All Questions
Tagged with pr.probability probability-distributions
1,384 questions
7
votes
0
answers
759
views
Product of two random Gaussian matrices - orthant probability
Let $X \in \mathbb{R}^{m \times n}$ and $Y \in \mathbb{R}^{n \times k} $ be two independent Gaussian random matrices, i.e., with entries independently sampled from $\mathcal{N}(0,1)$ (a normal ...
- 968
1
vote
0
answers
110
views
Approximating or calculating the mutual information of certain binary random vectors
In my studies of applied probability I have recently met the following problem on which I need help:
We consider two binary random (column) vectors $ X,Y \in \{0,1\}^d $ where the mutual ...
- 620
4
votes
1
answer
141
views
Fuzzy layers in graphs and neural networks
I wonder if the following statistical description of the layer architecture of finite graphs has been considered before and where I can find some references (especially under which name).
Consider a ...
CommunityBot
- 1
3
votes
2
answers
1k
views
Is there a notion of Convergence in PDF/PMF
I am learning about local limit theorems. The following example is probably why we don't have a "convergence in density/pmf."
Ex: $X_1,X_2,\ldots$ is a sequence of independent RVs with mean $a$ and ...
- 34.7k
1
vote
0
answers
58
views
A possible extension of the information bottleneck principle with added equality constraint on the conditional probability
This question is related to research on Tishby's information bottleneck principle as seen here, the problem at hand is inherently an optimization problem as seen directly on page 7 section 3.2, my ...
- 620
0
votes
1
answer
905
views
Parameter estimation distribution for hypergeometric distribution
Let the hypergeometric distribution is given by $h(k\mid N;M;n)$, where
$k$ is the number of observed successes,
$N$ is the population size,
$M$ is the number of success states in the population and
$...
1
vote
0
answers
77
views
Random variables with values in binary operations or in topologies of a certain set $X$
I wonder if the following situations have already been considered by mathematicians :
Random variables with values in a set of binary operations endowed
with a certain topology (or just with a $\...
- 129
4
votes
2
answers
314
views
Convexity of truncated expectation
Let $k, n$ be two positive integers with $k \leq n$, and let $P = \{ (x_1, \dots, x_n) \in [0, 1]^n : \sum_i x_i = k \}$.
Given $x = (x_1, x_2, \dots, x_n) \in P$, let $X_i$ be the random variable ...
CommunityBot
- 1
-1
votes
2
answers
512
views
Deriving the joint distribution of multivariate normal transformed into Bernoulli
Given a covariance matrix $\sum_{ij}$ and a mean vector $\mu$ I have sampled $N$ multivariate normal vectors $Z = (z_1,...z_n)$ My goal is to create a vector of Bernoulli random variables $Y = (y_1,......
- 37.7k
4
votes
0
answers
95
views
Approximating martingales given marginal distributions
Let $(\mu_0,\mu_1)$ be a vector of probability measures on $\mathbb R$ that are of finite first moment, i.e.
$$\int_{\mathbb{R}}|x|\mu_i(dx)~<~+\infty \mbox{ for } i=0,1$$
and increasing in ...
- 1,835
2
votes
0
answers
63
views
Sensitivity of a function against its random arguments
Let $g:R^{n+m} \to R$ be a deterministic function of some independent random variables $x_1,\ldots,x_n$ with distributions $f_{x_1}(x),\ldots,f_{x_n}(x)$ and some deterministic variables $z_1,\ldots,...
- 482
4
votes
1
answer
463
views
Variance and expectation of timed-change squared Bessel process
Let $X_t$ be a squared Bessel process satisfying the SDE:
$$
dX_t=\left(1-\frac{\beta}{(1-\beta)(1-\rho^2)} \right) dt +2\sqrt{X_t}dW^{(1)}_t
$$
and $v_t=v_0e^{-\alpha^2 t/2+\alpha W^{(2)}_t}$ be a ...
- 6,045
2
votes
1
answer
228
views
Clustering in Euclidean space
Let $P_1,\cdots,P_m$ be points in $\mathbb{R}^n$ such that the distance between any $P_i,P_j$ is less than 1.
Let $p_{i,j}$ be a probability distribution on pairs of these points, that is for $1\leq ...
- 23.7k
3
votes
1
answer
649
views
Bound on the total variation distance for multiple samples $d_{tv}(P^n,Q^n)$
Given two discrete distributions $P$ and $Q$, with computable total variation distance $d_{TV}(P,Q)=||P - Q||_1$, is there a precise bound for $d_{TV}(P^n,Q^n)=||P^n - Q^n||_1$, as need to estimate ...
CommunityBot
- 1
2
votes
0
answers
417
views
What is the Blumenthal-Getoor index of Student's distributions?
For infinitely divisible random variables, Blumenthal and Getoor introduced in [1] an index that allow to study for instance the local Hölder regularity of Lévy processes.
For a symmetric infinitely ...
- 2,306
2
votes
0
answers
80
views
Maximal and second max of chi-squared (normal squared) distribution
Suppose $X_i$ are i.i.d.~$\log \chi^2$ where $\chi\sim N(0,1)$ distribution, in which case
$$
F(x) = \mathbb{P}\left( \log \chi^2 \le x \right)
= 2\Phi(e^{x/2}) - 1,
$$
and
$$
f(x) = e^{x/2}\phi(e^{x/...
CommunityBot
- 1
2
votes
1
answer
295
views
The asymptotics of $\int_{-\infty}^{\infty} \phi(x) {\Phi(\frac{x}{a})}^{qa} dx $ for normal distribution using saddle point approximation
In my probability and numerical analysis research I have come across the following predicament:
If we have a standard normal random variable X with CDF $ \Phi $, and PDF $ \phi $ I am interested in ...
CommunityBot
- 1
3
votes
1
answer
961
views
Expected value: Stakes and coin tosses [closed]
Consider the following game, lets call it $G$. You flip a fair coin $100$ times, but instead of having a fixed stake, you can freely choose the stake for each flip, just before the flip.
You start ...
- 349
0
votes
1
answer
369
views
How to derive this change of measure identity in multi-armed bandits?
We have two Bernoulli distributions with success probabilities $\mu_1$ and $\mu_2$. We sample $n$ times from distribution 1 and the sequence we get is $X_1, \ldots, X_n$. Let
$
\hat{kl}_s = \sum_{t=...
- 14.2k
2
votes
1
answer
401
views
Reference on Probability theory on functional spaces (in special Hilbert spaces)
Currently, I am working on some sort of stochastic optimization problems defined over function spaces.
I am familiar with standard probability theory (R. Durrett, ''Probability: Theory and Examples")...
CommunityBot
- 1
1
vote
1
answer
510
views
Total variation distance between multinomial laws
Can someone help me with the following problem:
Let $P_n$ and $Q_n$ two multinomial laws with parameters $(p,n)$ and $(q,n)$, where $p$ and $q$ are two probability measures on some measurable space ...
- 1
8
votes
1
answer
552
views
Frobenius norm of the principal submatrix of a uniformly distributed random orthonormal matrix
Suppose that we have a uniformly distributed $d\times d$ random orthonormal matrix $\mathbf{X}$. Here "uniform" is defined in the sense of Haar measure, i.e., the distribution does not change up to ...
- 1,127
3
votes
0
answers
699
views
How does Jensen Shannon divergence and KL divergence correlate?
I am wondering if there is way to derive the correlation between Jensen Shannon divergence and KL divergence for two distributions: P and Q, in order to show that if JSD(P,Q) decreases, KLD(P,Q) ...
CommunityBot
- 1
3
votes
0
answers
83
views
Selecting the best choice for the smallest single appearing natural number
Assume we have $n$ players (each knows the number of competitors). Each has to chose a natural number and the player that has selected the smallest number, that appears uniquely, is going to win (if ...
- 749
5
votes
2
answers
559
views
A measure of how "spread out" a probability measure is
Consider a random variable $X$ whose variance is large. As a contrast to Markov's or Chebyshev's inequality, both of which measure the concentration of a probability distribution, is there a measure ...
- 128k
1
vote
0
answers
448
views
Largest possible variance for log-concave distributions on a bounded interval
Let $f$ be the density of a log-concave probability distribution on the interval $[0,1]$ (with respect to Lebesgue measure). To be concrete, suppose that $f(x) = \exp( - \varphi(x))$, for some convex ...
CommunityBot
- 1
2
votes
1
answer
376
views
Independent Decomposition of a Random Vector
Let $\vec{x} \in \mathbb{R}^n$ be a fixed vector and suppose that we are given an isotropic random vector $\vec{a} = (a_1, \dots, a_n)^T$ in $\mathbb{R}^n$ (i.e., the covariance matrix of $\vec{a}$ is ...
- 53
1
vote
1
answer
313
views
Finding the joint distribution from Poisson conditionals
Suppose that for two discrete random variables $X_1$ and $X_2$, we know their conditional distributions. Namely
$$X_1~|~X_2 = x_2 \sim \mathrm{Poisson}(\lambda_1 + ax_2),$$
$$X_2~|~X_1 = x_1 \sim \...
- 54.2k
0
votes
0
answers
96
views
How to get some information about a random variable if we know very little about its distribution
Suppose that $X, Y$ are random variables,both from a probability space to $(0,\infty]$, such that X and $1+Y$ have the same distribution, $Y=Z_1$ with probability equals half, otherwise $Y= \frac{...
- 228
2
votes
1
answer
207
views
Expectation of Truncated Bivariate Gaussian Random Variables
Suppose $Z , \epsilon \sim N(0, 1)$ are independent Gaussian random variables. Let $a \ll 1$ be a small positive number. Let $W = aZ + \epsilon$. It can be show that
\begin{align}
\mathbb{E} [ W^2 (Z^...
- 61.9k
0
votes
1
answer
65
views
Limit of iterative addition of a mean-preserving spread
Suppose I iteratively add a given mean-preserving spread to a random variable. In the limit, will exactly half the mass be above $0$?
Formally: Let $X$ be a random variable, and let $\varepsilon_1,\...
- 29.8k
1
vote
1
answer
147
views
Proving that an integral related to order statistics is increasing in a certain parameter
Let $f$ and $F$ denote, respectively, the pdf and cdf of a probability distribution on $\mathbb R$. Take any natural $n\ge3$ and any real $a$ and $c$ such that $a\le c$.
Does it always follow that
$$...
- 128k
2
votes
1
answer
271
views
How to compute bounding coefficients for McDiarmid's inequality?
I am trying to understand the proof in Sec. A2 of Gretton et al.. To make the question self-contained, I summarize below the key ingredients. At the end of the post, I state my question.
Given a ...
7
votes
2
answers
606
views
Uniform Concentration Bounds on Weighted Sum of i.i.d. Bernoulli Random Variables
Let $\delta_1,...,\delta_n$ be $n$ independent identically distributed Bernoulli random variables with $\mathbb{P}(\delta_1=1)=p$. We consider a set $\Omega = \{\mathbf{a}:=(a_1,...,a_n)~|~a_i\in [0,c/...
- 24.2k
6
votes
2
answers
735
views
Negative probabilities - what are two ordinary pgfs that correspond to the gf of a half-coin?
In Half of a Coin: Negative Probabilities, author considers pgf of a fair coin represented by random variable, $X = 1_H$:
$$G_X(z) = E[z^X] = \sum_{x=0,1} z^xP(X=x) = (z^0)(1/2) + (z^1)(1/2) = \frac{...
- 247
2
votes
1
answer
150
views
Probability of collision of some family of hash functions
Given $x$ and $y$ in $\mathbb{R}$, and let $\mathcal{H} = \{ h \mid \mathbb{R} \to \mathbb{N} \}$ be a family of hash functions where $ h(x) = \left\lfloor x + \sum^C_{i=1} U_i \right\rfloor$ for some ...
- 128k
4
votes
1
answer
681
views
Tail bound for product of normal distribution
Let $U, V$ be two standard normal random variables with covariance $cov(U,V) = \beta \in [0,1)$. Let $W = UV$ be the product of two RV's, and $W_1, W_2, \ldots, W_n$ be n i.i.d copies of $W$, what's ...
- 128k
2
votes
0
answers
278
views
Radon-Nikodym for continuous time processes
Likelihood theory for statistical inference concerning stochastic processes in continuous time are well used. How ever i've found no real literature concerning the fundamentals.
What is know from ...
- 257
1
vote
0
answers
171
views
Bounding a distribution using moments
Suppose $X$ is a non-negative random variable with bounded image. I was wondering if anybody knew of any results that could answer a question of the following type: Suppose the $n$-th moment satisfies ...
- 171
5
votes
0
answers
149
views
Distribution of Random Knots from Braids
Let $R_{2n,l}$ be a random braid word of length $l$, where each letter is chosen uniformly from the braid generators of $B_{2n}$, $\{\sigma_1,\ldots,\sigma_{2n-1},\sigma_1^{-1},\ldots,\sigma_{2n-1}^{-...
- 71
6
votes
1
answer
385
views
Functional limit theorem under random change of time
FINAL EDIT: There is one main question left: According to the answer, we have choosen $\theta=1$ , where we could choose $0<\theta<\infty$ as we like. His this sufficient, if we regarde the ...
- 61
1
vote
1
answer
64
views
Conditioned sum of n Poissons versus unconditioned Poissons
Let $\theta >1$ and take independent random variables $Z_k \sim \text{Poisson}(\theta/k)$ for $1 \leq k \leq n$ and let $Z_k^*$ have marginals like the $Z_k$ conditioned on $\sum_1^n k Z_k = n$:
$$\...
- 457
2
votes
1
answer
216
views
Ask for a special function related to the error function
I am wondering whether anyone knows the following integration has a named special function or a reference
$$
F_{a,b}(z) :=\frac{2}{\sqrt{\pi}} \int_0^z \text{erf}(a+b y)\: e^{-y^2} \text{d}y
$$
for ...
- 1,649
1
vote
1
answer
219
views
connection between the statistical properties of a scalar field and its columns
Consider a scalar field $s:[0,1]^3 \to \mathbb{R}$ and its "column" field
\begin{equation}
c: [0,1]^2 \to \mathbb{R}: (x,y) \mapsto \int_0^1 s(x,y,z) \,\mathrm{d}z.
\end{equation}.
What can be said ...
CommunityBot
- 1
2
votes
0
answers
71
views
Existence of probability distribution satisfying upper/lower bounds on events
Suppose we have a finite sample space $S$ and some events $A_1, \dots, A_k \subseteq S$. We would like to put a probability distribution on $S$ so that no element has probability greater than a ...
- 21
40
votes
1
answer
5k
views
When should we expect Tracy-Widom?
The Tracy-Widom law describes, among other things, the fluctuations of maximal eigenvalues of many random large matrix models. Because of its universal character, it obtained his position on the ...
- 141
1
vote
0
answers
90
views
The role of absolute continuity in stochastic ordering defined over sets of probability distributions
This question is about a claim given in this paper (page 261, the remark), but without any proof.
It simply says that if two sets of probability distributions, $\mathscr{P}_0$ and $\mathscr{P}_1$ (...
- 359
7
votes
1
answer
342
views
Upper Bound for the Difference of Even Probability and Odd Probability in Hypergeometric Distribution
Let $X$ be a random variable following the hypergeometric distribution with parameters $N,K,n$, where
\begin{equation}
Pr(X=k) = \frac{\binom{K}{k}\binom{N-K}{n-k}}{\binom{N}{n}}.
\end{equation}
To ...
- 3,574
2
votes
1
answer
191
views
Median of a uniform multinomial variable
Let $k\in\mathbb N^+$ be a positive integer.
Consider a set of i.i.d. random variables $X_1,X_2,\ldots, X_n$, each of which is distributed uniformly over $\{1,2,\ldots,2k+1\}$.
For $i\in \{1,2,\...
CommunityBot
- 1
5
votes
2
answers
200
views
Expected number of changes in the sign of a rolling sum of independent normal variables
Imagine we define $Y(t+n)=
X(t+1)+.....+X(t+n)$ where $X(i)$ is an independent normal (i.e. everyday we remove the starting observation and we add a new one). We have $n$ consecutive observations of $...
- 28k