modified Wishard matrix $Y'Y$ (where $i$th row of $Y$ is zero if $|x_i^\top u| \le \theta$; else it equals $x_i$)
Let $n$ and $d$ be positive integers tending to infinity such that $d/n \to \phi \in (0,\infty)$. Let $X$ be an $n \times d$ random matrix with iid rows $x_1,\ldots,x_n$ from $N(0, \Sigma)$, where $\...
- 6,853
0
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0
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89
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Stein's Lemma for conditional expectation?
Let $X=(X_1,\ldots,X_d)$ be a standard normal random vector in $\mathbb R^d$, let $m:\mathbb R^d \to \mathbb R$ be a function, and let $E=E_m$ denote the expectation operator conditioned on $m(X) > ...
- 6,853
0
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0
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37
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Compatibility of 2-copulas
An $n$-copula is the joint distribution function of a distribution on $[0,1]^n$ with uniform marginals. A family of 2-copulas $(C_{i,j})_{i<j\leq n}$ is compatible if there exists an $n$-copula $\...
- 677
0
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0
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42
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Limiting value of trace of resolvent matrix involving two independent Wishart random matrices
Let $n_1$, $n_2$, and $d$ be positive integers tending to infinity such that
$$
d/n_k \to \phi_k \in (0,\infty).
$$
Let $X_1 \in \mathbb R^{n_1 \times d}$ and $X_2^{n_2 \times d}$ be independent ...
- 6,853
0
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0
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73
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Asymptotic stochastic ordering for weighted sum of i.i.d. random variables
Are you aware of any literature focusing on the conditions such that for two i.i.d. sequences of discrete r.v.'s $\{X_n\}$ and $\{Y_n\}$,
\begin{equation}
a_1X_1+a_2X_2+\ldots+a_nX_n\geq_1 a_1Y_1+...
- 19
0
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0
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91
views
Some new questions on Rademacher complexity
For $A\subset R^n$,$A=(a_1,a_2,\dots, a_n)$, $\sigma_i$ are Rademacher random variable.
Is $|\mathbb{E}_\sigma \inf_{a\in A}\sum_{i=1}^n\sigma_ia_i| \le |\mathbb{E}_\sigma \sup_{a\in A}\sum_{i=1}^n\...
- 185
0
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0
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82
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High probability bound on number of sparse solutions to Gaussian linear system
Suppose we have a random matrix $A \in \mathbb{R}^{m \times n}$ with all entries i.i.d. from the standard Normal distribution $\mathcal{N}(0, 1)$. Suppose $k$ divides $n$, and let $S \subseteq \mathbb{...
- 43
0
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0
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19
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Analyzing point distributions in Voronoi tessellations from two probability sources $p$ and $\tilde{p}$
Let's suppose I have a probability distribution $p$ and another distribution $\tilde{p}$. Suppose I sample $K$ points from the distribution $p$ which will be my centroids for my Voronoi tessellation. ...
- 101
0
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0
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50
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Approximate CDF of integral using the Berry-Esseen theorem
I'm trying to approximate CDF of the integral $$\frac{1}{T}\int_0^T e^{\sigma W_t+\left(r-\frac{\sigma^2}{2}\right)t}dt,$$
where $W_t$ is the Wiener process, i.e. $W_t\sim N(0,t)$.
For this I use ...
- 49
0
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0
answers
55
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Modeling player interactions in multi-dimensional rating systems
In traditional rating systems (such as Elo), a player's strength is represented by a single scalar value, which is assumed to be consistent across different opponents. However, in some games, the ...
- 1
0
votes
0
answers
140
views
Bounding the difference of weighted-mean estimations
I recently asked a question related to a proof I'm working on as well as a follow-up to that. The detailed counter-examples I received elucidated a lot, but it quickly transpired (i) that I was ...
- 15
0
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1
answer
113
views
Inequality on conditional variance of a vector
I have a random vector $X$ and an event $\mathcal{E}$ such that $\mathbb{P}(\mathcal{E}) = p$. I am trying to show the following inequality :
\begin{equation}
p\mathbb{E}[\|X - \mathbb E [X \vert \...
- 11
0
votes
0
answers
52
views
Classifier-specific lower bounds on the misclassification rate in binary classification
Consider a binary classification problem for $(X,Y)$, and let $\hat{f}$ be a proposed classifier. We wish to bound the misclassification rate $P(\hat{f}(X)\ne Y)$. There are many known lower bounds on ...
- 13
0
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0
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49
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Gaussian white noise model in application
I am interested in applications (to data) of non-parametric statistics, and my question concerned the Gaussian white noise model defined by,
$$
X_{t_1, \ldots, t_d}=f\left(t_1, \ldots, t_d\right) d ...
- 73
0
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0
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60
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Norms of Wigner matrices under power law decay
Suppose $\Sigma=\operatorname{diag}(h)$ where $h=(1^{-p},2^{-p},3^{-p},\ldots,d^{-p})$ and $p> 1$
$X$ is a matrix with $b$ rows sampled independently from $\operatorname{Normal}(0,\Sigma)$
Suppose $...
- 2,252
0
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0
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78
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Kernel density estimation is sub-gaussian
Let $X_1, ..., X_n$ be i.i.d. samples drawn from a pdf $f(x)$ on the real line. The kernel density estimator is defined as follows,
$$\hat{f_n}(x) = \frac{1}{nh}\sum_1^n K(\frac{x-X_k}{h})$$
where $K:\...
- 81
0
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0
answers
29
views
k-means errors for a block Gaussian vector
Consider a standard centered Gaussian vector $(X_1,...,X_n)$ with an approximate block structure, i.e. there is $q$ and a partition of $\{1,...,n\}$ in $q$ classes such that if $i,j$ are in the same ...
- 1,352
0
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0
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103
views
Does such probability distribution exist?
I'm hunting for a probability distribution with the following properties:
The support is $(0,\infty)$.
Denote by $F(x)$ the CDF of this distribution.
If $X_1, X_2,...$ are independent random ...
- 9
0
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0
answers
36
views
How to recalculate the weights for an event that happens multiple times and requires all outcumes to be unique?
I think it's easiest to explain with an example.
I have a weighted probability list
A : 0.15
B : 0.15
C : 0.15
D : 0.1
E : 0.1
F : 0.1
G : 0.1
H : 0.075
I : 0.075
...
0
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0
answers
97
views
Verification of a certain computation of VC dimension
Disclaimer: I'm not very familiar with the concept of VC dimensions and how to manipulate such objects. I'd be grateful if expects on the subject (learning theory, probability), could kindly proof ...
- 6,853
0
votes
0
answers
195
views
Upper-bound for bracketing number in terms of VC-dimension
Let $P$ be a probability distribution on a measurable space $\mathcal X$ (e.g;, some euclidean $\mathbb R^m$) and let $F$ be a class of funciton $f:\mathcal X \to \mathbb R$. Given, $f_1,f_2 \in F$, ...
- 6,853
0
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0
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91
views
Spectrally-weighted Stieltjes transform of random matrix $Z=XX^\top$ in terms of Stieltjes transform of $Z$ and the weighting function
Let $n$ and $d$ positive integers going to infinity such that $d/n \to \gamma \in (0,\infty)$. Let $X$ be a random $n \times d$ iid rows from $N(0,\Sigma)$, where $\Sigma = diag(\lambda_1,\ldots,\...
- 6,853
0
votes
0
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95
views
Empirical estimation of Brenier map from data
Let $f:\mathbb R^d \to \mathbb R$ be a "nice" (say, continuous) function define $A = A_f := \{x \in \mathbb R^d \mid f(x) \ge 0\}$ and $B =B_f:= \{x \in \mathbb R^d \mid f(x) \le 0\}$, and ...
- 6,853
0
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1
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101
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Realizations of alternative configurations
Consider a discrete distribution $P(\mathbf{X},Y)$ with $\mathbf X = \{ X_1, \dotsc, X_N \}$. I use the shorthand notation $p(\mathbf{x}, y)$ for $P(\mathbf{X}=\mathbf{x}, Y=y)$. Consider $P_\text{ind}...
- 189
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0
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202
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$|\frac{1}{n}\sum_{i=1}^n X_i-E(X_1)|=O_P(\frac{1}{\sqrt{n}})$ under $E(|X_1|)<\infty$?
For i.i.d. random variables $X_1,\dots, X_n$ with $E(|X_1|)<\infty$. Does the following equation hold?
$$
\left|\frac{1}{n}\sum_{i=1}^n X_i-E(X_1)\right|=O_P\left(\frac{1}{\sqrt{n}}\right)
$$
I ...
- 193
0
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0
answers
330
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Lower-bound smallest eigenvalue of covariance matrix of $y = f(Ax)$, for $x$ uniform on unit-sphere
Let $A=(a_1,\ldots,a_)$ be a fixed $k \times d$ matrix (with $d$ large), and $x$ be a random vector uniformly distributed on the unit-sphere in $\mathbb R^d$. Let $f:\mathbb R \to \mathbb R$ be a ...
- 6,853
0
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0
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769
views
sub-exponential type upper bound on the Poisson probability
I posted this question on Math Stack Exchange, though I'm not satisfied with the answer I received.
Question:
For a Poisson random variable $Z$ with the parameter $\lambda,\,$ what would be a good ...
- 11
0
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0
answers
171
views
A basic property of maximal correlation
Let $𝑋$ and $𝑌$ be random variables. Then the maximal correlation $\rho_{m}(X;Y)$ is defined as:
$$\rho_{m}(X;Y):=\max_{f,g}\mathbb{E}[f(X)g(Y)],$$
where the maximization is taken over real-valued ...
- 11
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0
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86
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What probability distribution is this?
Thank you in advance for any suggestions or feedback.
I have a discrete 1D probability distribution represented as a vector $\textbf{p}$, $p_i = p(x_i)$.
I am interested in finding the Wasserstein (...
0
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0
answers
80
views
Can we construct a surjective mapping from $\mathbb{R}^{?}$ to this space?
(Note : I'm not sure about the tags, please re-tag this if you think you have the right tag).
I am optimising a certain function over a certain space (that i will describe), and to use non-constraint ...
- 686
0
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0
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173
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The reason why a test is undersized?
Now I have a statistic $T_n$ for testing $H_0 \leftrightarrow H_1$, and I have proved that:
$$n T_n \rightarrow_d \chi_K^2$$
under $H_0$. Then an asymptotic $\chi^2$ test can be used, an asymptotic ...
- 331
0
votes
0
answers
136
views
expectation of the exponential of the inverse of variable with Marchenko–Pastur distribution
This question is related to another answered before
distribution on the inverse Wishart matrix eigenvalues summation
my question is, is their finite expression for the expectation of
\begin{align}
{\...
- 377
0
votes
0
answers
250
views
Concentration (or two sided tail bounds around expectations) of maximum and minimum of $n$ iid, subgaussian random variables
I asked this on MSE, but got no answer, hence asking here now. Help appreciated!
My question is motivated by this question and this question, where the first was aimed for giving a one sided tail ...
- 1,467
0
votes
0
answers
141
views
What is the distribution of the norm of the multivariate $X \sim \mathcal{N}(\mu, \Sigma) \in \mathbb{R}^d?$
Let $X \sim \mathcal{N}(\mu, \Sigma) \in \mathbb{R}^d$ follow a multivariate normal distribution. Then what's the distribution (PDF, CDF etc.) of $X?$ When $\mu = 0, \Sigma = I_d,$ we know that $||X||...
- 1,467
0
votes
1
answer
552
views
Hypothesis testing for not identically distributed random variables conditioned on the outcome of a subset
I encountered the following problem (I give more details of the problem at the end of the post) and I am trying to figure out the best way of performing a null hypothesis testing. I looked for similar ...
- 9
0
votes
0
answers
221
views
Distance between two sample quantiles
Let $X_1,\dots X_n$ be i.i.d. samples from an unknown distribution. We know the distribution has uniformly bounded probability density function $f(x)$. Let $1>\tau_1>\tau_2>0$ be two quantile ...
- 323
0
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0
answers
58
views
Bounds on $\inf_{x,x' \in \mathbb B_X}TV(P+x,Q+x')$, where $P$ and $Q$ are distributions with density on the space $X=(\mathbb R^n,\ell_p)$
Let $n \ge 1$ be an integer, $p \in [1,\infty]$, and $P$ and $Q$ be two (probability) measures on the metric space space $X=(\mathbb R^n,\ell_p)$ which have densities w.r.t the Lebesgue measure on $X$,...
- 6,853
0
votes
0
answers
103
views
Expectation of maximal Wasserstein distance between empirical distribution and a pdf
Let $P$ be a continuous probability distribution on $R^d$, $X$ the random variable $\sim P$, and $
\hat{X}$ be n i.i.d samples drawn according to $P$. We have another variable $\mu \in S^{d-1}$.
Do ...
- 109
0
votes
0
answers
57
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{...
- 1,095