All Questions
Tagged with pr.probability probability-distributions
1,386 questions
7
votes
2
answers
1k
views
Conditional Expectation for $\sigma$-finite measures
Someone knows of some definition or reference of how to define conditional expectation for a measure space with $\sigma$-finite measure.
I think it should be as follows:
Let $(X,\mathcal{B},\nu)$ ...
- 37.8k
4
votes
0
answers
228
views
Minimization over a convex function of equal vs unequal success probabilities of Bernoulli random variables
Let $U_1,U_2,\ldots,U_n$ be $n\geq 2$ mutually independent Bernoulli random variables. There are two cases of interest:
$1.$ The random variables $U_1,U_2,\ldots,U_n$ are identically distributed;
$...
CommunityBot
- 1
7
votes
0
answers
774
views
Calculate the expectation of the maximum of averaged random walks
Let $X_1, X_2, \ldots$ be iid random variables with bounded second moment. The question is to calculate the exact value of $$\mathbb{E} \max_{1 \le j < \infty} \frac{X_1 + \cdots + X_j}{j}.$$
Is ...
- 773
8
votes
3
answers
935
views
Question about Wasserstein metric
Let $\mu$ and $\nu$ be two probability measures on $\mathbb R^n$ with finite first moment. Denote by $d:=W_1(\mu,\nu)$, where $W_1(\cdot,\cdot)$ stands for the Wasserstein distance of order $1$.
My ...
- 128k
3
votes
0
answers
125
views
Probability distributions with all positive cumulants
Is there a term for a distribution with all cumulants positive (or nonnegative)?
- 3,069
2
votes
0
answers
77
views
"Optimal" local limit theorems for densities vanishing at zero
Consider a nonnegative stable distribution with a density that vanishes at zero, such as
$$f(t)=\frac{e^{-1/2t}}{\sqrt{2\pi t^3}},\qquad t\geq0.$$
Suppose (for simplicity) that we have i.i.d copies $(...
- 3,085
3
votes
1
answer
1k
views
Normal approximation to the pointwise/Hadamard/Schur product of two multivariate Gaussian/normal random variables
Let $X \sim \mathcal{N}\left( {{\mu _x},\sigma _x^2} \right)$ and $Y \sim \mathcal{N}\left( {{\mu _y},\sigma _y^2} \right)$ be two univariate and independent Gaussian/normal random variables and let $...
- 1,026
6
votes
2
answers
312
views
maximal distance of nearby iid unifrom random variables
Question: Let $X_1, \ldots ,X_n$ be $n$ iid uniformly distributed random variables, i.e., $X_j \sim \mathcal{U}(0,1)$ for each $j=1,\ldots ,n$. What is the PDF of the maximal distance between to ...
- 128k
3
votes
2
answers
329
views
Log-concavity of the maximum of gaussians
Let $Z_1,\ldots, Z_n$ be independent standard gaussian random variables. Is it true that $X=\max\{Z_1,\ldots,Z_n\}$ has a log-concave distribution function?
- 128k
3
votes
1
answer
368
views
Minimising the f-divergence to a conditional probability constraint
Let $P$ be a probability distribution and let $A$ and $B$ be some events, and suppose that we want to minimise an $f$-divergence between $P$ and the set of all distributions $Q$ that satisfy that ...
- 128k
4
votes
1
answer
288
views
Radon-Nikodym derivative of the group action on the Furstenberg-Poisson boundary of lamplighter groups
Let $G_d$ be the Lamplighter group $G_d = \mathbb{Z}^d \wr \mathbb{Z}_2 $ and $\Gamma =\{(\bar{\eta},\tilde{0}),(\bar{0},\tilde{e_1}), \cdots,(\bar{0},\tilde{e_d})\}$ be the generator set of $G_d$ (...
- 2,090
2
votes
1
answer
3k
views
Discrete Maximum Entropy Distribution with given mean
For a given mean $\mu$, what is the entropy maximizing probability distribution on the nonnegative integers?
Different sources indicated either the geometric or the Poisson distribution for this. As ...
4
votes
0
answers
823
views
Total Variation distance of polynomials of Bernoulli R.V.s
Let $X_i, Y_i$ be i.i.d Bernoulli $0/1$ random variables with
$\mathbb{E}[X_i] = p$ and $\mathbb{E}[Y_i] = q$.
Let
\begin{align*}
X &= X_1 X_2 + Χ_2 Χ_3 + \ldots +X_{n-2} X_{n-1}+ X_{n-1} X_n\\...
- 175
-1
votes
1
answer
76
views
transformation of two measures on different space
Let $\{e_1,e_2,...,e_n\}=E $ be the standard bases of $\mathbb{R}^n$, and $U\subset\mathbb{R}^n$ be a linear space generated by $\{e_1,e_2,...,e_n\}$.
Let $\Sigma_U$ be the smallest $\sigma-$ field ...
- 3,085
5
votes
1
answer
231
views
CLT for Bernoulli RV with negative correlation
Suppose $X_1,X_2,...$ are Bernoulli random variables with $P(X_i=1)=p_i$ and $X_i$ have negative correlation. Is there a CLT in this case, i.e. does $\frac{Z_n-(\Sigma^n_{i=1}p_i)}{\sqrt{n}}$ converge ...
1
vote
1
answer
1k
views
Independence of stochastic processes [closed]
Suppose that $(X_t)$ and $(Y_t)$ are stochastic processes defined on the same probability space whose sample paths belong to some Hilbert space $K$ (or more generally some function space). We may view ...
- 331
6
votes
1
answer
365
views
uniquely determining a distribution using moments
Let $A$ be a parametric family of probability distributions that include all distributions in the form of $\phi(X)$ where $X\sim\mathcal{N}(0,\mathbf{I})$ is jointly Gaussian and $\phi:\mathbb{R}^d\to ...
- 24.8k
7
votes
1
answer
466
views
Martingale version of Bernstein-type inequality for (slightly) heavy-tailed distributions?
It is known that for sub-exponentially distributed martingale difference sequence, the following Bernstein-type inequality holds:
$$
ℙ\left(\left|
\sum_{i=1}^N a_i X_i
\right| \ge t \right)
\le
2\...
- 4,713
9
votes
2
answers
8k
views
What is the expected maximum out of a sample (size N) from a geometric distribution?
Lets say I have a geometric distribution (of the number X of Bernoulli trials needed to get a success) with parameter p (success probability of a trial).
Assume I ...
- 101
2
votes
1
answer
599
views
Cantelli's inequality: the original source
Does anyone know where and when Cantelli's inequality was originally published? Strangely enough, I have not been able to find this information online.
- 188k
2
votes
0
answers
799
views
Calculating Wasserstein's distance between an empirical distribution and a combination of normal distributions
Context of the problem
Let $\xi$ be a random variable (with real value) with support $\Xi=\mathbb{R}$ and $\xi_{1},\ldots,\xi_{N}$ be a sample of $\xi$. We consider the empirical probability
$$\...
- 159
4
votes
0
answers
867
views
For what sub-$\sigma$-algebra are these two measures equivalent?
In two statistics papers (linked inline below) I have come across two definitions of certain probability measures. I conjecture that for particular choices of the construction that they are ...
- 1,228
0
votes
0
answers
102
views
Lower bound for the probability that $X=\omega\left(\mathbb E[X]\right)$ for $X\sim Bin(n,p)$
Let $X\sim Bin(n,p)$ be a binomial variable and let $\delta\in (0,1)$.
I'm looking for a lower bound of the form $\Pr[X > f(\delta)] \ge \delta$.
Specifically, if $\delta,p=o(1)$ are not ...
- 618
4
votes
1
answer
322
views
Asymptotic form of pdf of Escape Time of arithmetic fBm
I am trying to apply the Girsanov formula and Doobs optional sampling theorem to obtain an asymptotic form of first passage density of an fbm process with drift, but the answer i am getting seems ...
- 6,045
3
votes
2
answers
566
views
Moments of Matrix Gamma distribution
Matrix gamma distribution (defined for example in http://en.wikipedia.org/wiki/Matrix_gamma_distribution) is one way to generalize Wishart distribution. In our course work that distribution was used ...
- 2,600
3
votes
1
answer
461
views
Bounding the "spikiness" of a probability distribution
Are there any well-known conditions that guarantee that a probability distribution isn't too "spiky"?
I ask this question because I am interested in the families of probability distributions $f(x)$ ...
- 8,071
3
votes
1
answer
164
views
Large deviations for integrands
I am a physicist caught in the following situation:
I have two probability measures $\mathbb{P}_1$ and $\mathbb{P}_2$ and have to deal with the following integral where $X_i$ are random iid:
$$\int_{...
- 7,514
1
vote
1
answer
357
views
Does CLT hold for joint distribution of two dependent binomial variables?
Let $S_n$ and $T_m$ be two binomial variables satisfying $S_n\sim B(n,\frac12)$ and $T_m\sim B(m,\frac12)$. Define $\tilde{S}_n=\frac{2S_n-n}{\sqrt{n}}$ and define $\tilde{T}_m$ similarly. For any ...
CommunityBot
- 1
8
votes
1
answer
198
views
Tail bound of a distribution
Let $X_1, X_2, \ldots, X_n, Y_1, Y_2, \ldots, Y_n$ be independent binary random variables each being $1$ with probability $\frac{1}{k}$.
Let $Z = X_1(Y_1 + \cdots + Y_k) + X_2(Y_2 + \cdots + Y_{k+1})...
CommunityBot
- 1
4
votes
2
answers
1k
views
Given the joint probability distributions of $X$ and $Y$ for $Y = R\,X+C$, find the probability distributions of $R$ and $C$
Let $R$, $C$, and $X$ be independent random variables defined on $(0,\infty)$ and
$$Y=\underbrace{R\, X}_{Z}+C.$$
We are given the joint probability distribution of $X$ and $Y$, $P_{XY}(x,y)$ and ...
- 8,071
0
votes
0
answers
141
views
Probability of getting through with a phone-call
Alice is quite popular. She gets called on her cell-phone in a Poisson$(\lambda)$ manner. She answers her calls when possible, and ignores them when in the middle of conversation. Since you know her ...
- 396
8
votes
1
answer
1k
views
Uniqueness of a Solution for a Convex Optimization Problem
I have the following convex optimization problem:
$$\begin{array}{ll} \text{maximize}_{{f,g}} & \displaystyle\int_{\Omega} g^u{f}^{1-u}\mathrm{d}\mu\\ \text{subject to} & \displaystyle\int_{\...
CommunityBot
- 1
2
votes
0
answers
123
views
Modified Wigner semicircle law
The Wigner semicircle law states that for a random GOE-matrix $M^N \in \mathbb{R}^{N \times N}$ in the $N \rightarrow \infty$ limit for any $f \in C^b(\mathbb{R})$
$$\lim_{N \rightarrow \infty}\frac{...
CommunityBot
- 1
4
votes
0
answers
5k
views
E[ | X - Y | ] where X and Y are independent Poisson random variable
What is the expected value of the absolute difference of two independent Poisson variables?
$$E[ |X - Y| ]$$
Seems like an easy question but I haven't found an easy solution.
I've split the double ...
- 303
1
vote
1
answer
151
views
Noncentral matrix beta distributions of type I and II
In Gupta & Nagar's book Matrix variate distributions, the noncentral Beta type I(B) distribution with parameters $a$, $b$ and noncentrality parameter $\Theta$ is defined by $U={(S_1+S_2)}^{-\...
- 2,560
2
votes
1
answer
266
views
A question about finite free convolution
For any square matrix $Y$ let $\chi_x(Y) = det(xI -Y)$ denote its characteristic polynomial.
Say $A$ and $B$ are two $n-$dimensional symmetric matrices with constant row sums $a$ and $b$. Lets ...
- 128k
9
votes
1
answer
2k
views
Uniform sampling from general simplex with a twist
This is part of a question I had asked elsewhere, and then some of the links redirected me to CS stack exchange.
Given $0\leq a_1\leq\dots\leq a_D\leq1$ (all strictly positive), I want to draw points ...
CommunityBot
- 1
7
votes
4
answers
2k
views
Singular distributions: Applications and Instances
Singular distributions are special mathematical objects. They have an interesting property of not having a density function, defined on a set with Lebesgue measure zero. Cantor distribution is the ...
- 27.5k
2
votes
1
answer
161
views
Linking error probability based on total variation
Consider probability measure $\mu_{XY}$ defined on $\mathbb{R}^d \times \{1,2,3\}$, and sub-probability measures $\mu_1$, $\mu_2$, and $\mu_3$ as $\mu_1(A):=P(X\in A, Y=0)$ and $\mu_2(A):=P(X\in A, Y=...
- 482
4
votes
1
answer
155
views
How to simulate the fractional noncentral Wishart distribution?
I already asked this question on math.stackexchange but got no answer.
For a non-integer number of degrees of freedom $\nu > p-1$, one can simulate the central Wishart distribution $W_p(\nu, \...
- 188k
3
votes
2
answers
732
views
Concentration inequality for sum of iid random variables that involve KL distance
Conider $X \in \mathbb{R}^d$ and $Y \in \{0,1\}$, and a joint distribution $p_{XY}(x,y)$, and a set of $N$ i.i.d. samples $\{(X_i,Y_i)\}_{i=1}^{N}$. Define $p_{X0} = p_{XY}(x,0)$ and $p_{X1} = p_{XY}(...
- 128k
4
votes
1
answer
229
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Product of estimates of mean values - Concentration of measure inequality
Let $X_{1},...,X_{d} \in \{-1,1\}^d$ be random variables, with $E[X_j]=\mu_j$. Having $n$ i.i.d. samples $x^{(i)}_1,x^{(i)}_2,....,x^{(i)}_d$, $i=1,...,n $, let $\hat{\mu}_{j}=\frac{1}{n}\sum^{n}_{i=1}...
- 7,249
3
votes
1
answer
752
views
Wasserstein convergence of conditional measures
Suppose $W_r(\mu_n,\mu)\to0$, where $\mu_n$ and $\mu$ are discrete probability measures on some metric space $\Omega$, and that all measures have the same number of atoms $d$ (but not the same atoms):
...
- 710
1
vote
0
answers
55
views
Central Limit Like theorem for the distribution of F-statistics on all possible partitions?
I'd be happy for simply a reference or even search terms as I feel like this has to be known*.
Suppose we have a known probability distribution $X$ and a fixed integer $n$. I am interested in the ...
- 51
3
votes
1
answer
1k
views
Measurable functions in product space
I am reading a book by Billingsley (convergence of probability measures) and he makes a footnote on page 27 which I am struggling to understand. I'll explain the setup below.
Suppose $(X_n,Y_n)$ are ...
- 11.3k
1
vote
0
answers
66
views
Matrix variate t-distribution and product of Beta distributions
This is a reference request for the following result. Let $X$ be a random matrix following the matrix variate $t$-distribution $T_{p,m}(\nu, M, U, V)$ (as defined in Wikipedia). Then
$$
\frac{\det(U)}{...
- 2,560
6
votes
2
answers
257
views
Minimum probability that two Gaussian random variables are small
Let $X,Y$ be two centered Gaussian random variables each with variance at most $1$. Note that we do not assume independence. I would like to minimize
$$\mathbb{P}(|X|\leq 1, |Y|\leq 1).$$
Is it true ...
- 28k
8
votes
2
answers
4k
views
Lower bounds on Kullback-Leibler divergence
This was originally a question on Cross Validated.
Are there any (nontrivial) lower bounds on the Kullback-Leibler divergence $KL(f\Vert g)$ between two measures / densities?
Informally, I am ...
- 128k
3
votes
1
answer
113
views
maximum likelihood estimation of X is better than that of f(X)?
Consider a binary variable $C$ with $\Pr(C=0)=\Pr(C=1)=0.5$. Consider a random vector $X \in \mathbb{R}^d$, characterized by $C$, with PDF $p_m(x)$, $m\in\{0,1\}$. Define a maximum likelihood (ML) ...
- 128k
3
votes
1
answer
73
views
Maximizing the $\alpha$-moment of a distributution
Given $\alpha$ and constant $\mu$,
$$\begin{array}{ll} \text{maximize} & \displaystyle\int_0^\infty p(x)x^\alpha \,\mathrm d x\\ \text{subject to} & \displaystyle\int_0^\infty p(x)\,\mathrm d ...
- 128k