All Questions
Tagged with pr.probability probability-distributions
1,384 questions
0
votes
1
answer
207
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Copulas and marginals thereof
Hello everyone,
I recently became aware of the existence of the copula concept.
So, I have been reading a few things about copulas lately, but
I cannot seem to find information on the following ...
- 1,902
8
votes
2
answers
14k
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Sum of Squares of Normal distributions
Given $X_i \sim \mathcal{N}(\mu_i,\sigma_i^2)$, for $i = 1,\dots,n$. How does one find the distribution of $D = \sum_{i=1}^n X_i^2$? In the case that all the standard deviations are the same (i.e. $\...
- 96.4k
3
votes
2
answers
334
views
Scale random variables in a way they have equal probabilities of being minimal
I have several positive random variables $x_i,\ i=1,...,N$ taken from different unknown distributions (these distributions can be closely approximated by log-normal if needed). I can sample these ...
- 32.4k
4
votes
0
answers
221
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probabilistic terminology for polynomials with positive coefficients
Given a polynomial $P(x) = p_0 + p_1 x + p_2 x^2 + ... + p_n x^n$ with non-negative coefficients, is there a standard name for (the function of $p_1,...,p_n$ equal to) the variance of an integer-...
- 19.7k
0
votes
0
answers
165
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Joint Probability that contains a variable and its Fourier Transform
Given the vector $\mathbf{d}$, where $\mathbf{d}\in\mathbb{C}^{N\times 1}$, we have two variables
$X = \mid\mathrm{F}[d]\mid^2,\quad\quad X\ge 0$
$Y = a+b (\mathrm{d}^H\mathrm{d})\quad Y\ge 0$
...
- 447
0
votes
1
answer
577
views
Expectation of little o in probablity [closed]
If I have $Z=o_p(1)$ where $o_p$ is the little-o in probability. I'm interested in find some properties about $E(Z)$.
My first idea was
$E(Z)=E(Z (1_{Z>\varepsilon} + 1_{Z\leq\varepsilon}) ) \...
CommunityBot
- 1
13
votes
1
answer
815
views
2/3 power law in the plane
I've recently come across a particular problem whose solution turns out to be a probability distribution given by $f(x) = \alpha \|x\|^{-2/3}$ in the unit disk in $\mathbb{R}^2$ and zero elsewhere (I ...
CommunityBot
- 1
3
votes
0
answers
323
views
Is this probability distribution known in the literature?
In some work I was doing I derived a probability distribution that I do not recognize. Is it a known distribution?
$\Pr(X\le x)=\exp\left[-\frac{1}{2}\left(\frac{1}{2}x-\sqrt{1+\frac{x^{2}}{4}}\right)...
- 31
8
votes
1
answer
2k
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Eigenvalue distributions of finite dimensional Wishart matrices
I am trying to obtain the eigenvalue distribution of a finite dimensional Wishart matrix. Let $A_{n\times n}\sim\mathbb{W}(\Sigma_{n\times n},m)$ where $\mathbb{W}(\Sigma_{n\times n},m)$ denotes the ...
CommunityBot
- 1
1
vote
1
answer
3k
views
expected value of inner products of iid standard normal vectors
Hello,
I wish to calculate (or upper bound) expectations of the form $E[\langle x,y \rangle^2]$, where $x$ and $y$ are i.i.d standard gaussian vectors of length n. Are there any exponential type ...
- 96.4k
0
votes
1
answer
377
views
Robust entropy-like measure for analyzing uncertainity
I'm looking for a measure to analysis the uncertainty observed in a set of variables (with multivariate Gaussian distribution). So, I've tried conventional Shanon entropy (differential entropy) which ...
CommunityBot
- 1
3
votes
0
answers
171
views
Iterated Kumaraswamy distributions
The Kumaraswamy distribution has cdf $F(x;a,b) = 1-(1-x^a)^b$.
Does anyone know any formulas or properties relating to iterations of this on itself, meaning
$$ F_i(x;a,b) = 1-(1-F_{i-1}^a)^b$$
If ...
- 233
4
votes
2
answers
295
views
Distribution of the biggest gap
Randomly select $n$ numbers from the universe $\{1,2\dots,m\}$ without replacement, and sort the numbers in ascending order.
We can get a list of number $\{(a_1,a_2,\dots,a_n\)}$, and then we can ...
CommunityBot
- 1
8
votes
2
answers
1k
views
Order statistics (e.g., minimum) of infinite collection of chi-square variates?
Hi everyone,
This is my first time here, so please let me know if I can clarify my question in any way (incl. formatting, tags, etc.). (And hopefully I can edit later!) I tried to find references, ...
CommunityBot
- 1
5
votes
0
answers
1k
views
Compute the expected value of the next step of a sorted random walk
Here's what I'm thinking about. If you have a random walk (move +1 or -1 at each step) of some fixed length, then if you're at the maximum of the walk, the next step you take is -1 with probability 1. ...
6
votes
1
answer
1k
views
Probability distributions: The maximum of a pair of iid draws, where the minimum is an order statistic of other minimums?
General question: What is the distribution for the maximum of 2 independent draws from cdf F(x), when we know that the minimum of those same two draws is the kth order statistic of the minimum of n ...
CommunityBot
- 1
-1
votes
1
answer
696
views
Can singular measures be viewed as vanishing distributions? (Answer No!)
Hello,
Here is my original question: let $\mu$ be a singular measure with respect to the Lebesgue's measure on $R$. Is it true that $\int \psi \mu(d x)=0$ for any test function $\psi\in C_c^\infty(R)$...
CommunityBot
- 1
1
vote
1
answer
220
views
Estimating the Distribution of a Very Large Population of Known Size and Unknown Variance
I would like to estimate the distribution of a very large population of known size but unknown mean and variance. I cannot assume anything about the underlying distribution. The values of observations ...
- 196
1
vote
1
answer
321
views
"Bridging" uniform and "mass" distributions
Foreword. The original formulation of this problem was inaccurate; chamomille and Didier Piau came up with a simple example which would not solve the problem in its accurate formulation. Sorry for my ...
- 5,721
10
votes
2
answers
913
views
Random Trigonometric Polynomial
Let $t_{1},t_{2},\ldots, t_{n}$ be i.i.d. real Gaussian random variables of zero mean and variance one. Let $a_{1},a_{2},\ldots, a_{n}$ be positive and fixed real numbers and define the random ...
- 3,626
2
votes
0
answers
292
views
Seeking the normalizing constant (or any references) for a distribution over a subset of positive definite martrices
I'm interested in a probability distribution over the set of positive definite matrices with unit diagonal elements. That is, and $X$ such that:
$X \in S^{n+}, \forall_{i}X_{ii} = 1$ where $S^{n+}$ ...
- 379
10
votes
1
answer
1k
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Talagrand's concentration inequality with limited independence
Is there a version of Talagrand's concentration inequality known when the variables have limited independence. More precisely, Let $F:\mathbb{R}^n \rightarrow \mathbb{R}$ be a $1$-Lipschitz convex ...
- 8,512
5
votes
2
answers
3k
views
Probability of return at step $n$ of a Random walk to its starting vertex
Hi,
given a discrete simple Random walk on a symmetric graph, what is known about the probability of the random walker to return to a starting site at step $n$? Specifically, I am interested in the ...
- 15.4k
3
votes
1
answer
412
views
Sparse representation of a distribution with independent and correlated variables
Here's what I'm trying to do:
Imagine a probability distribution over $\mathbf{R}^2$, $P(x,y)$. I can approximate $P(x,y)$ with set of $N$ points $\{(x,y)_i\}$ drawn from $P$. By approximate, I mean ...
- 1,902
4
votes
2
answers
2k
views
finding numbers at k hamming distance
Guys,
I have N < 2^n randomly generated n-bit numbers stored in a file the lookup for which is expensive. Given a number Y, I have to search for a number in the file that is at most k hamming dist....
- 4,462
19
votes
0
answers
988
views
On random Dirichlet distributions
Fix a dimension $d\ge2$.
Let $Q_d$ denote the positive quadrant of $\mathbb{R}^d$, that is, $Q_d$ is the set of points $\mathbf{x}=(x_i)_i$ in $\mathbb{R}^d$ such that $x_i>0$ for every $i$.
For ...
- 5,721
2
votes
2
answers
419
views
when does inner product with fixed vectors determine joint distribution?
Given a random vector $(X_1,X_2)$. If $aX_1 + bX_2$ is Gaussian for all pairs $a,b$, then $(X_1,X_2)$ is jointly normal. More generally, is the following statement true?
If $aX_1 + bX_2$ has the same ...
- 11.4k
-1
votes
1
answer
1k
views
Approximating expectation [closed]
if we are given a finite number N of points drawn from a probability distribution, expectation can be approximated as a finite sum over these points:
E[f]=(1/N)(summation of f(x) over these N points).
...
- 4,952
2
votes
2
answers
959
views
Exist closed forms of the distribution of return time in markov chains?
Hi, I am interested in the distribution of return times in simple random walks on finite graphs.
Let $G$ be a connected finite graph with, with two independent random walks. If both random walks ...
- 23
3
votes
3
answers
2k
views
Statistics of a simple Markov chain
Imagine a two-state Markov chain which hops between the states $\pm 1$ with probability $p<1/2$, so that the autocorrelation function after $k$ steps is
$\rho_k = (2p-1)^k$
If I take an ...
- 2,135
0
votes
2
answers
294
views
Relationship between these two probability mass functions.
If I have two different discrete distributions of random variables X and Y, such that their probability mass functions are related as follows:
$P(X=x_i) = \lambda\frac{P (Y=x_i)}{x_i} $
what can ...
CommunityBot
- 1
5
votes
1
answer
349
views
How to calculate the probability of N normal variable being in increasing order?
Suppose we have $n$ normal variable $X_1,X_2,\dots,X_n$, with corresponding mean $\mu_1,\dots,\mu_n$ and sd $\sigma_1,\dots,\sigma_n$. What is the probability of $X_1 < X_2 < \dots < X_n$, i....
- 377
2
votes
1
answer
2k
views
Density function for a multivariate Bernoulli-like distribution
I'm looking for a distribution to model a vector of $k$ binary random variables, $X_1, \ldots, X_k$. Suppose I have observed that $\sum_i X_i = n$. In this case I do not want to treat them as ...
- 24.8k
0
votes
1
answer
2k
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kalman filter: understanding the mathematical part
i am currently reading the Probabilistic robotics book where the filters are discussed.
Such filters as kalman filter or particle filters.
Now I can understand one thing while reading about the ...
- 6,870