All Questions
Tagged with pr.probability probability-distributions
1,384 questions
1
vote
1
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131
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Large scale analysis of matrix multiplications
Let $\mathbf{A}_{m\times n}$ and $\mathbf{B}_{m\times n}$ be two random i.i.d matrices with zero mean and unit variance. Then, are the following large-scale analysis true (m,n go to infinity with ...
- 1,124
1
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0
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83
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Tracy Widom type results for asymptotic distribution of the $k$-th largest eigenvalue of the sample covariance when $n, p \to \infty$?
Earlier I asked a question: Distribution of the $k$-th largest eigenvalue of in the sample covariance matrix?, but I forgot to mention that I'd like results for asymtotic regime. So, I'm posting here ...
- 1,467
2
votes
1
answer
760
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Show the coordinate distribution has a very large sub-gaussian norm
Consider a random vector X with the coordinate distribution is uniformly distributed in the set $\{\sqrt{n}e_i : i = 1,..., n\}$, where $e_i$ denotes the n-element set of the canonical basis vectors ...
- 23
2
votes
2
answers
854
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Eigenvalue distribution of a random matrix
Is there any closed form distribution formula for the distribution of the eigenvalues of $\mathbf{X}^\mathrm{H}\mathbf{X}$ where the entries of $\mathbf{X}$ are independent Gaussian random variables ...
- 188k
0
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0
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275
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Is there any relation between moments of random matrix and its eigenvalue distribution?
Let $\mathbf{X}$ be a random matrix with independent Gaussian random variable entries with different variances $v_{ij}$. Also define $\mathbf{A}=\mathbf{X}^\mathrm{H}\mathbf{X}$. Is there any relation ...
- 287
1
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1
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104
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Limit of normalized sum of Dirac measures at first $\lfloor p/2\rfloor$ eigenvalues of the sample covariance matrix, with Marcenko-Pastur assumptions?
Let $\lfloor{*}\rfloor$ denotes the nearest integer $\le *$. I'm asking myself the question what's the limit of the part of the empirical spectral distribution corresponding to the first $\lfloor{p/2}...
- 188k
0
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1
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153
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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},...
- 17k
2
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1
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635
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Azuma's Inequality when the conditions hold with high probability?
In Azuma's Inequality, is the statement true when $|X_k - X_{k-1}| < c_k$ almost surely rather than with probability 1? If not, is there another result which gives strong concentration when the ...
1
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1
answer
193
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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 ...
- 63.9k
0
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1
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320
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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 ...
- 7,514
6
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1
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611
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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
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1
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96
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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 ...
- 128k
4
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1
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385
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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 ...
- 128k
1
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0
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105
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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
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2
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267
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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 \{+...
- 54.2k
0
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1
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86
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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 $\...
- 128k
0
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2
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210
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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 ...
- 17.2k
3
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3
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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) = \...
- 4,713
1
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1
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448
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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.
...
- 128k
9
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1
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556
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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
3
votes
1
answer
667
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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 ...
- 63.9k
4
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3
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300
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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 ...
- 128k
2
votes
0
answers
67
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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
1
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1
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117
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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
3
votes
2
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421
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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 ...
- 188k
2
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1
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759
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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 $n \ge 1$ ...
- 128k
2
votes
1
answer
464
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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
1
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2
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190
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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 ...
- 85.6k
2
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2
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185
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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 ...
- 1,338
1
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1
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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 $1$, ...
CommunityBot
- 1
3
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3
answers
483
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$H(p) \le H(q) + KL(p, q)$?
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
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1
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1k
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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 $U_{n:1}\le\cdots\le U_{n:n}$ ...
CommunityBot
- 1
0
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1
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112
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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 ...
- 128k
7
votes
1
answer
627
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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 ...
- 128k
11
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4
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3k
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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:
...
- 2,272
3
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0
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253
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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
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1
answer
3k
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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
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1
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798
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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?
- 364
6
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0
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150
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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 ...
- 63.9k
4
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1
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469
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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
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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 ...
- 128k
2
votes
1
answer
129
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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 ...
- 128k
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 $\mathbb R^2$ ...
- 128k
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,\...
- 128k
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 ...
- 128k
0
votes
1
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295
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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
$$
\mathbb E[X\ | X\le \tau]?
$$
- 3,486
2
votes
0
answers
204
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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
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]$...
- 128k
7
votes
3
answers
3k
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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 ...
- 128k
1
vote
1
answer
303
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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 ...
- 128k