All Questions
Tagged with pr.probability probability-distributions
1,384 questions
4
votes
2
answers
315
views
Sampling from a recursively defined distribution
I'd like to know if there are techniques for sampling from a recursively defined probability distribution, assuming that solving the recursion for a formula for the distribution is too difficult.
As ...
- 41
1
vote
1
answer
368
views
Product of probability densities of the form x^{-t} exp (-ax)
I have two probability distributions $p(x) = N_1 x^{-\tau} \exp(-\frac{x}{x_0})$ and $p(y) = N_2 y^{-\kappa} \exp(-\frac{y}{y_0})$. $N_1$ and $N_2$ are just normalization constants and $x>0$, $y>...
0
votes
0
answers
104
views
Proving that a property holds for random sequences with given marginal distribution by rearrangement
I am currently investigating the property of random sequences with a special marginal distribution function $F(x)$. Given any random sequence $X_1, X_2, \cdots, X_n$, supposing their joint ...
- 21
4
votes
2
answers
8k
views
Upper bound on expectation value of the product of two random variables [closed]
Hello,
I am trying to find an upper bound on the expectation value of the product of two random variables.
So suppose x, y are two non-independent random variables, given that I know the distribution ...
- 343
1
vote
2
answers
147
views
limit of functionals on weak convergent random variables
Suppose real value random variables satisfy
$X_{n} \Rightarrow X$ (convergence in distribution)
as $n\to \infty$ in the same probability space
$(\Omega, \mathcal F, \mathbb P)$.
It is well known that ...
- 1,399
1
vote
0
answers
501
views
Distribution of random vectors
Two positive numbers $\alpha$ and $\beta$ are given. We are going to describe a process of choosing a random vector on the unit sphere $S$ in $\mathbb R^3$ (given by $x^2+y^2+z^2=1$).
A vector $u\in ...
- 143
3
votes
1
answer
453
views
When can you describe a population and its component subpopulations with the same parametric family of distributions?
I believe that it is often the case that you are trying to select the best probability distribution to use to describe some phenomenon you are studying, and you have data not only for a population, ...
- 133
2
votes
2
answers
2k
views
The probability distribution of random variable of random variable
In my understanding, random variable is a measurable function from a probability space to a measurable space. Suppose $X$ is a random variable from $(A, \sigma_{A},P_A)$ to $(B,\sigma_{B})$. And $Y$ ...
- 131
4
votes
3
answers
3k
views
What is the name for a non-normalized distribution?
For some analysis work with probability distributions, I remember a common trick being to drop the "integrate to 1" requirement, so the set becomes closed under addition and is more convenient to work ...
4
votes
1
answer
580
views
Tracking down locality assumption in CHSH inequality
CHSH inequality requires both locality and realism. I will equate here realism with counterfactual definiteness.
Now counterfactual definiteness tells us that given two different measurements on the ...
0
votes
0
answers
111
views
Stationarity of an Integral Process
Let $f$ be a continous deterministic function defined on $\left[0,c\right]$ and $(B_{t}^{H})_{t\geq 0}$ be a fBM with $H\in \left(0,1\right)$. We define a Process $\left(X_{t}\right)_{t\geq 0}$ with
$$...
3
votes
1
answer
520
views
Results regarding $E[\min X,Y]$. when $X$ and $Y$ are independent, of given distributions.
Working on fairly unrelated stuff, I needed to prove the following, fairly easy results, and I wonder if anyone can provide references to the literature. Not being a probabilist I wouldn't know where ...
- 577
1
vote
0
answers
223
views
Why this two model have same probability distribution?
(1)
Consider the following method of generating a random tree with $n$ nodes.
First expand the root node into two branches.
Then expand one of the two terminal nodes at random.
At time $k$, ...
- 351
2
votes
1
answer
635
views
Azuma's Inequality when the conditions hold with high probability?
In Azuma's Inequality, is the statement true when $|X_k - X_{k-1}| < c_k$ almost surely rather than with probability 1? If not, is there another result which gives strong concentration when the ...
- 201
4
votes
0
answers
5k
views
E[ | X - Y | ] where X and Y are independent Poisson random variable
What is the expected value of the absolute difference of two independent Poisson variables?
$$E[ |X - Y| ]$$
Seems like an easy question but I haven't found an easy solution.
I've split the double ...
- 41
1
vote
1
answer
902
views
Product of densities of a wrapped normal distribution
The density of a wrapped normal distribution is given by
$$\frac{1}{\sigma \sqrt{2\pi} }\sum _{k=-\infty }^{\infty }\text{Exp}\left[\frac{-(\theta -\mu -2\pi k)^2}{2\sigma^2}\right]$$
Considering two ...
- 73
2
votes
2
answers
943
views
measuring distance between probability measures only at the tail
Is there any official (i.e., to be found in probability books) metric for the distance between two probability measures, defined only on a subset of their support?
Take, for example, the total ...
- 171
3
votes
1
answer
376
views
The degrees in a random subgraph
Fix some positive integers $N$ and $d_k$, $k=1,2,\dots$ with $N=\sum_{k=1}^\infty d_k$.
Suppose you have a graph $G$ taken randomly uniformly among the set of all (unoriented) graphs with $N$ ...
15
votes
2
answers
10k
views
Convergence of moments implies convergence to normal distribution
I have a sequence $\{X_n\}$ of random variables supported on the real line, as well as a normally distributed random variable $X$ (whose mean and variance are known but irrelevant). I know that the ...
- 12.8k
2
votes
1
answer
272
views
Derivative of the CDF of a family of random variables
Suppose I have a r.v. $Z = X + \alpha Y$ and that $F_Z$ is the probability distribution function of $Z$. If we think of the probability $p = F_Z(q) = \mathbb{P}(X+\alpha Y < q)$ as a function $p = ...
- 322
3
votes
2
answers
350
views
Continuity of hitting distributions
Hi everybody
Let $U$ be the domain (as shown in the picture) and $\bar{U}$ its closure, further more set $\partial_r U$ to be the reflecting boundary and $\partial_a U$ the absorbing one. The process ...
- 279
0
votes
1
answer
329
views
Is it known that every PDF continuous in all $R^n$ has a maximum? [closed]
I'm working with maximum a posteriori estimation and managed to show that every probability density function that is continuous in all $R^n$ always has at least one global maximum. I've search around ...
2
votes
2
answers
2k
views
Multivariate power law distributions?
Is there a text books or publications that describes multivariate power law/pareto distributions?
- 21
6
votes
2
answers
410
views
If Mean Residual Lifetime is approximately constant, Residual Lifetime is Approximately Exponential in a Strong Sense
Suppose the "mean residual lifetime," $\mathbb{E}[X-x|X≥x]$ is approximately constant for large $x$. Then, I believe that the conditional tail distribution is approximately exponential, in the sense ...
- 99
2
votes
1
answer
521
views
Limit of a rescaled random sum of i.i.d. random variables
Consider a sequence of i.i.d. random variables $(X_i)_{i \in \mathbb N}$ and let $S_n=X_1+\dots+X_n$
For every $\alpha \in ]0,+\infty[$, let $N(\alpha)$ be a discrete random variable on $\mathbb N$, ...
- 418
6
votes
1
answer
404
views
References for this game
I would like to know how the following game is known in the literature and, possibly, to have references for related papers.
Description of the game: Fix a space $X$ and two Borel probability ...
user11618
1
vote
2
answers
2k
views
Variance of exponential random variable
For a random variable $\xi$, what bounds can be achieved for Var $e^{\xi}$ in terms of E$\xi$ and Var $\xi$?
- 11
0
votes
1
answer
207
views
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 ...
- 103
8
votes
2
answers
14k
views
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. $\...
- 231
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 ...
4
votes
0
answers
221
views
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
views
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
8
votes
3
answers
789
views
A Variance-Tail Description for Continuous Probability Distributions
Start with a continuous probability distribution given by a density function f(x). Let X be a real random variable whose distribution is given by the probability distribution.
I would like to ask ...
- 24.7k
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 ...
- 1,125
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
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}) ) \...
3
votes
1
answer
528
views
Cover a line segment randomly with smaller line segments
Covering a circle randomly with arcs has been well studied in the past (Geometric Probability - Solomon).
But the problem when the circle is changed to a line segment doesn't seem to have been ...
- 161
8
votes
1
answer
2k
views
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 ...
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 ...
- 119
1
vote
1
answer
142
views
Does a definition for delta sequences in the multidimensional case exist?
does anybody know a good book on multidimensional delta sequences?
- 47
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 ...
- 177
1
vote
2
answers
570
views
An inequality on Difference of Entropies
Hi,
I have the following problem that came up. It is not a homework problem or something similar. I did my simulations and it seems to hold but i was unable to prove it.## Heading ##
Let $P$ and $Q$ ...
- 199
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 ...
- 5
3
votes
1
answer
651
views
What conditions on a probability distribution defined by long-time averaging do I need to satisfy a central limit theorem?
For integer $n$, $1 \le n \le N$, consider the random variables
$X_n = \cos[t \omega_n]$
For any fixed $N$, we can take the mean
$Y_N = \frac{1}{N} \sum_{n=1}^N X_n$
and define a (cumulative) ...
- 846
21
votes
1
answer
32k
views
How to compute KL-divergence when PMF contains 0s?
From the Wikipedia page on Kullback-Leibler divergence, the way to compute this metric is to utilize the following formula:
The way I understand this is to compute the PMFs of two given sample sets ...
- 439
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, ...
- 101
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 ...
- 2,135
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 ...
- 233