All Questions
Tagged with pr.probability probability-distributions
1,384 questions
0
votes
1
answer
153
views
Probability distribution of random products of elements of a generating set of a finite non-abelian group
Let $G$ be a finite non-abelian group, and consider a choice of $N$ distinct elements $g_{0},g_{1},\ldots,g_{N-1}\in G$ that generate $G$. Now, let $t$ be an arbitrary positive integer, and let $d_{1},...
- 1,284
0
votes
1
answer
320
views
Marcenko Pastur law when the dimensionality/sample size ratio $p/n \to 0, \infty$? Lack of resources?
Let $X: \Omega \to \mathbb{R}^{p \times n}$ be a random matrix so that each entry $X_{ij}$ is a random variable with $\mathbb{E}X_{ij}=0, \mathbb{E}X_{ij}^2=\sigma^2$
I was wondering what would ...
- 1,467
6
votes
1
answer
611
views
The "Chaos Game" as a particular series of i.i.d. random variables
Fix a parameter $\alpha\in(0,1)$ and take an i.i.d. sequence $X_0,X_1,\ldots$ of $\mathbb{R}^n$ valued random variables. Construct the limiting random variable
$X_\infty = (1-\alpha)\sum_{k=0}^\infty ...
- 646
4
votes
1
answer
96
views
Identifications between different phase spaces
I've discovered Adam's lecture notes on statistical mechanics after posting my first question about Minlo's discussion on continuous Gibbs measures. Adam's lecture notes are really good, but there is ...
- 1,305
1
vote
0
answers
105
views
Measure on a set and its value on $\emptyset$
After my first post here, I have one more doubt which is bothering me. It concerns Minlos's book Introduction to mathematical statistical physics again. To fix the notation, we have $\Lambda \subset \...
- 1,305
4
votes
2
answers
267
views
Grand-canonical Gibbs measure for continuous systems
Let's consider a bounded (maybe compact) set $\Lambda \subset \mathbb{R}^{d}$ with particles interacting on it. Suppose, for each $N \in \mathbb{N}$, $U_{N}: (\mathbb{R}^{d})^{N} \to \mathbb{R}\cup \{+...
- 3,515
4
votes
1
answer
385
views
Lower-bound for $\Pr[X \geq m]$ subject to $E[X]>m$ where $X$ is a binomial random variable
Given an integer number $m>0$ and a real number $\alpha\in [1, 2]$, I am interested in finding a lower-bound for $\Pr[X\geq m]$ subject to $X \sim \text{Binomial}(n, m\alpha/n)$.
For large values ...
- 189
0
votes
1
answer
86
views
Renormalization group map on hierarchical models
I have already addressed this problem on my previous question but I still have trouble understanding Brydges' RG maps on his lecture notes, so I'll try to elaborate my question a little better.
Let $\...
- 3,515
2
votes
1
answer
161
views
Expected value of global functions in renormalization group
This is related to my previous question. I'm having some problems understanding the local to global program discussed in Brydge's lecture notes. We are assuming $C=C_{1}+\cdots+C_{N}$ is a covariance ...
- 3,515
3
votes
3
answers
219
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When do $\phi^2$ and $\phi’^2$ have the same expectation under a Gaussian random variable?
I am looking for a function $\phi(x)$ such that
$\mathbb{E}_{x\sim\mathcal{N}(0,1)}[\phi(x)^2] = \mathbb{E}_{x\sim\mathcal{N}(0,1)}[\phi'(x)^2]$.
Obvious solutions are $\phi(x) = x$ and $\phi(x) = \...
- 175
0
votes
2
answers
210
views
Limited sum for whole sum approximation
Let $d_n, n\in\{1,2,\cdots,N\}$ be $N$ realizations drawn independent and identically from uniform distribution on $(0,L)$ where $L=\gamma\sqrt{N}$ with constant $\gamma$. Suppose that we need to ...
- 287
1
vote
1
answer
448
views
Law of large numbers for random Dirac measures
Suppose $\{X_1,...X_n\}:\Omega \to \mathbb{R}^p$ be i.i.d. random vectors with common probability law/measure $p$, i.e. $Prob(X_i^{-1}(E))=p(E) \forall E \subset \mathbb{R}^p $ Borel measurable.
...
- 1,467
9
votes
1
answer
556
views
A non-recursive, explicit formula for the Fabius function
The Fabius function $F\colon\mathbb R\to[-1,1]$ may be defined as the unique solution of the functional integral equation
$F(x)=\int_0^{2x}F(t)\,dt$ for all real $x$ such that $F(1)=1$.
The recent ...
- 128k
1
vote
1
answer
193
views
Random matrix properties
Let $\mathbf{H}_{N,K}$ be a random matrix whose entries are i.i.d complex Gaussian random variables with variance $1$. Then, we know from the law of large number that if $N,K\rightarrow\infty$, we ...
- 287
3
votes
1
answer
667
views
Characteristic function and moments
Let $X\in L^1(\Omega)$ and $\phi_X$ the corresponding characteristic function.
We know that: $\phi_X$ is $n$ times differentiable (at $u=0$) iff $\mathbb{E}[X^n]<\infty$. (This depends a bit on ...
- 255
2
votes
0
answers
67
views
Less regular version of the Gaussian free field
One can define (continuous) Gaussian free field as follows: one can consider some orthonormal basis $(\psi_k)_{k=1}^{\infty}$ in the Sobolev space $H^1(\Omega)$ (here $\Omega \subset \mathbb{R}^d$) ...
- 9,330
4
votes
3
answers
300
views
Reconstructing probability distribution with high probability
Sample $m$ times from unknown probability distribution $p=(p_1,p_2,\cdots,p_n)$, we can construct a probability distribution $q=(q_1.q_2,\cdots,q_n)$.
How large $m$ should be to achieve that the ...
- 1,503
2
votes
1
answer
759
views
History of the name "subexponential distribution" in probability
In probability theory, the term subexponential distribution has historically been used for a distribution whose CDF $F(x)$ satisfies the relation
$$
n(1-F(x)) \sim 1 - F^{*n}(x)
$$ for any $n \ge 1$ ...
- 1,124
1
vote
1
answer
117
views
Proximity in terms of characteristic functions for $n$-dimensional distributions
Let $X\in \mathbb{R}^n$ and $Y\in \mathbb{R}^n$ be random variables with characteristic functions $\phi_X(t)$ and $\phi_Y(t)$, respectively.
Suppose that
\begin{align}
\sup_{t \in \mathbb{R}^n} \...
- 671
2
votes
1
answer
464
views
Lower-bound for $E[\min(X, k)]$ where $X$ is sum of Bernoulli random variables with $E[X]$ being a linear function of $k$
Given a real number $\alpha \in [0.5, 1.5]$, an integer number $k>1$, and a set of independent Bernoulli random variables $x_1, \dots, x_n$, I am interested to find a lower-bound for $F(\alpha, k)= ...
- 189
3
votes
3
answers
483
views
$H(p) \le H(q) + KL(p, q)$?
Let $H(p) = \sum_i p_i\log\frac{1}{p_i}$ be the entropy of $p$
and $KL(p, q) = \sum_i p_i\log\frac{p_i}{q_i}$ be the KL divergence between $p$ and $q$. Does it hold that $H(p) \le H(q) + KL(p, q)$?
...
- 143
2
votes
2
answers
185
views
Independence depth of linearly dependent random variables
Suppose, $\Xi$ is a collection of random variables. We call $\Xi$ $k$-independent, iff any $k$ distinct elements of $\Xi$ are mutually independent. For example, $2$-independence is pairwise ...
- 2,618
0
votes
1
answer
112
views
PDF of $z = \exp(j\varphi)$, where $\varphi \sim \mathcal{U}[-a, +a]$ [closed]
How can I find the PDF of $z = \exp(j\varphi)$, where $\varphi \sim \mathcal{U}[-a, +a]$, $i.e.$, a uniformly distributed r.v.?
My difficulty here is that it involves complex numbers and I don't know ...
7
votes
1
answer
627
views
Do there exist three pairwise independent random variables, such that their sum is zero?
Do there exist such three non-constant pairwise independent random variables $X, Y, Z$ such that $X + Y + Z = 0$?
I managed only to prove the following two facts:
If such $X, Y, Z$ exist, they are ...
- 2,618
3
votes
0
answers
253
views
Metric ($f$-divergence) on space of probability measures that satisfies pythagorean theorem
Let $E$ be a polish space, $\mathcal{P}(E)$ the Borel probability measures on $E$ with the topology of weak convergence and $\mathcal{Q} \subset \mathcal{P}(E)$ a convex and compact set.
First, the ...
- 1,095
1
vote
1
answer
3k
views
Tail bound regime for Binomial distribution in concentration paper
In paper 'Concentration Inequalities and Martingale Inequalities:A Survey' gives the following inequality:
My question is whether the inequality holds in regime $\lambda$ being $o(\sqrt n)$ (say $\...
- 1,826
1
vote
1
answer
798
views
Which distributions of $X$ and $Y$ yield a Gaussian $Z=XY$?
Let $Z=XY$ where $X$, $Y$ are random variables with support of non-trivial measure. For what distributions of $X$ and $Y$ can $Z$ be guaranteed to be Gaussian?
- 409
3
votes
2
answers
421
views
PDF of $ | \sum_{k=1}^{n}{|h_k||g_k|\exp\left( j \theta_k \right)} |^2$ for small values of $n$ and $Q$?
Given the following function of random variables
$$f = \left|\sum_{k=1}^{n}{|h_k||g_k|\exp\left( j \theta_k \right)} \right|^2,$$
where $h_1, \cdots, h_n$ and $g_1, \cdots, h_n$ are i.i.d. random ...
1
vote
2
answers
190
views
PDF of $g = \frac{1}{n} \sum_{k=1}^{n}{|h_k|\exp\left( j \theta_k \right)}$?
Given the following function of random variables
$$g = \frac{1}{n} \sum_{k=1}^{n}{|h_k|\exp\left( j \theta_k \right)},$$
where $h_1, \cdots, h_n$ are i.i.d. random variables following the complex ...
1
vote
1
answer
55
views
Distribution limit of a jump process
Divide the interval $[0,1]$ in $n$ subintervals with length $\frac{1}{n}$.
The $n$ subintervals are numerated from $1$ to $n$.
We have a particle that, after an exponential time of parameter $1$, ...
- 139
6
votes
0
answers
150
views
Delayed Pólya's urn process
The standard Pólya's urn process can be stated as follows:
You have an urn with red and green balls. At any time unit you choose one ball at random, note the colour, and give the ball back. At the ...
- 396
4
votes
1
answer
469
views
Probability of achieving the maximum among absolute value of Gaussians
Yesterday the following question was asked by user sigmatau:
I'm interested in the following question:
given $n$ i.i.d. random variables $X_i \sim \mathcal{N}(0,\sigma^2_1), i=1,\ldots,n$ ...
- 128k
1
vote
1
answer
635
views
Convergence in probability of Cesaro means
Suppose that $(X_n)_{n\geq 1}$ is a sequence of (non-negative) random variables on a probability space ($\Omega, \mathcal{A}, P)$ such that $X_n = o_P(n^{-\beta})$ for some $\beta \in (0,1)$.
Does it ...
- 301
2
votes
1
answer
1k
views
Order statistics on the spacings between order statistics for the uniform distribution
For any natural $n$, let $U_1,\dots,U_n$ be independent identically distributed
random variables each uniformly distributed on the interval $[0,1]$. As usual, let $U_{n:1}\le\cdots\le U_{n:n}$ ...
- 128k
4
votes
1
answer
478
views
Order statistic - Rate of convergence of a p-quantile to the expectation
Fix some $k\in\mathbb N$ and some probability $p\in[0,1]$. Denote with $F_n$ the cdf of the k-th highest oder statistic (i.e. the distribution of the k-th highest draw) of $n$ draws from a uniform ...
- 43
5
votes
0
answers
797
views
How many balls should we throw into $m$ bins so that at least $k$ bins get at least $r$ balls, with probability $1-\delta$?
Let $m,k,r\in\mathbb N$ and $\delta\in(0,1)$, such that $k\le m$.
Suppose that we throw balls uniformly and independently into $m$ bins.
I am looking for an upper bound $N_{m,k,r,\delta}$ on the ...
- 51
5
votes
1
answer
392
views
comparing Gaussian to order statistic of Gaussian
I would like to compute the probability of
$$\mathbb{P}[Y > \max(X_i)], Y\sim N(0, 1), X_i \sim N(0, \sigma_i)$$
All the random variables have zero mean, but the variances are different.
My ...
- 151
3
votes
3
answers
410
views
Statistical moments of $\frac X{X + Y}$ when $X$ and $Y$ are two independent random variables with a Beta distribution
I'm trying to find the moments (or the pdf but I'm less confident there's a closed form) of $\frac X{X + Y}$ where $X$ and $Y$ are two independent random variables with a Beta distribution. There's a ...
- 133
1
vote
1
answer
82
views
Expectation value of multilinear forms over independent Gaussian vectors
Let $A$ be a symmetric multilinear form on $\left(\mathbb{R}^d\right)^{\otimes n}\times \left(\mathbb{R}^d\right)^{\otimes n}$ and consider the random variable:
\begin{align*}
X=A(g_1,\ldots,g_n,g_1,\...
- 111
1
vote
1
answer
191
views
Hitting time estimates
In a number of different contexts, I have wanted to estimate hitting times for a monotonic process $(T_n)$ taking values in the reals (or sometimes a process $(T_n,X_n)$ taking values in $\mathbb R^2$ ...
- 23.2k
2
votes
1
answer
129
views
Maximizing entropy of summation of unknown distributions
Let the random variable $Y = X_1+X_2$, where $X_1$ follows an unknown distribution and $Y$ has finite variance.
Assuming as measurement of normality the entropy, is it correct to support that the ...
0
votes
1
answer
295
views
Are there known bounds on the expectation of the truncated Beta distribution?
Let $X\sim beta(\alpha,\beta)$ be a random variable and let $\tau\in(0,1)$.
Are there any known closed-form bounds (I'm specifically interested in lower bounds) on
$$
\mathbb E[X\ | X\le \tau]?
$$
- 618
4
votes
1
answer
124
views
The behavior of a uniform order statistic near zero
Let $X_{(k)}$ be the $k$th order statistic out of $n$ uniform $[0,1]$ random variables. Let $q$ be the location of the $p$ quantile of $X_{(k)}$, i.e. $\Pr[X_{(k)}\leq q] = p$. For small $p$, Is it ...
- 43
2
votes
0
answers
204
views
Do there exist iid random variables $X$, $Y$ with countable support such that $X + Y$ and $X Y$ are also distributed with the same parameterisation?
This is perhaps a well-known result and I'd appreciate a reference if that's the case. Let $X$, $Y$ be iid random variables with support $S \subset \mathbb{Z}$ and let
$$P(X = x) = f(x, \theta)$$
...
- 89
11
votes
4
answers
3k
views
If the sum of two independent random variables is discrete uniform on $\{a, \dots,a + n\}$, what do we know about $X$ and $Y$?
Basically I want to know whether the sum being discrete uniform effectively forces the two component random variables to also be uniform on their respective domains.
To be a bit more precise:
...
- 121
2
votes
1
answer
90
views
A probability inequality: $p+(1-p)E[v|v\geq a] \geq E[v|v \geq p+(1-p)a]$
There is a random variable $v \sim F(\cdot)$ with support $[0,1]$. For a parameter $p \in (0,1)$ and $a \in (0,1)$. Define $A$ and $B$ as the following:
$$A=p+(1-p)E[v|v\geq a], B=E[v|v \geq p+(1-p)a]$...
- 121
7
votes
3
answers
3k
views
expected value of squared infinity norm of vector of iid gaussians
Given a random vector
\begin{equation}
x=(x_1, \ldots, x_n)
\end{equation}
with independent and identically distributed entries $x_i \sim \mathcal{N}(0,\sigma^2)$, I would like to find a lower ...
- 237
1
vote
1
answer
303
views
Minimum of Pareto Random Variables given Harmonic Mean
I have the following problem. Assume I have $n$ independent Pareto random variables $X_1,...,X_n$, with the CDF of $X_i$ being $Pr(X_i \leq x_i) = F(x_i) = 1 - (\frac{b_i}{x_i})^{\alpha_i}$. For ...
- 371
0
votes
1
answer
339
views
Expectations, double integrals and Jensen's inequality
$\def\anonfunc#1{#1(\cdot)}$Consider two random variables distributed $v\sim \anonfunc G$ and
$c \sim \anonfunc F$ with pdfs $\anonfunc g$ and $\anonfunc f$. Let the supports of $c$ and
$v$ be $[x,y]$....
- 13
1
vote
1
answer
158
views
Computing probability of ultimate absorption in B&D processes
Consider a B&D process with infinitely many states, State 0 absorbing. Probability of ultimate absorption when starting in State $n$ (denoted $a_n$) is computed by solving $$a_n=\frac{\lambda_na_{...
- 419