All Questions
Tagged with pr.probability probability-distributions
1,384 questions
1
vote
0
answers
65
views
Shift invariance for the distribution of quadratic polynomials
For a probability distribution $X$, supported on integers, define the shift-invariance of $X$, denoted by $shift(X)$ = total variation distance between the random variable $X$ and $X+1$.
Let $p:\...
- 563
5
votes
1
answer
176
views
Asymptotic behavior of $X_n$ in a Dirichlet vector $(X_1, ..., X_n)$
Let $(\alpha_k)$ be a sequence of positive numbers and let $(Y_k)$ be a sequence of independent random variables $Y_k \sim \text{Gamma}(\alpha_k,1)$. Set $X_n=\dfrac{Y_n}{\sum_{i=1}^nY_i}$.
(edit) ...
- 2,560
3
votes
0
answers
286
views
Inequality with CDF of order statistics
here is a problem I have been struggling with for a while now. This is for a paper I am working on. Any help would be appreciated! Here we go:
Each bidder's valuation $\theta _{i},$ $i=1,...,N$, is ...
- 31
6
votes
2
answers
3k
views
Weak convergence of random measures
Let $\mu_n,n\in \mathbb N$ be a random probability measures and let $\mu$ be a deterministic probability measure on $\mathbb R$. That is to say, that the $\mu_n$ are measurable maps from a probability ...
- 255
7
votes
2
answers
460
views
Rate of Convergence of Compound Poisson Laws to Infinitely Divisible Laws
It is known that every infinitely divisible random variable is the limit in law of a sequence of compound Poisson random variables (see for instance Theorem 1.2.18 of Lévy Processes and Stochastic ...
- 2,306
4
votes
1
answer
196
views
Error for the convergence by distribution
A sequence of random variables $X_n$ converges in distribution to $X$, if there is pointwise convergence of its characteristic functions, i.e. $\lim_{n\rightarrow\infty}\phi_{X_n}(\lambda) = \phi_X(\...
- 497
0
votes
3
answers
690
views
An inequality based on expectation of continuous random variables
I am trying to prove the following statement:
$$
E[g(X)] E[X^2g(X)]\ge E[Xg(X)] E[Xg(X)]
$$
where $X$ is a random variable, $E[\cdot]$ denotes the expectation operator with respect to $X$, and ...
- 1
5
votes
3
answers
133
views
Random partitions with prescribed pairwise membership probabilities
Let $(p_{ij}) \in [0,1]^{n \times n}$ be a given symmetric matrix, with $1$ on the diagonal. Suppose $\pi$ is a partition of $[n]=\{1,\dots,n\}$ and let us write $i \stackrel{\pi}{\sim} j$ if $i$ and $...
- 1,731
1
vote
1
answer
524
views
Convergence in the Wasserstein metric and the square root function
Let $f$ be a smooth probability distribution on the unit square $S$ such that $f(x)>0$ on $S$. Let $\{g_i\}$ be a sequence of smooth probability distributions such that $g_i(x)>0$ on $S$ as ...
- 11
5
votes
1
answer
3k
views
1-wasserstein distance v.s. total variation distance
Suppose that $\mu_1$ and $\mu_2$ are two distributions defined on $\mathbb{R}^n$ and $\gamma$ is a symmetric distribution (around $0$) on $\mathbb{R}^n$ with compact support. Let $\gamma_x$ denote the ...
- 640
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 ...
- 1,282
3
votes
1
answer
1k
views
Book on Convergence Concepts in Probability without Measure Theory [closed]
I am looking for a comprehensive book on Probability which discusses Convergence of Random Variables in detail, excluding portions of Measure Theory. Allan Gut's "Probability: A Graduate Course" seems ...
- 319
2
votes
0
answers
1k
views
Probability question involving drawing balls from an urn
Suppose there's an urn containing $r$ red balls and $b$ blue balls. At each trial, I'm drawing a ball at random from the urn, without replacement. Let $R$ denote the event of drawing a red ball, and ...
- 285
1
vote
0
answers
255
views
Multiple Bipartite graphs and matchings
I've been told recently that it's better i just for help regarding my 'specific' problem rather than lots of little questions around the same topic which appear somewhat unclear. I would first like to ...
- 903
4
votes
1
answer
821
views
Does bounding moments make distributions close in total variation distance?
Let $W\sim\mathcal{N}(0,\sigma^2)$ be a "reference" Gaussian random variable.
Suppose I have a set of distributions, $\mathcal{W}$, where $W_a\in\mathcal{W}$ if it satisfies the following criteria:
...
- 907
2
votes
0
answers
160
views
Is it possible to improve the order of convergence of averages of random variables if they are not identically distributed?
Let $X_n$ be a sequence of independent random variables (but not necessarily identically distributed)
taking values in $[-1,1]$ that have the following property:
1) The average $A_n := \frac{(X_1+ \...
- 3,245
5
votes
2
answers
1k
views
PDF of the product of normal and Cauchy distributions
I am having trouble in finding out the resulting PDF of the product of normal and Cauchy distributions. It turns out that we have a general formula for calculating the PDF of product of two random ...
- 51
3
votes
1
answer
476
views
distribution discretization
Let $\mu$ be a distribution on $\mathbb{R}^n$. We partition $\mathbb{R}^n$ into small cubes congruent with $[0,\delta)^n$, parallel to the axes. In each cube, pick a point $x$ (for instance, the ...
- 640
14
votes
4
answers
2k
views
Gaussian distributions as fixed points in Some distribution space
I'm taking a course on topology and probabily. Today, the professor remarked something along the lines of:
If you look at the space of probability distributions with $0$ mean and variance $1$, ...
- 935
0
votes
2
answers
136
views
What are some examples of isotrophic sets?
What are some examples of isotrophic sets? and is there a "good" way to describe them?
Isotrophic meaning that a random vector X uniformly distributed in the set has the isotrophic property for all $...
- 111
6
votes
1
answer
1k
views
Properties of a finite random walk
Consider the simplest random walk - $X_0 = 0$ and from there on (i.i.d), $X_i=X_{i-1}+1$ with probability $p$ or $X_{i-1}-1$ otherwise.
Let $Y_N$ be the highest point $X$ have reached on the first $N$...
- 618
1
vote
0
answers
1k
views
How to show that two linear combinations of Bernoulli random variables have jointly Gaussian distribution (and more)
Let $X_1,\ldots,X_n$ be independent Bernoulli random variables such that $\mathbb{P}(X_i=\pm 1)=1/2$ and consider two collections of real numbers $a_1,\ldots,a_n, b_1,\ldots, b_n$. For the moment let ...
- 2,288
1
vote
0
answers
273
views
A natural sum over multisets (expectation over multinomial)
I think this is a natural question but am not sure where to find resources.
Consider the possible multisets arising from choosing $n$ times an item from one of $k$ categories. We can represent one ...
- 4,529
-1
votes
1
answer
287
views
Property of relative entropy [closed]
For $X$ a measurable space and $P,Q$ two probability measures on $X$ s.t. $Q$ is absolutely continuous with respect to $P$, the relative entropy is defined as
$$D(Q\|P)=\int_X \log(\frac{dQ}{dP})dQ,$$
...
- 35
0
votes
0
answers
216
views
Computation on Random Bipartite graphs
I'm looking at a random bipartite graph $K_{\omega(n)}*K_{\omega(n)}$ where $\mathrm{log}(n)\leq \omega(n) \leq n^{1/2}$, in which each of the $\omega(n)^{2}$ edges is placed randomly with probability ...
- 903
1
vote
0
answers
63
views
A series with long-tailed terms
Let's consider the following series:
$$
\zeta = \sum_{k=1}^{\infty} a_k \xi_k,
$$
where the sum is understood as the limit in $L_2(\Omega)$, $a_k \in \mathbb{R}$,
$\sum_{k=1}^{\infty} a_k^2< \...
- 195
9
votes
2
answers
531
views
Geometric interpretation of the average of two independent Cauchy distributions
Let me state two facts:
(1) It is well known that if one takes a point uniformly distributed on the unit circle, and then takes it stereographic projection, the corresponding measure induced on the ...
2
votes
0
answers
737
views
Probability question involving simulations of picking balls from a bag
I’m working on a chemistry problem, which essentially translates to finding the answer to a related probability problem. However, my knowledge in probability is very limited and I'd be grateful if ...
- 285
12
votes
2
answers
1k
views
lower-bound for $Pr[X\geq EX]$
Given n random variables, $X_1, ..., X_n$, each takes value 0 or $a_i \in[0, 1]$. $X = \sum_{i=1}^n X_i$ and $EX \geq 1$ is the expected value of $X$. Can we get a lower-bound for $Pr[X \geq EX]$? It ...
- 121
7
votes
1
answer
487
views
A note on Doob's theorem
I have faced the following problem, regarding to the Martingale Theory. Because this area far from my area I don't know whether this problem is in literature or this can be simple question for ...
- 103
8
votes
1
answer
3k
views
Algorithm to produce random number with a gamma distribution
I'd like to produce pseudo-random numbers with different distributions for a Monte Carlo simulation.
I've got the poisson distribution working nicely with an algorithm from Knuth. I'm having trouble ...
- 183
0
votes
1
answer
309
views
Probability of the maximum of a throw of an infinite number of $n$-sided dice being $k$ [closed]
Let $X$ be the random variable obtained as the maximum of a throw of $m$ dice (each of which is $n$-sided). In other words, $X = \max\{l_1,\cdots, l_m\}$ where $l_i$ can take any value between $1$ and ...
- 761
3
votes
0
answers
96
views
A 1-D random variable from a random distribution
I have a random variable $X$ that is drawn from the pdf
$$
f(x; \mu, \sigma, \sigma_{\mu}, \sigma_{\sigma}) = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \frac{1}{|\hat{\sigma}|\sqrt{2\pi }} \...
- 229
4
votes
0
answers
1k
views
Total variation and Hellinger distance inequality between truncated Gaussians
We know that the total variation distance, $d_{TV}(P,Q) = \frac{1}{2}\left|\left|P-Q\right|\right|_1$, between any two distributions $P$ and $Q$ is lower bounded by their squared Hellinger distance, $...
- 41
3
votes
1
answer
688
views
Is it possible to construct any random variable on the Euclidean Probability space?
Let $(\Omega,\mathscr A,P)$ be an arbitrary probability space,
and let $X:\Omega\to\mathbb R$ be a random variable.
Then,
one can generate a random variable $Y$ from the probability space $\big([0,1],\...
user34424
1
vote
1
answer
304
views
Entropy on a draw from a random distribution.
Suppose I am attempting to calculate the entropy of a continuous, normally distributed random variable $X$, from the distribution $\mathcal{N}(\mu, \sigma)$. This is easy to to do - I just calculate
$...
- 229
3
votes
1
answer
326
views
Two matrix Fisher distributions on SO(3)?
After the uniform distribution (normalized Haar measure), the matrix Fisher distribution seems to be the most popular probability distribution on the Lie group SO(3). The density is proportional to ...
- 41
0
votes
1
answer
220
views
Behavior of the integral of products of probability densities
Assume $z \in \mathbb{R}^m$ and $x \in \mathbb{R}^n$. Assume we have proper density function $P(z)$ and proper conditional density function $P(x|z)$. We give the definition
$$
T(x_1,\ldots,x_n) := \...
- 65
1
vote
0
answers
171
views
An inequality for moments of a random variable
I'm interested in a class C of $R^1$-valued random variables $\xi$ which satisfy
an inequality of the type
$$
(1) \qquad E|\xi|^p \leq F(E|\xi|^2),
$$
where $p>2$, $F$ is a certain non-...
- 195
1
vote
2
answers
221
views
A special class of random variables
I'm interested in classes C of $R^1$-valued random variables which possess the following properties:
1) the sum of two independent random variables from class C belongs to class C;
2) for any $\...
- 195
6
votes
1
answer
1k
views
How to check if a symmetric random variables is the difference of two iid symmetric random variables
I have the continuous symmetric random variable $X$ in $\mathbb{R}$. If I know its distribution function $F(x)$ what are the conditions on $F(x)$ so that $X=Y_1 - Y_2$ where $Y_i$ are also iid ...
6
votes
3
answers
2k
views
Estimating the variance of a discrete normal distribution
Let $f(x; \sigma) = \frac{1}{\sigma\sqrt{2\pi}}\cdot e^{-\frac{x^2}{2\sigma^2}}$ be the probability density function of a normal distribution $\mathcal{N}(0, \sigma^2)$. We consider a discrete normal ...
- 167
3
votes
2
answers
589
views
Measure concentration for law of large numbers
The classical law of large numbers states that
$$\frac1k\sum_{i=1}^k X_i \rightarrow \mathbb{E} X_1$$
for i.i.d. $X_1, X_2, \ldots$ with finite $L^1$ norm.
I was wondering whether is it possible to ...
- 773
0
votes
1
answer
215
views
Residual lifetime of heavy-tailed random variable
The residual life time distribution of a random variable $X$ with distribution function $F$ is given by the formula
\begin{equation}R(t)=P[X_\text{res}\leq t] = 1-\frac{1}{\mathbb{E}[X]}\int_{y=0}^\...
- 230
0
votes
0
answers
94
views
Dominating Poisson with parameter depending on a Bernoulli
Fix $\mu >0$ and take $\lambda \geq 0$. Let $B_p \sim \text{Ber}(p)$ with $p = \exp(-\mu - \frac{\lambda}2) $. Define the random variable $Y$ which is Poisson with parameter depending on the value ...
- 191
1
vote
1
answer
352
views
Double Markovity
Suppose we have a double Markov relation for three random variables $X$, $Y$ and $W$ as follows
$$X\to W\to Y,$$ and $$X\to Y\to W.$$
How to prove that there exist functions $f$ and $g$ such that
$$...
- 1,109
5
votes
2
answers
155
views
Approximate Moment Conditions
It is known in classical probability that if two random variables $X$ and $Y$ obeys
$$\mathbb{E} X^k = \mathbb{E}Y^k, \ \forall \ k \geq 1$$
with additional condition that $\mathbb{E}X^k$ does not ...
- 773
1
vote
1
answer
278
views
Laplace transform of : $t^{\gamma-1} F(\alpha,\beta,\delta,t)$, where $F$ is the Gauss' hypergeometric function
What is the Laplace transform of : $t^{\gamma-1} F(\alpha,\beta,\delta,t)$, where $\gamma >0 $ and $F$ is the Gauss' hypergeometric function.
Thanks!
- 233
2
votes
1
answer
663
views
Using a probability measure, P, defined on uncountable sets to construct a probability measure, P' on singleton P-null sets
Let $\Omega$ be an uncountable set and $(\Omega, \mathcal{F},P)$ be a probability space built on $\Omega$.
Let $S \subset \{A \in \mathcal{F}: P(A)=0,\;|A|=1\}:|S|<\infty$ be a finite subset of ...
user42192
2
votes
1
answer
263
views
Probability distribution of uAv…
Consider the complex domain ℂ. If U and V are 2 unitary random matrices and A is a deterministic matrix.
What is the distribution of $u^HAv$ ( or $||u^HAv||^2$)
where : u is a column vector of U. v ...
- 233